## Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

- Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
- Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
- Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

- Doing a PhD in Mathematics
- Doing a PhD

## What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

- Number theory
- Numerical analysis
- String theory
- Random matrix theory
- Graph theory
- Quantum mechanics
- Statistical forecasting
- Matroid theory
- Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

- Biomechanics and transport processes
- Evidence-based medicine
- Fluid dynamics
- Financial mathematics
- Machine learning
- Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

- Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
- Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
- Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
- Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
- Attend a wide range of lectures, seminars and events.

## Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

## Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

## What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

## Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

## How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1] for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

## What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

## What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

- Logical ability to consider and analyse complex issues,
- Commitment and persistence towards reaching research goals,
- Outstanding verbal and written skills,
- Strong attention to detail,
- The ability to liaise with others from unique disciple backgrounds and work as part of a team
- Holistic deduction and reasoning skills,
- Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
- Exceptional numeracy skills.

## What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

- Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
- Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
- Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
- Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
- Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

## What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

- Education – All UK and international universities
- Governments – STFC and Department for Transport
- Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
- Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
- Computing – IBM, Microsoft and Facebook
- Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

## How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

- Approximately £30,000 – £35,000 starting salary
- Approximately £40,000 with a few years experience
- Approximately £45,000 – £55,000 with 10 years experience
- Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

- Approximately £35,000 starting salary
- Approximately £45,000 – £55,000 with a few years experience
- Approximately £70,000 and over with 10 years experience
- Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

- Approximately £28,000 starting salary
- Approximately £35,000 – £40,000 with a few years experience
- Approximately £60,000 and over with 10 years experience

Data Analyst

- Approximately £45,000 – £50,000 with a few years experience
- Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

## Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

## Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

## Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

## Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

## Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

- http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
- https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
- https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
- https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
- http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
- https://www.surrey.ac.uk/postgraduate/mathematics-phd
- https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
- https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
- https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
- https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
- https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
- https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

## Browse PhDs Now

Join thousands of students.

Join thousands of other students and stay up to date with the latest PhD programmes, funding opportunities and advice.

## PhD in Mathematics

The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

## Degree Requirements

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin .

- Courses : The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here . Half may be at the MA 500 level or above, but the rest must be at the MA 700 level or above. Students can also request to use courses from other departments to satisfy some of these requirements. Please contact your advisor for more information about which courses can be used in this way. All courses must be passed with a grade of B- or higher.
- Analysis (examples include MA 711, MA 713, and MA 717)
- PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
- Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
- Topology (examples include MA 721, MA 722, and MA 727)
- Geometry (examples include MA 725, MA 731, and MA 745)
- Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
- Applied Mathematics (examples include MA 750, MA 751, and MA 770)
- Comprehensive Examination : This exam has both a written and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. On completion of the paper, the student takes an oral exam given by a three-person committee, one of whom is the student’s advisor. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. A student who does not pass the MA Comprehensive Examination may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year.
- Oral Qualifying Examination: The topics for the PhD oral qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not pass the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year.
- Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree .

Admissions information can be found on the BU Arts and Sciences PhD Admissions website .

## Financial Aid

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

## More Information

Please reach out to us directly at [email protected] if you have further questions.

The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 Ph.D. students, about 2/3 are in Pure Mathematics, 1/3 in Applied Mathematics.

The two programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, mathematics of data, and the theory of computation. In addition, many mathematically-oriented courses are offered by other departments. Students in Applied Mathematics are especially encouraged to take courses in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty , and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

## Degree Requirements

Degree requirements consist of:

- Oral qualifying exam
- Classroom teaching
- Original thesis and defense

Prospective students are invited to consult the graduate career timeline for more information, and to read about the application procedure .

## Graduate Co-Chairs

Graduate Student Issues, math graduate admissions

Jonathan Kelner , Davesh Maulik , and Zhiwei Yun

## Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series. Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

## Chenyang Xu

Jill leclair.

## Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

- Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
- Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
- Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
- Complete a seminar, giving a talk of at least one-hour duration
- Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
- Meet the University residence requirement of two years or four semesters

## Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

## Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

## Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

## Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members. All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

## PhD in Mathematics

The PhD in Mathematics consists of preliminary coursework and study, qualifying exams, a candidacy exam with an adviser, and creative research culminating in a written dissertation and defense. All doctoral students must also do some teaching on the way to the PhD. There are minimal course requirements, and detailed requirements and procedures for the PhD program are outlined in the PhD Handbook .

Please note that our department alternates recruiting in-coming classes that are focused on either applied or pure mathematics. For the Fall 2024 admissions (matriculation in September 2024), we are focusing on students interested in areas of applied mathematics.

All our professors are active in research, and are devoted to teaching and mentoring of students. Thus, there are many opportunities to be involved in cutting-edge research in pure and applied mathematics. Moreover, the seven other research universities in the Boston area are all within easy reach, providing access to many more classes, seminars and colloquia in diverse areas of mathematical research.

Teaching assistantships are available for incoming PhD students, as well as a limited number of University-wide fellowships. Tufts has on-campus housing for graduate students, but many choose to live off-campus instead.

In addition to the above, PhD students often:

- Mentor undergraduates as teaching assistants and course instructors, and through graduate-student run programs like the Directed Reading Program.
- Meet with advisors and fellow students to share research and collaborate with scholars across disciplines
- Attend professional development workshops and present research at conferences

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## Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras – Low order models of storm track variability

Ed Clark – Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes

Chiara Cecilia Maiocchi – Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions

Jennifer E. Israelsson – The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi – Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo – Local-global principles for norms

Tsz Yan Leung – Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli – Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart – On the treatment of correlated observation errors in data assimilation

Chris Davies – Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev – Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark – Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker – Path Properties of Levy Processes

Hasen Mekki Öztürk – Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro – Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn – The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

## Mathematics, PhD

Zanvyl krieger school of arts and sciences.

The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and go on to exciting careers in mathematics both inside and outside of academia.

Faculty research interests in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

## Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022-2023 academic year are $33,000, and exceptional applicants are also considered for supplementary fellowships. Students making satisfactory progress can expect to be supported for six years.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the requirements page .

All graduate students are invited to attend weekly research seminars in a variety of topic areas as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

## Financial Aid

Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.

## William Kelso Morrill Award

The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.

## Excellence in Teaching Awards

Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.

## Admission Requirements

Admission to the PhD program is based on primarily on academic records, letters of recommendation, and a personal statement. The Department of Mathematics values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

Via the online application , applicants should submit:

- A Statement of Purpose
- An optional Personal Statement
- Transcripts from all institutions attended
- Three letters of recommendation
- Official GRE scores for both the general and the subject test
- Official TOEFL scores (if English is not your first language)

The required Statement of Purpose discusses your academic interests, objectives, and preparation. The optional Personal Statement describes your personal background, and helps us create a more holistic understanding of you as an applicant. If you wish you may also discuss your personal background in the Statement of Purpose (e.g. if you have already written a single essay addressing both topics), instead of submitting separate statements.

Application fee waivers are available based on financial need and/or participation in certain programs .

Many frequently asked questions about the graduate admission process are answered here .

No application materials should be mailed to the department. All application materials are processed by the Graduate Admissions Office .

## Undergraduate Background

The following is an example of what the math department would consider a good background for a student coming out of a four-year undergraduate program at a college or university in the U.S. (assuming a semester system):

- Calculus in one variable (two semesters, or AP credits)
- Multivariable Calculus (one semester)
- Linear Algebra (one semester)
- Complex analysis (one semester)
- Real analysis (two semesters)
- Abstract algebra (two semesters)
- Point-set topology (one semester)

Many admitted students have taken upper-level undergraduate mathematics courses or graduate courses. Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model. In such cases, we strongly recommend that the student start to close the gap over the summer, before arriving for the start of the fall semester.

## Financial Support

Students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022–2023 academic year are $33,000. Students making satisfactory progress can expect to be supported for six years. Exceptional applicants are considered for supplementary fellowships of $6,000 each year for three years.

Students from underrepresented groups may be eligible for other university-wide supplemental fellowships. Summer teaching is available for students seeking extra income.

## Additional Information for International Students

Student Visa Information: The Office of International Services at Homewood will assist admitted international students in obtaining a student visa.

