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Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2023 2023.

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

Statistical Analysis of Health Habits for Incoming College Students , Wendy Isamara Lizarraga Noriega

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez

SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Samar Mikhail Kasouha

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS , Katelyn Sage Risinger

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

DNA COMPLEXES OF ONE BOND-EDGE TYPE , Andrew Tyler Lavengood-Ryan

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

ASSESSING STUDENT UNDERSTANDING WHILE SOLVING LINEAR EQUATIONS USING FLOWCHARTS AND ALGEBRAIC METHODS , Edima Umanah

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

VANISHING LOCAL SCALAR INVARIANTS ON GENERALIZED PLANE WAVE MANIFOLDS , Brian Matthew Friday

Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.

THE EFFECTIVENESS OF DYNAMIC MATHEMATICAL SOFTWARE IN THE INSTRUCTION OF THE UNIT CIRCLE , Edward Simons

CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM , Ty Stockham

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

THE STRUGGLE WITH INVERSE FUNCTIONS DOING AND UNDOING PROCESS , Jesus Nolasco

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING , Gina Marie Cheney

Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis

BIO-MATHEMATICS: INTRODUCTION TO THE MATHEMATICAL MODEL OF THE HEPATITIS C VIRUS , Lucille J. Durfee

ANALYSIS AND SYNTHESIS OF THE LITERATURE REGARDING ACTIVE AND DIRECT INSTRUCTION AND THEIR PROMOTION OF FLEXIBLE THINKING IN MATHEMATICS , Genelle Elizabeth Gonzalez

LIFE EXPECTANCY , Ali R. Hassanzadah

PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS , Sean M. Hearon

A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS , Leonard B. Lamp

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field , Nolberto Rezola

ALGEBRA 1 STUDENTS’ ABILITY TO RELATE THE DEFINITION OF A FUNCTION TO ITS REPRESENTATIONS , Sarah A. Thomson

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Home > USC Columbia > Arts and Sciences > Mathematics > Mathematics Theses and Dissertations

Mathematics Theses and Dissertations

Theses/dissertations from 2023 2023.

Extreme Covering Systems, Primes Plus Squarefrees, and Lattice Points Close to a Helix , Jack Robert Dalton

On the Algebraic and Geometric Multiplicity of Zero as a Hypergraph Eigenvalue , Grant Ian Fickes

Widely Digitally Delicate Brier Primes and Irreducibility Results for Some Classes of Polynomials , Thomas David Luckner

Deep Learning Methods for Some Problems in Scientific Computing , Yuankai Teng

Theses/Dissertations from 2022 2022

Covering Systems and the Minimum Modulus Problem , Maria Claire Cummings

The Existence and Quantum Approximation of Optimal Pure State Ensembles , Ryan Thomas McGaha

Structure Preserving Reduced-Order Models of Hamiltonian Systems , Megan Alice McKay

Tangled up in Tanglegrams , Drew Joseph Scalzo

Results on Select Combinatorial Problems With an Extremal Nature , Stephen Smith

Poset Ramsey Numbers for Boolean Lattices , Joshua Cain Thompson

Some Properties and Applications of Spaces of Modular Forms With ETA-Multiplier , Cuyler Daniel Warnock

Theses/Dissertations from 2021 2021

Simulation of Pituitary Organogenesis in Two Dimensions , Chace E. Covington

Polynomials, Primes and the PTE Problem , Joseph C. Foster

Widely Digitally Stable Numbers and Irreducibility Criteria For Polynomials With Prime Values , Jacob Juillerat

A Numerical Investigation of Fractional Models for Viscoelastic Materials With Applications on Concrete Subjected to Extreme Temperatures , Murray Macnamara

Trimming Complexes , Keller VandeBogert

Multiple Frailty Model for Spatially Correlated Interval-Censored , Wanfang Zhang

Theses/Dissertations from 2020 2020

An Equivariant Count of Nodal Orbits in an Invariant Pencil of Conics , Candace Bethea