English Proficiency: Johns Hopkins University requires students to have adequate English proficiency for their course of study. Students must be able to read, speak, and write English fluently upon their arrival at the university. Applicants whose native language is not English must submit proof of their proficiency in English before they can be offered admission and before a visa certificate can be issued. Proficiency can be demonstrated by submitting results from either the Test of English as a Foreign Language (TOEFL) or the IELTS . Johns Hopkins prefers a minimum score of 100 on the TOEFL or a Band Score of 7 on the IELTS. Results should be sent to Johns Hopkins directly by TOEFL or IELTS. Applicants taking the IELTS must additionally upload a copy of their score through the application system. However, do not send the student copy or a photocopy of the TOEFL.

## Program Requirements

Course requirements.

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. The first-year algebra and analysis requirement can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. In addition, PhD candidates must take Algebraic Topology (110.615) and Riemannian Geometry (110.645) by their second year. Students having sufficient background can substitute an advanced topology course for 110.615, or an advanced geometry course for 110.645 with the permission of the instructor.

Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and if they have not passed their oral qualifying exam, in the first semester of their third year.

## Qualifying Exams

Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real & Complex) and in Algebra. Exams are scheduled for September and May of each academic year, and the dates are announced well in advance.

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10 of the third year. The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student’s thesis advisor, who will also be involved in administering the exam.

## PhD Dissertation

Candidates must produce a written dissertation based upon independent and original research. After completion of the thesis research, the student will defend the dissertation by means of the Graduate Board Oral exam . The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

Our PhD program does not have a foreign language requirement.

- Admission Policies
- Financial Support
- Ph.D. in Atmosphere Ocean Science
- M.S. at Graduate School of Arts & Science
- M.S. at Tandon School of Engineering
- Current Students

## Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements.

## NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

## The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

## The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

## The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

## Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

## Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

## Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs page.

## Department of Mathematics

Graduate program.

Application deadline is December 15th, 2023.

Test requirements:

GRE Subject Test: GRE Subject Math Test scores are OPTIONAL.

GRE General Test: GRE General Test scores are OPTIONAL.

TOEFL or IELTS: Scores are REQUIRED (the link below contains answers to common questions on these exams including who has to take them).

Standardized Test Questions: Yale Graduate School of Arts & Sciences

Fee waiver: if you wish to apply to waive the application fee (105$) please apply for the waiver here: Application Fees & Fee Waivers | Yale Graduate School of Arts & Sciences . We recommend to do this as early as possible and, at least, several days before the deadline of January 2, 2023. Please note that the department has no control over the waivers.

Program in Applied Mathematics . Note that there is a separate program in Applied Mathematics. You cannot apply for both programs. Follow Welcome | Applied Mathematics Program (yale.edu) for the general information about that program and https://applied.math.yale.edu/graduate-program-0 for the information about admissions, requirements, etc.

## Welcome to the Yale graduate program in Mathematics.

The transition from mathematics student to working mathematician depends on ability, hard work and independence, but also on community. Yale’s graduate program provides an excellent environment for this, and we are proud of the talented students who come here and the leading faculty with whom they learn the profession.

In their first two years, students focus on building their general knowledge and passing the qualifying exams , but are also encouraged to use the time to think about their areas of interest, work together to explore them, and begin making connections with faculty advisors. There are few formal requirements and this flexibility allows students to develop independence, formulating and following their own goals.

Mathematics, while requiring intense individual focus, also thrives on collaborative work. Students form study groups and seminars together, and also benefit from our excellent cohort of Gibbs Assistant Professors and other Postdoctoral Fellows, who are a source of fresh mathematical perspectives and camaraderie.

Research, and the contribution of new ideas and results to the body of mathematical knowledge, naturally form the main focus of the next few years, and typically students complete their PhD by the end of the 5th (sometimes 6th) year. During this time they also get to know the faculty better, and continue building intellectual and personal connections, horizontally across the discipline and through time to our shared intellectual history and tradition.

Teaching is an important component of our profession, and the department provides support and training to graduate students. Teaching assignments proceed from individual coaching to classroom teaching, with careful mentoring provided by our dedicated team of lecturers. The Lang Lunch Seminar, in the second year, provides in-depth training to graduate students before they begin to lecture.

Director of Graduate Studies : Van Vu .

Inquiries concerning the graduate program in mathematics should be sent to Van Vu .

Registrar of Graduate Studies: TBA

Some useful links:

- The mathematics department page in the Graduate School catalog.
- Graduate school homepage for general information.
- Admissions information from the graduate school.
- Mathematics Graduate Program Advising Guidelines

## PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later.

## Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis.

## Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas:

- real analysis
- geometry and topology
- applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

## Topics Covered on the Exams:

- Algebra Syllabus
- Real Analysis Syllabus
- Geometry and Topology Syllabus
- Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn.

## Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn.

## Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter: The exams are held during the week prior to the first week of the quarter. Spring Quarter: The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the Academic Calendar

## Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates:

## Monday, April 1st

Analysis Exam, Room 384H

## Wednesday, April 3rd

Algebra Exam, Room 384I

## Thursday, April 4th

Geometry & Topology Exam, Room 384I

## Friday, April 5th

Applied Math Exam, Room 384I

## Doctor of Philosophy (PhD) in Mathematics Education

Graduate Programs

The Ph.D. program emphasizes research and requires a written dissertation for completion. The program is individualized to meet the needs of graduate students. The student must develop, with the guidance from the major professor and committee, a program that is applicable to their background and interest. The average Ph.D. program requires 4-6 years beyond a master’s degree. The program is comprised of coursework in four major areas.

- Mathematics Education
- Mathematics or a related area
- Cognate Area
- Research Core

This residential program has rolling admission Applications must be fully complete and submitted (including all required materials) and all application fees paid prior to the deadline in order for applications to be considered and reviewed. For a list of all required materials for this program application, please see the “Admissions” section below.

- July 1 is the deadline for Fall applications.
- November 15 is the deadline for Spring applications.
- March 15 is the deadline for Summer applications.

This program does not lead to licensure in the state of Indiana or elsewhere. Contact the College of Education Office of Teacher Education and Licensure (OTEL) at [email protected] before continuing with program application if you have questions regarding licensure or contact your state Department of Education about how this program may translate to licensure in your state of residence.

## APPLICATION PROCEDURE

Application Instructions for the Mathematics Education PhD program from the Office of Graduate Studies:

In addition to a submitted application (and any applicable application fees paid), all completed materials must be submitted by the application deadline in order for an application to be considered complete and forwarded on to faculty and the Purdue Graduate School for review.

Here are the materials required for this application:

- Transcripts (from all universities attended, including an earned bachelor’s degree from a college or university of recognized standing)
- Minimum undergraduate GPA of 3.0 on a 4.0 scale
- 3 Recommendations
- Academic Statement of Purpose
- Personal History Statement
- Writing Sample
- International Applicants must meet English Proficiency Requirements set by the Purdue Graduate School

We encourage prospective students to submit an application early, even if not all required materials are uploaded. Applications are not forwarded on for faculty review until all required materials are uploaded.

When submitting your application for this program, please select the following options:

- Select a Campus: Purdue West Lafayette (PWL)
- Select your proposed graduate major: Curriculum and Instruction
- Please select an Area of Interest: Mathematics Education
- Please select a Degree Objective: Doctor of Philosophy (PhD)
- Primary Course Delivery: Residential

## Program Requirements

I. mathematics education courses (15 – 18 hours).

In mathematics education, students engage in courses that cover topics in the cognitive and cultural theories of learning and teaching mathematics, and the role of curriculum in mathematics education. A three (3) course sequence is required that consists of:

- EDCI 63500 – Goals and Content in Mathematics Education
- EDCI 63600 – The Learning of Mathematics: Insights and Issues
- EDCI 63700 – The Teaching of Mathematics: Insights and Issues

In addition, students are encouraged to take (6 – 9) hours of EDCI 620: Developing as a Mathematics Education Researcher

## II. Related Course Work (minimum 6 hours)

All students should have appropriate course work in mathematics, statistics, educational technology, or a related field. Students without a master’s level background in mathematics may be required to take more courses in mathematics. This will be determined by the student’s major professor and advisory committee.