Finite Axiomatisability in Nilpotent Varieties , Joshua Thomas Grice

Rationality Questions and the Derived Category , Alicia Lamarche

Counting Number Fields by Discriminant , Harsh Mehta

Distance Related Graph Invariants in Triangulations and Quadrangulations of the Sphere , Trevor Vincent Olsen

Diameter of 3-Colorable Graphs and Some Remarks on the Midrange Crossing Constant , Inne Singgih

Two Inquiries Related to the Digits of Prime Numbers , Jeremiah T. Southwick

Windows and Generalized Drinfeld Kernels , Robert R. Vandermolen

Connections Between Extremal Combinatorics, Probabilistic Methods, Ricci Curvature of Graphs, and Linear Algebra , Zhiyu Wang

An Ensemble-Based Projection Method and Its Numerical Investigation , Shuai Yuan

Variable-Order Fractional Partial Differential Equations: Analysis, Approximation and Inverse Problem , Xiangcheng Zheng

Theses/Dissertations from 2019 2019

Classification of Non-Singular Cubic Surfaces up to e-invariants , Mohammed Alabbood

On the Characteristic Polynomial of a Hypergraph , Gregory J. Clark

A Development of Transfer Entropy in Continuous-Time , Christopher David Edgar

Moving Off Collections and Their Applications, in Particular to Function Spaces , Aaron Fowlkes

Finding Resolutions of Mononomial Ideals , Hannah Melissa Kimbrell

Regression for Pooled Testing Data with Biomedical Applications , Juexin Lin

Numerical Methods for a Class of Reaction-Diffusion Equations With Free Boundaries , Shuang Liu

An Implementation of the Kapustin-Li Formula , Jessica Otis

A Nonlinear Parallel Model for Reversible Polymer Solutions in Steady and Oscillating Shear Flow , Erik Tracey Palmer

A Few Problems on the Steiner Distance and Crossing Number of Graphs , Josiah Reiswig

Successful Pressing Sequences in Simple Pseudo-Graphs , Hays Wimsatt Whitlatch

On The Generators of Quantum Dynamical Semigroups , Alexander Wiedemann

An Examination of Kinetic Monte Carlo Methods with Application to a Model of Epitaxial Growth , Dylana Ashton Wilhelm

Dynamical Entropy of Quantum Random Walks , Duncan Wright

Unconditionally Energy Stable Linear Schemes for a Two-Phase Diffuse Interface Model with Peng-Robinson Equation of State , Chenfei Zhang

Theses/Dissertations from 2018 2018

Theory, Computation, and Modeling of Cancerous Systems , Sameed Ahmed

Turán Problems and Spectral Theory on Hypergraphs and Tensors , Shuliang Bai

Quick Trips: On the Oriented Diameter of Graphs , Garner Paul Cochran

Geometry of Derived Categories on Noncommutative Projective Schemes , Blake Alexander Farman

A Quest for Positive Definite Matrices over Finite Fields , Erin Patricia Hanna

Comparison of the Performance of Simple Linear Regression and Quantile Regression with Non-Normal Data: A Simulation Study , Marjorie Howard

Special Fiber Rings of Certain Height Four Gorenstein Ideals , Jaree Hudson

Graph Homomorphisms and Vector Colorings , Michael Robert Levet

Local Rings and Golod Homomorphisms , Thomas Schnibben

States and the Numerical Range in the Regular Algebra , James Patrick Sweeney

Thermodynamically Consistent Hydrodynamic Phase Field Models and Numerical Approximation for Multi-Component Compressible Viscous Fluid Mixtures , Xueping Zhao

Theses/Dissertations from 2017 2017

On the Existence of Non-Free Totally Reflexive Modules , J. Cameron Atkins

Subdivision of Measures of Squares , Dylan Bates

Unconditionally Energy Stable Numerical Schemes for Hydrodynamics Coupled Fluids Systems , Alexander Yuryevich Brylev