## III. Cognate (9 hours)

Students will take three graduate courses in a self-selected cognate area. Cognate area selection should be discussed with the student’s major professor and advisory committee. Possible cognate areas include: mathematics, psychology, philosophy, sociology, technology.

## IV. Research Core Courses (15 hours)

All doctoral students in the Department of Curriculum and Instruction must complete five (5) courses from areas in research methodology and analysis before beginning their dissertation:

- EDPS 53300 – Introduction to Research in Education
- EDCI 61500 – Qualitative Research Methods in Education
- MA 51200 – Introductory Statistics
- Advance electives in either quantitative or qualitative methods
- Ackerman Center
- Serious Games
- CnI Online Fac
- Curriculum Studies
- Education for Work and Community
- Elementary Education
- English Education
- English Language Learning
- Learning Design and Technology
- Literacy and Language Education
- Science Education
- Social Studies Education
- Applied Behavior Analysis
- Counseling and Development
- Educational Leadership and Policy Studies
- Educational Psychology and Research Methodology
- Gifted Education
- Special Education

## Laura Bofferding

Amber brown, signe kastberg, rachael kenney, jill newton.

Course Registration, payment, drops/withdraws, and removing holds: [email protected] Career accounts: ITaP (765) 494-4000

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## Georgia Institute of Technology College of Sciences

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- Doctoral Programs

## PhD in Mathematics

Here are the requirements for earning the PhD degree in Mathematics offered by the School of Math. For requirements of other PhD programs housed within the School, please see their specific pages at Doctoral Programs . The requirements for all these programs consist of three components: coursework , examinations , and dissertation in accordance to the guidelines described in the GT Catalogue .

Completion of required coursework, examinations, and dissertation normally takes about five years. During the first one or two years, students concentrate on coursework to acquire the background necessary for the comprehensive examinations. By the end of their third year in the program, all students are expected to have chosen a thesis topic, and begin work on the research and writing of the dissertation.

The program of study must contain at least 30 hours of graduate-level coursework (6000-level or above) in mathematics and an additional 9 hours of coursework towards a minor. The minor requirement consists of graduate or advanced undergraduate coursework taken entirely outside the School of Mathematics, or in an area of mathematics sufficiently far from the students area of specialization.

Prior to admission to candidacy for the doctoral degree, each student must satisfy the School's comprehensive examinations (comps) requirement. The first phase is a written examination which students must complete by the end of their second year in the graduate program. The second phase is an oral examination in the student's proposed area of specialization, which must be completed by the end of the third year.

Research and the writing of the dissertation represent the final phase of the student's doctoral study, and must be completed within seven years of the passing of comps. A final oral examination on the dissertation (theses defense) must be passed prior to the granting of the degree.

## The Coursework

The program of study must satisfy the following hours , minor , and breadth requirements. Students who entered before Fall 2015 should see the old requirements , though they may opt into the current rules described below, and are advised to do so.

Hours requirements. The students must complete 39 hours of coursework as follows:

- At least 30 hours must be in mathematics courses at the 6000-level or higher.
- At least 9 hours must form the doctoral minor field of study.
- The overall GPA for these courses must be at least 3.0.
- These courses must be taken for a letter grade and passed with a grade of at least C.

Minor requirement. The minor field of study should consist primarily of 6000-level (or higher) coursework in a specific area outside the School of Math, or in a mathematical subject sufficiently far from the student’s thesis work. A total of 9 credit hours is required and must be passed with a grade of B or better. These courses should not include MATH 8900, and must be chosen in consultation with the PhD advisor and the Director of Graduate Studies to ensure that they form a cohesive group which best complements the students research and career goals. A student wishing to satisfy the minor requirement by mathematics courses must petition the Graduate Committee for approval. Courses used to fulfill a Basic Understanding breadth requirement in Analysis or Algebra should not be counted towards the doctoral minor. Upon completing the minor requirement, a student should immediately complete the Doctoral Minor form .

Breadth requirements. The students must demonstrate:

- Basic understanding of 2 subjects must be demonstrated through passing the subjects' written comprehensive exams. At least 1 of these 2 exams must be in Algebra or Analysis.
- Basic understanding of the third subject may be demonstrated either by completing two courses in the subject (with a grade of A or B in each course) or by passing the subject's written comprehensive exam.
- A basic understanding of both subjects in Area I (analysis and algebra) must be demonstrated.
- Earning a grade of A or B in a one-semester graduate course in a subject demonstrates exposure to the subject.
- Passing a subject's written comprehensive exam also demonstrates exposure to that subject.

The subjects. The specific subjects, and associated courses, which can be used to satisfy the breadth requirements are as follows.

- Area I subjects:
- Area II subjects:

Special Topics and Reading Courses.

- Special topics courses may always be used to meet hours requirements.
- Special topics courses may be used to meet breadth requirements, subject to the discretion of the Director of Graduate Studies.
- Reading courses may be used to meet hours requirements but not breadth requirements.

## Credit Transfers

Graduate courses completed at other universities may be counted towards breadth and hours requirements (courses designated as undergraduate or Bachelors' level courses are not eligible to transfer for graduate credit). These courses do not need to be officially transferred to Georgia Tech. At a student’s request, the Director of Graduate Studies will determine which breadth and hours requirements have been satisfied by graduate-level coursework at another institution.

Courses taken at other institutions may also be counted toward the minor requirement, subject to the approval of the Graduate Director; however, these courses must be officially transferred to Georgia Tech.

There is no limit for the transfer of credits applied toward the breadth requirements; however, a maximum of 12 hours of coursework from other institutions may be used to satisfy hours requirements. Thus at least 27 hours of coursework must be completed at Georgia Tech, including at least 18 hours of 6000-level (or higher) mathematics coursework.

Students wishing to petition for transfer of credit from previous graduate level work should send the transcripts and syllabi of these courses, together with a list of the corresponding courses in the School of Math, to the Director of Advising and Assessment for the graduate program.

## Comprehensive Examinations

The comprehensive examination is in two phases. The first phase consists of passing two out of seven written examinations. The second phase is an oral specialty examination in the student's planned area of concentration. Generally, a student is expected to have studied the intended area of research but not necessarily begun dissertation research at the time of the oral examination.

Written examinations. The written examinations will be administered twice each year, shortly after the beginning of the Fall and Spring semesters. The result of the written examination is either pass or fail. For syllabi and sample exams see the written exams page .

All students must adhere to the following rules and timetables, which may be extended by the Director of Graduate Studies, but only at the time of matriculation and only when certified in writing. Modifications because of leaves from the program will be decided on a case-by-case basis.

After acceptance into the PhD Program in Mathematics, a student must pass the written examinations no later than their fourth administration since the student's doctoral enrollment. The students can pass each of the two written comprehensive exams in separate semesters, and are allowed multiple attempts.

The Director of Graduate Studies (DGS) will be responsible for advising each new student at matriculation of these rules and procedures and the appropriate timetable for the written portion of the examination. The DGS will also be responsible for maintaining a study guide and list of recommended texts, as well as a file of previous examinations, to be used by students preparing for this written examination.

Oral examination. A student must pass the oral specialty examination within three years since first enrolling in the PhD program, and after having passed the written portion of the comprehensive exams. The examination will be given by a committee consisting of the student's dissertation advisor or probable advisor, two faculty members chosen by the advisor in consultation with the student, and a fourth member appointed by the School's Graduate Director. The scope of the examination will be determined by the advisor and will be approved by the graduate coordinator. The examining committee shall either (1) pass the student or (2) fail the student. Within the time constraints of which above, the oral specialty examination may be attempted multiple times, though not more than twice in any given semester. For more details and specific rules and policies see the oral exam page .

## Dissertation and Defense

A dissertation and a final oral examination are required. For details see our Dissertation and Graduation page, which applies to all PhD programs in the School of Math.

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## PhD projects

Several School members offer supervision for PhD research projects in the School of Mathematics and Statistics.

Navigate via the tabs below to view project offerings by School members in the areas of Applied Mathematics, Pure Mathematics and Statistics. (This list was updated September 2022.)

Please note that this is not an exhaustive list of all potential projects and supervisors available in the School.

Information about PhD research offerings and potential supervisors can be found in various locations. It's worth browsing the current research students list to see what research our PhD students are currently working on, and with whom.

There is also a past research students list which provides links to the theses of former students and the names of their supervisors.

It's also recommended to browse our Staff Directory , where our staff members' names are linked to their research profiles which provide details about their areas of research and often include the topics they are open to supervising students in.

We host PhD information sessions in the School of Mathematics and Statistics twice a year. Keep an eye on our events page for session information.

- Applied mathematics
- Pure mathematics
- Real world problem solving using dynamical systems, stochastic modelling and queueing theory for stochastic transport and signalling in cells.
- Real & Computational Algebraic Geometry: Possible subjects include nonnegativity of real polynomials, polynomial system solving, semialgebraic sets, and algorithmic aspects of real algebraic & convex geometry.
- Polynomial & Convex Optimization: Potential topics include convex relaxations, designing algorithms, exploiting structure (e.g. sparsity), and applications in science & engineering.
- Dynamical Systems and Ergodic Theory: Projects that combine techniques from nonlinear dynamics, ergodic theory, functional analysis, differential geometry, or machine learning and can range from pure mathematical theory through to numerical techniques and applications (including ocean/atmosphere/fluids/blood flow), depending on the student.
- Optimisation: Projects are occasionally available in optimisation, mainly using either techniques from mixed integer programming to solve applied problems (e.g. transport, medicine,…) or mathematical problems arising from dynamics.
- Modelling and analysis of ocean biogeochemical cycles including isotope dynamics, inverse modelling of hydrographic data to detect climate-driven circulation changes, and analysis of large-scale ocean transport. PhD students should be highly motivated, have a strong background in applied mathematics and/or theoretical physics, and will have the opportunity to contribute to shaping their project.
- Data-Driven Multi-stage Robust Optimization: The aim of this study is to develop mathematical principles for multi-stage robust optimization problems, which can identify true optimal solutions and can readily be validated by common computer algorithms, to design associated data-driven numerical methods to locate these solutions and to provide an advanced optimization framework to solve a wide range of real-life optimization models of multi-stage technical decision-making under evolving uncertainty.
- Semi-algebraic Global Optimization: The goal of this study is to examine classes of semi-algebraic global optimization problems, where the constraints are defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles and numerical methods which can identify and locate the globally best solutions.
- Detection and cloaking of surface water waves created by submerged objects
- Decomposition of ocean currents into wave-like and eddy-like components
- Theory and application of Quasi-Monte Carlo methods: for high dimensional integration, approximation, and related problems.
- Computational Mathematics: with specialised topics in radial basis functions, random fields, uncertainty quantification, partial differential equations on spheres and manifolds, stochastic partial differential equations.
- Discrete Integrable Systems: These are birational maps with particularly ordered dynamics and their study is a nice motivation for using algebraic geometry, symmetry, ideal theory and number theory in the study of dynamical systems.
- Arithmetic Dynamics: This field is the study of iterated rational maps over the integers or rationals or over finite fields, rather than the complex or real numbers. I am particularly interested in how the usual structures present in dynamical systems over the continuum manifest themselves over discrete spaces.
- Convex geometry: Focused on the study of the facial structure of convex sets and the relations between the geometry of convex optimisation problems and performance of numerical methods. The project can be oriented towards convex algebraic geometry, experimental mathematics or classical convex analysis.
- Algebraic and Geometric Aspects of Integrable Systems: The ubiquitous nature of integrable systems is reflected in their (apparent or disguised) presence in a wide range of areas in both mathematics and (mathematical) physics. Projects focus on the algebraic and/or geometric aspects of discrete and/or continuous integrable systems, depending on the individual student's background and preferences.
- Analysis of multiscale problems in stochastic systems: These projects will involve an analytical study of certain multiscale problems arising in Markov chains and stochastic differential equations. These projects are suited for those interested in both analysis and probability, and will employ tools from differential equations, functional analysis and stochastic processes.
- Numerical methods for sampling constrained distributions: These projects are aimed at sampling problems arising in molecular dynamics. They will deal with designing and analysing numerical schemes to sample constrained probability distributions using stochastic differential equations.
- Fluid flow in channels with porous walls
- Mathematics education
- Nonlinear differential equations
- Difference equations
- Dynamic equations on time scales
- How many oceans are there? Using novel statistical and machine learning techniques to characterise oceanic zones and provide a blueprint for quantifying the ocean's role in a changing climate.
- How does heat get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate.
- Making climate models work better: Developing new methods to validate and improve the inner workings of numerical climate models and improve their projections of global warming and its impacts.
- Will it mix? New perspectives on turbulence in rotating fluid flows and how we estimate mixing from observations.
- Combinatorics
- Graph theory
- Coding theory
- Extremal set theory
- Operator algebras (von Neumann algebras)
- Mathematical physics (quantum field theory)
- Group theory
- Jones subfactor theory
- Vaughan Jones' connection between conformal field theory, Richard Thompson's groups and knot theory.
- Noncommutative algebra
- Algebraic geometry
- Quantum groups/supergroups
- The Schur-Weyl duality
- Representation Theory
- Random graphs
- Asymptotic enumeration
- Randomized combinatorial algorithms

Extremal and probabilistic combinatorics: Possible subjects therein include Ramsey theory, random graphs, positional games and hypergraphs.

- Unlikely Intersection in Number Theory and Diophantine Geometry: These are problems of showing that arithmetic “correlations" between specialisations of algebraic functions are rare unless there is some obvious reason why they happen. These “correlations” may refer to common values or to values factored into essentially the same set of prime ideals and similar.
- Arithmetic Dynamics: This area is concerned with algebraic and arithmetic aspects of iterations of rational functions over domains of number theoretic interest.
- Isometries, conformal mappings, and other special mappings on metric Lie groups
- Complex structures on Lie groups and their Lie algebras
- Counting integral and rational solutions to Diophantine equations and congruences. The goal is to obtain upper bounds on the number of integer solutions to some multivariate equations and congruences in variables from a given interval [M, M+N]. Similarly, for rational solutions one restricts both numerators and denominators to certain intervals.
- Kloostermania: Kloosterman and Salie sums and their applications. A classical direction in analytic number theory where the goal is to obtain new bounds on bilinear sums of Kloosterman and Salie sums and apply them to various arithmetic problems, such as the Dirichlet divisor problem in progressions.

Exponential sums and applications. This topic is about understanding the behaviour (e.g. extreme and typical values) of some most important exponential sums, in particular of Weyl sums.

- Non-commutative functional analysis and its applications to non-commutative geometry, particularly those related to quantised calculus and index theorems.
- Singular (Dixmier) traces and their applications
- Non-commutative integration theory
- Non-commutative probability theory
- Various aspects of Banach space geometry and its applications
- Algebraic geometry (birational geometry and moduli)
- Hodge theory
- Transcendental methods in algebraic geometry
- Motivic cohomology and algebraic K-theory - an intersection of algebraic geometry and algebraic topology
- Equivariant algebraic topology
- Extreme Value Analysis: Projects available on the modelling of the dependence of multivariate and spatial extremes, spatio-temporal modelling, high-dimensional inference. Interests in environmental/climate applications.
- Symbolic Data Analysis: Projects available on symbol design, distributional symbols and others. Applications in big and complex data analysis.
- Ancient river systems and landscape dynamics with Bayeslands framework
- Bayesian inference and machine learning for reef modelling
- Deep learning for the reconstruction of 3D ore-bodies for mineral exploration
- Bayesian deep learning for protein function detection
- COVID-19 modelling with deep learning
- Variational Bayes for surrogate assisted deep learning
- Bayesian deep learning for incomplete information
- Computational Statistics
- Event sequence data analysis
- Hidden Markov Models and State-Space Models and their inference and applications
- Financial data analysis and modelling
- Point processes and their inference and applications
- Semi- and non-parametric inference
- Bayesian statistical inference
- Computational statistics and algorithms
- Approximate Bayesian inference
- Quantile regression method
- Statistical text analyses
- Applications to climate science, social science, image analyses

*Yanan Fan is an Adjunct A/Prof in the School and is able to co-supervise students (not as primary supervisor)

- Nonparametric and semiparametric statistics: Nonparametric dependence modelling (copulas) and nonparametric functional data analysis.
- Social network analysis for epidemiology, social sciences, defence, national security, and other areas
- Statistical models for dependent categorical data
- Survey sampling (design and inference), particularly for network data
- Statistical computing, particularly MCMC-based methods
- Dependence measures
- Complex-valued random variables
- Goodness-of-fit tests
- Machine learning (with potential applications in medical imaging)
- Time series analysis
- Real-time analytics with the Raspberry Pi
- For some examples of my current projects, have a look at my personal webpage .
- Fast and efficient model selection for high-dimensional data
- Development of efficient estimation and sampling algorithms for random graphs and spatial point processes
- Development of model compression methods for deep neural networks.