Convergence and Rate of Convergence of Approximate Greedy-Type Algorithms , Anton Dereventsov

Covering Subsets of the Integers and a Result on Digits of Fibonacci Numbers , Wilson Andrew Harvey

Nonequispaced Fast Fourier Transform , David Hughey

Deep Learning: An Exposition , Ryan Kingery

A Family of Simple Codimension Two Singularities with Infinite Cohen-Macaulay Representation Type , Tyler Lewis

Polynomials Of Small Mahler Measure With no Newman Multiples , Spencer Victoria Saunders

Theses/Dissertations from 2016 2016

On Crown-free Set Families, Diffusion State Difference, and Non-uniform Hypergraphs , Edward Lawrence Boehnlein

Structure of the Stable Marriage and Stable Roommate Problems and Applications , Joe Hidakatsu

Binary Quartic Forms over Fp , Daniel Thomas Kamenetsky

On a Constant Associated with the Prouhet-Tarry-Escott Problem , Maria E. Markovich

Some Extremal And Structural Problems In Graph Theory , Taylor Mitchell Short

Chebyshev Inversion of the Radon Transform , Jared Cameron Szi

Modeling of Structural Relaxation By A Variable-Order Fractional Differential Equation , Su Yang

Theses/Dissertations from 2015 2015

Modeling, Simulation, and Applications of Fractional Partial Differential Equations , Wilson Cheung

The Packing Chromatic Number of Random d-regular Graphs , Ann Wells Clifton

Commutator Studies in Pursuit of Finite Basis Results , Nathan E. Faulkner

Avoiding Doubled Words in Strings of Symbols , Michael Lane

A Survey of the Kinetic Monte Carlo Algorithm as Applied to a Multicellular System , Michael Richard Laughlin

Toward the Combinatorial Limit Theory of free Words , Danny Rorabaugh

Trees, Partitions, and Other Combinatorial Structures , Heather Christina Smith

Fast Methods for Variable-Coefficient Peridynamic and Non-Local Diffusion Models , Che Wang

Modeling and Computations of Cellular Dynamics Using Complex-fluid Models , Jia Zhao

Theses/Dissertations from 2014 2014

The Non-Existence of a Covering System with all Moduli Distinct, Large and Square-Free , Melissa Kate Bechard

Explorations in Elementary and Analytic Number Theory , Scott Michael Dunn

Independence Polynomials , Gregory Matthew Ferrin

Turán Problems on Non-uniform Hypergraphs , Jeremy Travis Johnston

On the Group of Transvections of ADE-Diagrams , Marvin Jones

Fake Real Quadratic Orders , Richard Michael Oh

Theses/Dissertations from 2013 2013

Shimura Images of A Family of Half-Integral Weight Modular Forms , Kenneth Allan Brown

Sharp Bounds Associated With An Irreducibility Theorem For Polynomials Having Non-Negative Coefficients , Morgan Cole

Deducing Vertex Weights From Empirical Occupation Times , David Collins

Analysis and Processing of Irregularly Distributed Point Clouds , Kamala Hunt Diefenthaler

Generalizations of Sperner's Theorem: Packing Posets, Families Forbidding Posets, and Supersaturation , Andrew Philip Dove

Spectral Analysis of Randomly Generated Networks With Prescribed Degree Sequences , Clifford Davis Gaddy

Selected Research In Covering Systems of the Integers and the Factorization of Polynomials , Joshua Harrington

The Weierstrass Approximation Theorem , LaRita Barnwell Hipp

The Compact Implicit Integration Factor Scheme For the Solution of Allen-Cahn Equations , Meshack K. Kiplagat

Applications of the Lopsided Lovász Local Lemma Regarding Hypergraphs , Austin Tyler Mohr

Study On Covolume-Upwind Finite Volume Approximations For Linear Parabolic Partial Differential Equations , Rosalia Tatano