Topics include regression to the mean, interrupted time series, meta-analysis, and population attributable fractions.

Monte Carlo and Uncertainty Quantification

- Projects on the stochastic analysis and development of modern Monte Carlo methods for uncertainty quantification, sequential Bayesian inference, high dimensional sampling, particle based Variational Inference (knowledge/experience with stochastic analysis and SDE & PDE theory highly desired).

Machine learning and generative modelling

- Projects with a focus on (but not restricted to) medical imaging and machine learning methods for uncertainty quantification of image segmentation
- Theoretical analysis of modern machine learning methods (knowledge/experience with functional and stochastic analysis highly desired).

Mathematics of sustainability

- Projects on stochastic games, agent based models, network science and their applications in sustainability science.
- Automating data analyses via natural language queries
- Bayesian statistics, algorithms and applications
- Building software tools, services and packages
- Data privacy and synthetic data
- Data science, theory and application
- Defence applications (nationality restrictions may apply)
- Extreme value theory and applications
- Machine learning
- Symbolic data analysis
- Developing statistical methods for point processes
- Financial data modeling
- Computational statistics
- Analysis of capture-recapture data
- Estimation of animal abundance
- Measurement error modelling
- Model selection for multivariate data
- Non-parametric smoothing
- Statistical ecology
- High-dimensional data analysis
- Simulation-based inference
- Eco-Stats project ideas

## Study Postgraduate

Phd in mathematics (2024 entry).

Course code

30 September 2024

3-4 years full-time

Qualification

Mathematics Institute

University of Warwick

## Explore our PhD in Mathematics

The PhD in Mathematics offers an intellectually stimulating and dynamic research course. Study at the University of Warwick's Mathematics Institute, an international centre of research excellence, ranked 3rd for research power and 3rd for the number of 4* research outputs in REF 2021 (amongst UK universities).

## Course overview

Mathematics at Warwick covers the full spectrum of mathematics and its applications. The Mathematics Postgraduate Degrees are appropriate for students with a strong and broad mathematical background who wish to engage in advanced mathematical techniques and attack mathematical research problems in their postgraduate work.

All students are required to undergo training in Year One and are encouraged to make use of further training opportunities available in subsequent years. Training ranges from gaining a broader knowledge of mathematics through taught modules, seminars and workshops, to enhancing your professional and transferrable skills. Our PhD students undertake high quality original research and are being well-prepared for a career, either in academia or elsewhere.

## Teaching and learning

Students are required to complete a series of modules in their first year (from a very wide selection of bespoke modules), with assessment including an oral examination component. Upon the successful completion of these modules, students are required to complete a research project before being formally upgraded to a PhD at the end of the first year.

Training will be supplemented with attendance to seminars, cohort building activities, and additional transferable skills training.

## General entry requirements

Minimum requirements.

First Class Honours undergraduate integrated Master's (4-year) degree from a UK university in Mathematics or a science degree with high mathematical content, or the equivalent qualification and grade from a non-UK university.

Alternatively, applicants who have a Bachelor's degree AND a Distinction in a postgraduate Master's degree would be considered.

## English language requirements

You can find out more about our English language requirements Link opens in a new window . This course requires the following:

- IELTS overall score of 6.5, minimum component scores not below 6.0

## International qualifications

We welcome applications from students with other internationally recognised qualifications.

For more information, please visit the international entry requirements page Link opens in a new window .

## Additional requirements

There are no additional entry requirements for this course.

## Our research

The mathematics department covers a wide range of research areas in mathematics and its applications.

You may also wish to explore the research interest of current Warwick academics .

Find a supervisor

The 'Find A Supervisor' link below will allow you to explore the research interests of academics within the department. Please include in your application the names of potential supervisors, with interests aligned with yours, or people you would like to work with.

The mathematics department, unlike some other departments, does not require students to make any arrangements with any potential supervisors before applying, though of course you are welcome to contact them directly and discuss your interests and any potential projects they may offer.

## Tuition fees

Tuition fees are payable for each year of your course at the start of the academic year, or at the start of your course, if later. Academic fees cover the cost of tuition, examinations and registration and some student amenities.

Find your research course fees

## Fee Status Guidance

We carry out an initial fee status assessment based on the information you provide in your application. Students will be classified as Home or Overseas fee status. Your fee status determines tuition fees, and what financial support and scholarships may be available. If you receive an offer, your fee status will be clearly stated alongside the tuition fee information.

Do you need your fee classification to be reviewed?

If you believe that your fee status has been classified incorrectly, you can complete a fee status assessment questionnaire. Please follow the instructions in your offer information and provide the documents needed to reassess your status.

Find out more about how universities assess fee status

## Additional course costs

As well as tuition fees and living expenses, some courses may require you to cover the cost of field trips or costs associated with travel abroad.

For departmental specific costs, please see the Modules tab on the course web page for the list of core and optional core modules with hyperlinks to our Module Catalogue (please visit the Department’s website if the Module Catalogue hyperlinks are not provided).

Associated costs can be found on the Study tab for each module listed in the Module Catalogue (please note most of the module content applies to 2022/23 year of study). Information about module department specific costs should be considered in conjunction with the more general costs below:

- Core text books
- Printer credits
- Dissertation binding
- Robe hire for your degree ceremony

## Scholarships and bursaries

## Scholarships and financial support

Find out about the different funding routes available, including; postgraduate loans, scholarships, fee awards and academic department bursaries.

## Mathematics Funding Opportunities

Find out more about the various funding opportunities that are available in our department.

## Living costs

Find out more about the cost of living as a postgraduate student at the University of Warwick.

## Mathematics at Warwick

Our challenging Mathematics degrees will harness your strong mathematical ability and commitment, enabling you to explore your passion for mathematics.

Find out more about us on our website Link opens in a new window

## Our courses

- Interdisciplinary Mathematics (Diploma plus MSc)
- Interdisciplinary Mathematics (MSc)
- Mathematics (Diploma plus MSc)
- Mathematics (MSc)
- Mathematics (PhD)

## How to apply

The application process for courses that start in September and October 2024 will open on 2 October 2023.

For research courses that start in September and October 2024 the application deadline for students who require a visa to study in the UK is 2 August 2024. This should allow sufficient time to complete the admissions process and to obtain a visa to study in the UK.

How to apply for a postgraduate research course

## After you’ve applied

Find out how we process your application.

## Applicant Portal

Track your application and update your details.

## Admissions statement

See Warwick’s postgraduate admissions policy.

## Join a live chat

Ask questions and engage with Warwick.

## Warwick Hosted Events Link opens in a new window

Postgraduate fairs.

Throughout the year we attend exhibitions and fairs online and in-person around the UK. These events give you the chance to explore our range of postgraduate courses, and find out what it’s like studying at Warwick. You’ll also be able to speak directly with our student recruitment team, who will be able to help answer your questions.

Join a live chat with our staff and students, who are here to answer your questions and help you learn more about postgraduate life at Warwick. You can join our general drop-in sessions or talk to your prospective department and student services.

## Departmental events

Some academic departments hold events for specific postgraduate programmes, these are fantastic opportunities to learn more about Warwick and your chosen department and course.

See our online departmental events

## Warwick Talk and Tours

A Warwick talk and tour lasts around two hours and consists of an overview presentation from one of our Recruitment Officers covering the key features, facilities and activities that make Warwick a leading institution. The talk is followed by a campus tour which is the perfect way to view campus, with a current student guiding you around the key areas on campus.

## Connect with us

Learn more about Postgraduate study at the University of Warwick.

We may have revised the information on this page since publication. See the edits we have made and content history .

## Why Warwick

Discover why Warwick is one of the best universities in the UK and renowned globally.

9th in the UK (The Guardian University Guide 2024) Link opens in a new window

67th in the world (QS World University Rankings 2024) Link opens in a new window

6th most targeted university by the UK's top 100 graduate employers Link opens in a new window

(The Graduate Market in 2024, High Fliers Research Ltd. Link opens in a new window )

## About the information on this page

This information is applicable for 2024 entry. Given the interval between the publication of courses and enrolment, some of the information may change. It is important to check our website before you apply. Please read our terms and conditions to find out more.

Alternatively, use our A–Z index

Attend an open day

Discover more about postgraduate research

## PhD Pure Mathematics / Overview

Year of entry: 2024

- View full page

The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).

Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.

Full entry requirements

Apply online

In your application you’ll need to include:

- The name of this programme
- Your research project title (i.e. the advertised project name or proposed project name) or area of research
- Your proposed supervisor’s name
- If you already have funding or you wish to be considered for any of the available funding
- A supporting statement (see 'Advice to Applicants for what to include)
- Details of your previous university level study
- Names and contact details of your two referees.

## Programme options

Programme description.

The The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.

Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.

For entry in the academic year beginning September 2024, the tuition fees are as follows:

- PhD (full-time) UK students (per annum): Band A £4,786; Band B £7,000; Band C £10,000; Band D £14,500; Band E £24,500 International, including EU, students (per annum): Band A £28,000; Band B £30,000; Band C £35,500; Band D £43,000; Band E £57,000
- PhD (part-time) UK students (per annum): Band A £2393; Band B £3,500; Band C £5,000; Band D £7,250; Band E 12,250 International, including EU, students (per annum): Band A £14,000; Band B £15,000; Band C £17,750; Band D £21,500; Band E £28,500

Further information for EU students can be found on our dedicated EU page.

The programme fee will vary depending on the cost of running the project. Fees quoted are fully inclusive and, therefore, you will not be required to pay any additional bench fees or administration costs.

All fees for entry will be subject to yearly review and incremental rises per annum are also likely over the duration of the course for Home students (fees are typically fixed for International students, for the course duration at the year of entry). For general fees information please visit the postgraduate fees page .

Always contact the Admissions team if you are unsure which fees apply to your project.

## Scholarships/sponsorships

There are a range of scholarships, studentships and awards at university, faculty and department level to support both UK and overseas postgraduate researchers.

To be considered for many of our scholarships, you’ll need to be nominated by your proposed supervisor. Therefore, we’d highly recommend you discuss potential sources of funding with your supervisor first, so they can advise on your suitability and make sure you meet nomination deadlines.

For more information about our scholarships, visit our funding page or use our funding database to search for scholarships, studentships and awards you may be eligible for.

## Contact details

Our internationally-renowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality. Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .

## Programmes in related subject areas

Use the links below to view lists of programmes in related subject areas.

- Mathematics

## Regulated by the Office for Students

The University of Manchester is regulated by the Office for Students (OfS). The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. More information can be found at the OfS website .

You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .

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+ - Algebraic geometry Click to collapse

Geometric Invariant Theory : Faculty: Santosha Pattanayak

The geometry of algebraic varieties: :

The geometry of algebraic varieties with reductive group actions, including flag varieties, toric varieties, torus actions on algebraic varieties, and spherical varieties. I am also interested in studying the structure of algebraic groups and GIT (Geometric Invariant Theory) quotients of algebraic varieties and exploring toric degenerations of algebraic varieties. Moreover, my interest extends to areas such as the Seshadri constants of algebraic varieties and the symplectic invariants of smooth projective algebraic varieties, including the Gromov width. Faculty: Narasimha Chary Bonala

+ - Commutative Algebra Click to collapse

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula. Faculty: A. K. Maloo

+ - Complex Analysis & Operator Theory Click to collapse

I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory. Faculty: P. Muthukumar

+ - Computational Acoustics and Electromagnetics Click to collapse

The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few. Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research: 1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners. 2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools. 3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing. 4. High performance computing. Faculty: Akash Anand , B. V. Rathish Kumar

+ - Computational Fluid Dynamics Click to collapse

Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems. Faculty: B. V. Rathish Kumar , Saktipada Ghorai

+ - Differential Equations Click to collapse

Semigroups of linear operators and their applications, Functional differential equations, Galerkin approximations

Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.

Faculty: D. Bahuguna

Homogenization and Variational methods for partial differential equation

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$- convergence.

Faculty: T. Muthukumar , B.V. Rathish Kumar

Functional inequalities on Sobolev space

Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated

Asymptotic analysis on changing domains

Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems. Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed. Faculty: Prosenjit Roy , Kaushik Bal , Indranil Chowdhury

Analysis of Nonlinear PDEs involving fractional/nonlocal operator:

Fully nonlinear elliptic and parabolic equation involving nonlocal operators. Equations motivated from stochastic control/ Game problems including Hamilton Jacobi Bellman Equations, Isaacs Equations, Mean Field Games problems -

- Viscosity Solution theory, Comparison Principle, Wellposedness theory, Stability, Continuous Dependence.
- Numerical Analysis –Wellposedness, Convergence, Error estimates of Finite difference method, Semi-lagrangian Method.

Faculty: Indranil Chowdhury

Control Theory and its applications:

We study several aspects of controllability, say exact controllability, null controllability, approximate controllability, controllability to the trajectories of a given system of ordinary and partial differential equations (both linear and nonlinear). We study stabilizability (exponential, asymptotic) of a system of differential equations and construct feedback control for that system. Currently we are studying controllability of reaction diffusion systems of partial equations using Carleman inequalities and fixed point technique. Multiplier techniques are also used to show controllability of system of hyperbolic partial differential equations. Several mixed systems (hyperbolic and parabolic) are also been studied.

Faculty: Mrinmay Biswas

+ - Functional Analysis & Operator Theory Click to collapse

Banach space theory

Geometric and proximinality aspects in Banach spaces. Faculty: P. Shunmugaraj

Function-theoretic and graph-theoretic operator theory

The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. Faculty: Sameer Chavan

Non-commutative geometry

The main emphasis is on the metric aspect of noncommutative geometry. Faculty: Satyajit Guin

Bounded linear operators

A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations. Faculty: Sasmita Patnaik + - Harmonic Analysis Click to collapse

Operator spaces

The main emphasis is on operator space techniques in abstract Harmonic Analysis.

In the Euclidean setting

Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.

Faculty: Parasar Mohanty

On Lie groups

Problems related to integral geometry on Lie groups are being studied.

Faculty: Rama Rawat

+ - Homological Algebra Click to collapse

Cohomology and Deformation theory of algebraic structures

Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids.

This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.

Faculty: Ashis Mandal + - Image Processing Click to collapse

TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.

Faculty: B. V. Rathish Kumar + - Mathematical Biology Click to collapse

There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions.

Mathematical ecology

1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape.

2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants.

Mathematical epidemiology

1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors.

2. Mathematical Modeling of HIV Dynamics in vivo

Bioconvection

Bioconvection is the process of spontaneous pattern formation in a suspension of swimming microorganisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques.

Bio-fluid dynamics

Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.

Faculty: Malay Banerjee , Saktipada Ghorai , B.V. Rathish Kumar

Cardiac electrophysiology

Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc

Faculty: B.V. Rathish Kumar + - Number Theory & Arithmetic Geometry Click to collapse

Algebraic number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.

Faculty: Sudhanshu Shekhar

Analytic number theory

L-functions, sub-convexity problems, Sieve method

Faculty: Saurabh Kumar Singh

Number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Selmer groups

Faculty: Somnath Jha

Number theory, Dynamical systems, Random walks on groups

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme.