Coloring Pythagorean Triples and a Problem Concerning Cyclotomic Polynomials , Daniel White

Theses/Dissertations from 2012 2012

A Computational Approach to the Quillen-Suslin Theorem, Buchsbaum-Eisenbud Matrices, and Generic Hilbert-Burch Matrices , Jonathan Brett Barwick

Mathematical Modeling and Computational Studies for Cell Signaling , Kanadpriya Basu

Fast Solution Methods For Fractional Diffusion Equations and Their Applications , Treena Basu

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Digital Commons @ USF > College of Arts and Sciences > Mathematics and Statistics > Theses and Dissertations

Mathematics and Statistics Theses and Dissertations

Theses/dissertations from 2023 2023.

Classification of Finite Topological Quandles and Shelves via Posets , Hitakshi Lahrani

Applied Analysis for Learning Architectures , Himanshu Singh

Rational Functions of Degree Five That Permute the Projective Line Over a Finite Field , Christopher Sze

Theses/Dissertations from 2022 2022

New Developments in Statistical Optimal Designs for Physical and Computer Experiments , Damola M. Akinlana

Advances and Applications of Optimal Polynomial Approximants , Raymond Centner

Data-Driven Analytical Predictive Modeling for Pancreatic Cancer, Financial & Social Systems , Aditya Chakraborty

On Simultaneous Similarity of d-tuples of Commuting Square Matrices , Corey Connelly

Symbolic Computation of Lump Solutions to a Combined (2+1)-dimensional Nonlinear Evolution Equation , Jingwei He

Boundary behavior of analytic functions and Approximation Theory , Spyros Pasias

Stability Analysis of Delay-Driven Coupled Cantilevers Using the Lambert W-Function , Daniel Siebel-Cortopassi

A Functional Optimization Approach to Stochastic Process Sampling , Ryan Matthew Thurman

Theses/Dissertations from 2021 2021

Riemann-Hilbert Problems for Nonlocal Reverse-Time Nonlinear Second-order and Fourth-order AKNS Systems of Multiple Components and Exact Soliton Solutions , Alle Adjiri

Zeros of Harmonic Polynomials and Related Applications , Azizah Alrajhi

Combination of Time Series Analysis and Sentiment Analysis for Stock Market Forecasting , Hsiao-Chuan Chou

Uncertainty Quantification in Deep and Statistical Learning with applications in Bio-Medical Image Analysis , K. Ruwani M. Fernando

Data-Driven Analytical Modeling of Multiple Myeloma Cancer, U.S. Crop Production and Monitoring Process , Lohuwa Mamudu

Long-time Asymptotics for mKdV Type Reduced Equations of the AKNS Hierarchy in Weighted L 2 Sobolev Spaces , Fudong Wang

Online and Adjusted Human Activities Recognition with Statistical Learning , Yanjia Zhang

Theses/Dissertations from 2020 2020

Bayesian Reliability Analysis of The Power Law Process and Statistical Modeling of Computer and Network Vulnerabilities with Cybersecurity Application , Freeh N. Alenezi

Discrete Models and Algorithms for Analyzing DNA Rearrangements , Jasper Braun

Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data , Kun Bu

On the p(x)-Laplace equation in Carnot groups , Robert D. Freeman

Clustering methods for gene expression data of Oxytricha trifallax , Kyle Houfek

Gradient Boosting for Survival Analysis with Applications in Oncology , Nam Phuong Nguyen

Global and Stochastic Dynamics of Diffusive Hindmarsh-Rose Equations in Neurodynamics , Chi Phan

Restricted Isometric Projections for Differentiable Manifolds and Applications , Vasile Pop

On Some Problems on Polynomial Interpolation in Several Variables , Brian Jon Tuesink

Numerical Study of Gap Distributions in Determinantal Point Process on Low Dimensional Spheres: L -Ensemble of O ( n ) Model Type for n = 2 and n = 3 , Xiankui Yang

Non-Associative Algebraic Structures in Knot Theory , Emanuele Zappala

Theses/Dissertations from 2019 2019

Field Quantization for Radiative Decay of Plasmons in Finite and Infinite Geometries , Maryam Bagherian

Probabilistic Modeling of Democracy, Corruption, Hemophilia A and Prediabetes Data , A. K. M. Raquibul Bashar

Generalized Derivations of Ternary Lie Algebras and n-BiHom-Lie Algebras , Amine Ben Abdeljelil

Fractional Random Weighted Bootstrapping for Classification on Imbalanced Data with Ensemble Decision Tree Methods , Sean Charles Carter

Hierarchical Self-Assembly and Substitution Rules , Daniel Alejandro Cruz

Statistical Learning of Biomedical Non-Stationary Signals and Quality of Life Modeling , Mahdi Goudarzi

Probabilistic and Statistical Prediction Models for Alzheimer’s Disease and Statistical Analysis of Global Warming , Maryam Ibrahim Habadi

Essays on Time Series and Machine Learning Techniques for Risk Management , Michael Kotarinos

The Systems of Post and Post Algebras: A Demonstration of an Obvious Fact , Daviel Leyva

Reconstruction of Radar Images by Using Spherical Mean and Regular Radon Transforms , Ozan Pirbudak

Analyses of Unorthodox Overlapping Gene Segments in Oxytricha Trifallax , Shannon Stich

An Optimal Medium-Strength Regularity Algorithm for 3-uniform Hypergraphs , John Theado

Power Graphs of Quasigroups , DayVon L. Walker

Theses/Dissertations from 2018 2018

Groups Generated by Automata Arising from Transformations of the Boundaries of Rooted Trees , Elsayed Ahmed

Non-equilibrium Phase Transitions in Interacting Diffusions , Wael Al-Sawai

A Hybrid Dynamic Modeling of Time-to-event Processes and Applications , Emmanuel A. Appiah

Lump Solutions and Riemann-Hilbert Approach to Soliton Equations , Sumayah A. Batwa

Developing a Model to Predict Prevalence of Compulsive Behavior in Individuals with OCD , Lindsay D. Fields

Generalizations of Quandles and their cohomologies , Matthew J. Green

Hamiltonian structures and Riemann-Hilbert problems of integrable systems , Xiang Gu

Optimal Latin Hypercube Designs for Computer Experiments Based on Multiple Objectives , Ruizhe Hou

Human Activity Recognition Based on Transfer Learning , Jinyong Pang

Signal Detection of Adverse Drug Reaction using the Adverse Event Reporting System: Literature Review and Novel Methods , Minh H. Pham

Statistical Analysis and Modeling of Cyber Security and Health Sciences , Nawa Raj Pokhrel

Machine Learning Methods for Network Intrusion Detection and Intrusion Prevention Systems , Zheni Svetoslavova Stefanova

Orthogonal Polynomials With Respect to the Measure Supported Over the Whole Complex Plane , Meng Yang

Theses/Dissertations from 2017 2017

Modeling in Finance and Insurance With Levy-It'o Driven Dynamic Processes under Semi Markov-type Switching Regimes and Time Domains , Patrick Armand Assonken Tonfack

Prevalence of Typical Images in High School Geometry Textbooks , Megan N. Cannon

On Extending Hansel's Theorem to Hypergraphs , Gregory Sutton Churchill

Contributions to Quandle Theory: A Study of f-Quandles, Extensions, and Cohomology , Indu Rasika U. Churchill

Linear Extremal Problems in the Hardy Space H p for 0 p , Robert Christopher Connelly

Statistical Analysis and Modeling of Ovarian and Breast Cancer , Muditha V. Devamitta Perera

Statistical Analysis and Modeling of Stomach Cancer Data , Chao Gao

Structural Analysis of Poloidal and Toroidal Plasmons and Fields of Multilayer Nanorings , Kumar Vijay Garapati