Faculty: Arijit Ganguly + - Numerical Analysis and Scientific Computing Click to collapse

The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high-quality research in the areas that include (but are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, Computational Fluid Dynamics, Computer-Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis, and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Nyström Method, Spline and Wavelet approximations, etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.

Faculty: B. V. Rathish Kumar , Akash Anand , Indranil Chowdhury + - Operator Algebra Click to collapse

Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.

Faculty: Keshab Chandra Bakshi + - Representation Theory Click to collapse

Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)

Faculty: Santosh Nadimpalli

Representations of finite and arithmetic groups

Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.

Faculty: Pooja Singla

Representation theory of Lie algebras and algebraic groups

Faculty: Santosha Pattanayak

Representation theory of infinite dimensional Lie algebras

Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.

Faculty: Sachin S. Sharma

Representation theory and Invariant theory

Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.

Faculty: Preena Samuel

Combinatorial representation theory

String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically

Faculty: Amit Kuber

Representation Theory of Algebraic groups:

Representation theory of Algebraic groups and Lie algebras, and its applications to Invariant theory and Algebraic geometry.

Faculty: Narasimha Chary Bonala + - Set Theory and Logic Click to collapse

Set theory (MSC Classification 03Exx)

We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/

Faculty: Ashutosh Kumar

Rough set theory and Modal logic

Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.

Faculty: Mohua Banerjee + - Several Complex Variables Click to collapse

Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.

Faculty: Vikramjeet Singh Chandel + - Topology and Geometry Click to collapse

Algebraic topology and Homotopy theory

The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.

Faculty: Debasis Sen

Algebraic topology, Combinatorial topology

I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.

Faculty: Nandini Nilakantan

Differential geometry

Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.

Faculty: G. Santhanam

Low dimensional topology

The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.

Faculty: Aparna Dar

Geometric group theory and Hyperbolic geometry

Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored.

Faculty: Abhijit Pal

Manifolds and Characteristic classes

We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.

Faculty: Ajay Singh Thakur

Moduli spaces of hyperbolic surfaces

The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.

Faculty: Bidyut Sanki

Systolic topology and Geometry

We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths.

Topological graph theory

We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.

Faculty: Bidyut Sanki + - Tribology Click to collapse

Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.

Faculty: B. V. Rathish Kumar

## Research Areas in Statistics and Probability Theory

Here are the areas of Statistics in which research is being done currently.

+ - Bayesian Nonparametric Methods Click to collapse

Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process.

Faculty: Minerva Mukhopadhyay

+ - Data Mining in Finance Click to collapse

Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners.

Faculty: Amit Mitra , Sharmishtha Mitra

+ - Econometric Modelling Click to collapse

Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric

Faculty: Shalabh , Sharmishtha Mitra

+ - Entropy Estimation and Applications Click to collapse

Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists.

Faculty: Neeraj Misra

+ - Environmental Statistics Click to collapse

The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission

Faculty: Arnab Hazra

+ - Estimation in Restricted Parameter Space Click to collapse

In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem.

+ - Game Theory Click to collapse

The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems.

Faculty: Soumyarup Sadhukhan

+ - Machine Learning and Statistical Pattern Recognition Click to collapse

Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms.

Faculty: Subhajit Dutta

+ - Markov chain Monte Carlo Click to collapse

Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target.

Faculty: Dootika Vats

+ - Non-Parametric and Robust Statistical methods Click to collapse

Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shaperestricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest.

Faculty: Subhra Sankar Dhar

+ - Optimal Experimental Design Click to collapse

The area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few.

Faculty: Satya Prakash Singh

+ - Ranking and Selection Problems Click to collapse

About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems.

+ - Regression Modelling Click to collapse

The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be auto correlated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area.

Faculty: Shalabh

+ - Robust Estimation in Nonlinear Models Click to collapse

Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of Mestimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models.

Faculty: Debasis Kundu , Amit Mitra

+ - Rough Paths and Regularity structures Click to collapse

The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of `pathwise' solutions of stochastic differential equations and stochastic partial differential equations.

Faculty: Suprio Bhar

Numerical analysis of differential equation driven by rough noise:

Developing numerical scheme for differential equations driven by rough noise and studying its convergence, rate of convergence etc.

Faculty: Mrinmay Biswas and Suprio Bhar

+ - Spatial statistics Click to collapse

The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice.

+ - Statistical Signal Processing Click to collapse

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators.

+ - Step-Stress Modelling Click to collapse

Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc.

Faculty: Debasis Kundu , Sharmishtha Mitra

+ - Stochastic Partial Differential Equations Click to collapse

The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few.

+ - Theory of Stochastic Orders and Aging and Applications Click to collapse

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.

DEPARTMENT OF Mathematics & Statistics

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Honors in mathematics.

Students desiring to graduate with honors must satisfy two requirements: they must have at least a 3.7 GPA in upper-division math courses (300-level and above) at the time of graduation, and they must write an honors thesis.

- 2.1 Thesis Advisor and Topic:
- 2.2 Thesis Committee:
- 2.3 Thesis Completion and Oral Presentation:
- 2.4 Optional LaTeX style file for honors theses:
- 2.5 Recent Honors Theses:

Students must write a thesis under the direction of an advisor, covering an advanced topic in mathematics. The student must give an oral presentation of the thesis to a committee, who must give their final approval.

Writing a thesis involves learning about an advanced topic in mathematics. To prepare yourself to be able to do this, it is a good idea to take one or more 400-level courses during your junior year. This is not a requirement, but it is strongly recommended.

## Timeline and Suggestions

Thesis advisor and topic:.

Honors theses from past years are available for perusal in Hilbert Space (University 107-108). These will give you an idea of what students have done in the past, and they might even inspire ideas for what you would like to do.

If you decide to pursue honors, then near the end of your junior year you should investigate possible advisors. Occasionally students have an idea right away of what topic they want to write their thesis on, and then they have to search for an advisor who is capable and willing to supervise such a thesis. What happens more often is that students know a professor who they would like to work with, and the professor recommends a topic (in consultation with the student). In either scenario, spring of your junior year is a good time to shop around for possible advisors. Most professors are happy to talk with students about possibilities for a thesis. If all else fails, you can always talk to the head undergraduate advisor or the director of undergraduate affairs (see Useful Contacts in the sidebar of this page).

It is important to have chosen an advisor before you start your senior year. This means making the arrangements during the spring of your junior year, or at the very latest during the summer after your junior year. Your work on your thesis will take place during the fall and winter of your senior year, with much of the writing done during the spring. The “work” usually involves the student reading an advanced piece of mathematics and meeting weekly (sometimes biweekly) with his or her advisor for discussion.

At the beginning of fall of your senior year, you should send an email to both the director of undergraduate studies and the head undergraduate advisor informing them that you will be working on an honors thesis, and naming your topic and your advisor.

## Thesis Committee:

In addition to your advisor you must select a thesis committee; this will consist of your advisor together with two other professors (usually math professors, but this is not required). The thesis committee does not have to be chosen until the winter of your senior year, and usually your advisor will make suggestions of people who are familiar with the area of your thesis.

## Thesis Completion and Oral Presentation:

Once your written thesis is completed, you must give a copy to all the people on your committee and then schedule a thesis defense. The defense must be held by the end of week 9 of spring term, and your committee members must get a copy of the thesis at least two weeks before the defense. So the written thesis must be completed by the end of week 7 of spring term. Usually committee members will make suggestions for improvements, and so this is not necessarily the “final” form of the thesis, but it should be very close to the final form.

When a date and time for the oral defense has been decided, work with the Undergraduate Coordinator (in the main math office) to schedule a room and post public announcements of the defense around Fenton and University. Oral defenses are open to the public! Announcements should be posted at least a week ahead of time.

The exact nature of the oral defense is worked out in consultation with the thesis advisor. Usually it involves the student giving a 30-45 minute presentation of the work, followed by questions from the thesis committee. At the end of the defense everyone except the thesis committee leaves the room, and the committee determines whether or not the thesis meets expectations.

At the end of the year, submit the final version of your thesis to the Director of Undergraduate Studies. This will be bound and added to the library in Hilbert Space.

## Optional LaTeX style file for honors theses:

Thanks to Seth Temple for providing the file.

## Recent Honors Theses:

Hannah (Qiaochu) Cui Using Information Theory to Understand Neural Representation in the Auditory Cortex

Sierra Nicole Battan, Orthogonal Structure on a Tripod

Ben Estevez, Random *-Cosquare Matrices and Self-Inverse Polynomials

Nathaniel B. Schieber, Random *-Cosquare Matrices and Self-Inverse Polynomials

Aleksander Shmakov, Galois Representations in Étale Fundamental Groups and the Profinite Grothendieck-Teichmüller Group

Seth David Temple, The Tweedie Index Parameter and Its Estimator

Lianjie Jiang, Higher Order Beliefs and Sequential Reciprocity

Dongmin Roh, Reflexive Polygons and Loops

Graham Simon, Hawkes Processes in Finance: A Review with Simulations

Department of Mathematics

202 Fenton Hall 1021 E. 13th, Eugene, OR 97403

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## Take Action

## Department of Mathematics

Phd thesis defence in mathematical statistics.

Thesis defence

Date: Wednesday 12 June 2024

Time: 13.00 – 15.00

Location: Department of Mathematics, Albano campus, auditorium 4, house 2.

Dongni Zhang will defend his PhD thesis in mathematical statistics

Title : Stochastic epidemic models with contact tracing

Opponent : Serik Sagitov (Chalmers/GU) Supervisor : Tom Britton

Abstract : This thesis consists of four papers, which all contain a certain amount of squares. In Paper I, we study compactly supported cohomology theories of varieties. These can be seen as functors with a nice descent property, out of a category whose objects are varieties, and whose morphisms are spans that consist of an open immersion and a proper map. Using the theory of cd-structures, which are sets of commutative squares that generate a topology, we show that a compactly supported cohomology theory can be uniquely extended from its restriction to smooth and complete varieties. In Paper II, we continue the study of cd-structures. If a morphism f between sites satisfies the conditions of the comparison lemma, then it induces an equivalence between the associated categories of (hyper)sheaves. If the topologies in question are generated by sufficiently nice cd-structures, then we show that f also induces an equivalence between the associated categories of symmetric monoidal hypersheaves. We use this to prove a variant of the main result of Paper I for symmetric monoidal hypersheaves. The highest degree of square-ness is reached in Paper III. Here, commutative squares are used to build K-theory spectra, most notably for the category of varieties. We reuse some of the square-y arguments from Paper I to show that the K-theory spectrum of the category of varieties is equivalent to the K-theory spectrum of the category of complete varieties. Moreover, exploiting the square-ness of the category of compactly supported cohomology theories that is demonstrated in Paper I, we can construct a new derived motivic measure. Paper III is joint with Jonathan Campbell, Mona Merling and Inna Zakharevich. In Paper IV, we build on the result of Paper I, and its variation proven in Paper II, to obtain a result about functors that encode six-functor formalisms. Squares show up again, not only in the form of cd-structures, but also in the form of adjointable squares, which play an important role in extending a six-functor formalism from the domain of complete varieties to all varieties.

Last updated: May 24, 2024

Source: Department of Mathematics

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## VIDEO

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Guide to Graduate Studies. The PhD Program. The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in ...

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below: Situation. Typical Fee (Median) Fee Range.

PhD Program. More information and a full list of requirements for the PhD program in Mathematics can be found in the University Bulletin. During their first year in the program, students typically engage in coursework and seminars which prepare them for the Qualifying Examinations . Currently, these two exams test the student's breadth of ...

PhD in Mathematics. The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

Graduate Students 2018-2019. The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests ...

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we ...

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements. During the first year of the Ph.D. program: Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics. Pass the six-hour written Preliminary Examination covering calculus ...

PhD In Mathematics PhD in Mathematics. The PhD in Mathematics consists of preliminary coursework and study, qualifying exams, a candidacy exam with an adviser, and creative research culminating in a written dissertation and defense. All doctoral students must also do some teaching on the way to the PhD. There are ...

Admission is limited and highly selective. Successful applicants have typically pursued an undergraduate major in mathematics. In the first year of PhD studies, students must pass written examinations in the areas of the basic first-year courses. In the second year an oral examination on two selected topics must be passed.

A selection of Mathematics PhD thesis titles is listed below, some of which are available online: 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991. 2023. Melanie Kobras - Low order models of storm track variability Ed Clark - Vectorial Variational Problems in L∞ and Applications ...

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. ... The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student's ...

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the NYU Shanghai Ph ...

Welcome to the Yale graduate program in Mathematics. The transition from mathematics student to working mathematician depends on ability, hard work and independence, but also on community. Yale's graduate program provides an excellent environment for this, and we are proud of the talented students who come here and the leading faculty with ...

Students who entered the program in Autumn 2023 or later. To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: algebra. real analysis. geometry and topology. applied mathematics. The exams each consist of two parts. Students are given three hours for each part.

The program is individualized to meet the needs of graduate students. The student must develop, with the guidance from the major professor and committee, a program that is applicable to their background and interest. The average Ph.D. program requires 4-6 years beyond a master's degree. The program is comprised of coursework in four major areas.

The students must complete 39 hours of coursework as follows: At least 30 hours must be in mathematics courses at the 6000-level or higher. At least 9 hours must form the doctoral minor field of study. The overall GPA for these courses must be at least 3.0. These courses must be taken for a letter grade and passed with a grade of at least C.

PhD projects. Several School members offer supervision for PhD research projects in the School of Mathematics and Statistics. Navigate via the tabs below to view project offerings by School members in the areas of Applied Mathematics, Pure Mathematics and Statistics. (This list was updated September 2022.)

The PhD in Mathematics offers an intellectually stimulating and dynamic atmosphere research in both pure and applied mathematics. Study at the University of Warwick's Mathematics Institute, an international centre of research excellence, ranked 3rd for research power and 3rd for the number of 4* research outputs in REF 2021 (amongst UK universities).

Opportunities for PhD research are available in a wide range of topics in Mathematics. For more information, please see the Pure Mathematics research areas and read advice on choosing a project or find out more about specific projects.

Research Areas in Mathematics. Here are the areas of Mathematics in which research is being done currently. + Algebraic geometry. Geometric Invariant Theory: Faculty: Santosha Pattanayak. The geometry of algebraic varieties:: The geometry of algebraic varieties with reductive group actions, including flag varieties, toric varieties, torus ...

PhD seminars at the Department of Mathematics. Essén Lectures. These lectures are presented by distinguished experts and directed to a general audience of PhD students. Students who complete the assignments get 3hp. The lecture series is named after Professor Matts Essén (1932-2003), who worked at Uppsala University until his retirement in 1997.

Baudhayana Sulba Sutra. Baudhayana; Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia.Though this was primarily a geometrical text, it also contained some important algebraic developments, including the list of Pythagorean triples discovered algebraically ...

Honors in Mathematics. Students desiring to graduate with honors must satisfy two requirements: they must have at least a 3.7 GPA in upper-division math courses (300-level and above) at the time of graduation, and they must write an honors thesis. Contents. 1 Thesis: 2 Timeline and Suggestions. 2.1 Thesis Advisor and Topic: 2.2 Thesis Committee:

Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability ...

A collaborative model for teaching and learning mathematics in secondary schools. Mathematics pass rates in South African schools, as in many developing nations, continue to be a source of concern for educators and policymakers alike. Improving mathematics performance is non-negotiable if Africa is to ...

Dongni Zhang will defend his PhD thesis in mathematical statistics. Title: Stochastic epidemic models with contact tracing. This thesis consists of four papers, which all contain a certain amount of squares. In Paper I, we study compactly supported cohomology theories of varieties. These can be seen as functors with a nice descent property, out ...

Mathematics Researcher. Mar 2014 - Dec 2015 1 year 10 months. - Worked on problems in matrix theory with applications to data analysis and quantum computing. - Participant in the NSF funded ...

A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: philosophiae doctor or doctor philosophiae) is the most common degree at the highest academic level, awarded following a course of study and research. The degree is most often abbreviated PhD (or, at times, as Ph.D. in North America).It is derived from the Latin Philosophiae Doctor, pronounced as three separate letters (/ p iː eɪ tʃ ˈ d ...