Dynamics of Multicultural Social Networks , Kristina B. Hilton

Cybersecurity: Stochastic Analysis and Modelling of Vulnerabilities to Determine the Network Security and Attackers Behavior , Pubudu Kalpani Kaluarachchi

Generalized D-Kaup-Newell integrable systems and their integrable couplings and Darboux transformations , Morgan Ashley McAnally

Patterns in Words Related to DNA Rearrangements , Lukas Nabergall

Time Series Online Empirical Bayesian Kernel Density Segmentation: Applications in Real Time Activity Recognition Using Smartphone Accelerometer , Shuang Na

Schreier Graphs of Thompson's Group T , Allen Pennington

Cybersecurity: Probabilistic Behavior of Vulnerability and Life Cycle , Sasith Maduranga Rajasooriya

Bayesian Artificial Neural Networks in Health and Cybersecurity , Hansapani Sarasepa Rodrigo

Real-time Classification of Biomedical Signals, Parkinson’s Analytical Model , Abolfazl Saghafi

Lump, complexiton and algebro-geometric solutions to soliton equations , Yuan Zhou

Theses/Dissertations from 2016 2016

A Statistical Analysis of Hurricanes in the Atlantic Basin and Sinkholes in Florida , Joy Marie D'andrea

Statistical Analysis of a Risk Factor in Finance and Environmental Models for Belize , Sherlene Enriquez-Savery

Putnam's Inequality and Analytic Content in the Bergman Space , Matthew Fleeman

On the Number of Colors in Quandle Knot Colorings , Jeremy William Kerr

Statistical Modeling of Carbon Dioxide and Cluster Analysis of Time Dependent Information: Lag Target Time Series Clustering, Multi-Factor Time Series Clustering, and Multi-Level Time Series Clustering , Doo Young Kim

Some Results Concerning Permutation Polynomials over Finite Fields , Stephen Lappano

Hamiltonian Formulations and Symmetry Constraints of Soliton Hierarchies of (1+1)-Dimensional Nonlinear Evolution Equations , Solomon Manukure

Modeling and Survival Analysis of Breast Cancer: A Statistical, Artificial Neural Network, and Decision Tree Approach , Venkateswara Rao Mudunuru

Generalized Phase Retrieval: Isometries in Vector Spaces , Josiah Park

Leonard Systems and their Friends , Jonathan Spiewak

Resonant Solutions to (3+1)-dimensional Bilinear Differential Equations , Yue Sun

Statistical Analysis and Modeling Health Data: A Longitudinal Study , Bhikhari Prasad Tharu

Global Attractors and Random Attractors of Reaction-Diffusion Systems , Junyi Tu

Time Dependent Kernel Density Estimation: A New Parameter Estimation Algorithm, Applications in Time Series Classification and Clustering , Xing Wang

On Spectral Properties of Single Layer Potentials , Seyed Zoalroshd

Theses/Dissertations from 2015 2015

Analysis of Rheumatoid Arthritis Data using Logistic Regression and Penalized Approach , Wei Chen

Active Tile Self-assembly and Simulations of Computational Systems , Daria Karpenko

Nearest Neighbor Foreign Exchange Rate Forecasting with Mahalanobis Distance , Vindya Kumari Pathirana

Statistical Learning with Artificial Neural Network Applied to Health and Environmental Data , Taysseer Sharaf

Radial Versus Othogonal and Minimal Projections onto Hyperplanes in l_4^3 , Richard Alan Warner

Ensemble Learning Method on Machine Maintenance Data , Xiaochuang Zhao

Theses/Dissertations from 2014 2014

Properties of Graphs Used to Model DNA Recombination , Ryan Arredondo

Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability , Jonathan Burns

On the Classification of Groups Generated by Automata with 4 States over a 2-Letter Alphabet , Louis Caponi

Statistical Analysis, Modeling, and Algorithms for Pharmaceutical and Cancer Systems , Bong-Jin Choi

Topological Data Analysis of Properties of Four-Regular Rigid Vertex Graphs , Grant Mcneil Conine

Trend Analysis and Modeling of Health and Environmental Data: Joinpoint and Functional Approach , Ram C. Kafle

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Main areas of research activity are algebra, including group theory, semigroup theory, lattice theory, and computational group theory, and analysis, including fractal geometry, multifractal analysis, complex dynamical systems, Kleinian groups, and diophantine approximations.

For more information please visit the School of Mathematics and Statistics home page.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.

Recent Submissions

Rearrangement groups of connected spaces , modern computational methods for finitely presented monoids , finiteness conditions on semigroups relating to their actions and one-sided congruences , on constructing topology from algebra , interpolating between hausdorff and box dimension .

Senior Thesis

This page is for Undergraduate Senior Theses.  For Ph.D. Theses, see here .

So that Math Department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online (while all theses are available through Harvard University Archives , it would be more convenient to have them online). It is absolutely voluntary, but if you decide to give us your permission, please send an electronic version of your thesis to cindy@math. The format can be in order of preference: DVI, PS, PDF. In the case of submitting a DVI format, make sure to include all EPS figures. You can also submit Latex or MS word source files.

If you are looking for information and advice from students and faculty about writing a senior thesis, look at this document . It was compiled from comments of students and faculty in preparation for, and during, an information session. Let Wes Cain ([email protected]) know if you have any questions not addressed in the document.

Dissertations and Placements 2010-Present

Kimoi Kemboi Thesis: Full exceptional collections of vector bundles on linear GIT quotients Advisor: Daniel Halpern-Leistner First Position: Postdoc at the Institution for Advanced Study and Princeton

Max Lipton Thesis: Dynamical Systems in Pure Mathematics Advisor: Steven Strogatz First Position: NSF Mathematical Sciences Postdoctoral Fellow at Massachusetts Institute of Technology

Elise McMahon Thesis: A simplicial set approach to computing the group homology of some orthogonal subgroups of the discrete group  Advisor: Inna Zakharevich First Position: Senior Research Scientist at Two Six Technologies

Peter Uttenthal Thesis: Density of Selmer Ranks in Families of Even Galois Representations Advisor: Ravi Kumar Ramakrishna First Position: Visiting Assistant Professor at Cornell University

Liu Yun Thesis: Towers of Borel Fibrations and Generalized Quasi-Invariants Advisor: Yuri Berest First Position: Postdoc at Indiana University Bloomington

Romin Abdolahzadi Thesis: Anabelian model theory Advisor: Anil Nerode First Position: Quantitative Analyst, A.R.T. Advisors, LLC

Hannah Cairns Thesis: Abelian processes, and how they go to sleep Advisor: Lionel Levine First Position: Visiting Assistant Professor, Cornell University

Shiping Cao Thesis: Topics in scaling limits on some Sierpinski carpet type fractals Advisor: Robert Strichartz (Laurent Saloff-Coste in last semester) First Position: Postdoctoral Scholar, University of Washington

Andres Fernandes Herrero Thesis: On the boundedness of the moduli of logarithmic connections Advisor: Nicolas Templier First Position: Ritt Assistant Professor, Columbia University

Max Hallgren Thesis: Ricci Flow with a Lower Bound on Ricci Curvature Advisor: Xiaodong Cao First Position: NSF Postdoctoral Research Fellow, Rutgers University

Gautam Krishnan Thesis: Degenerate series representations for symplectic groups Advisor: Dan Barbasch First Position: Hill Assistant Professor, Rutgers University

Feng Liang Thesis: Mixing time and limit shapes of Abelian networks Advisor: Lionel Levine

David Mehrle Thesis: Commutative and Homological Algebra of Incomplete Tambara Functors Advisor: Inna Zakharevich First Position: Postdoctoral Scholar, University of Kentucky

Itamar Sales de Oliveira Thesis: A new approach to the Fourier extension problem for the paraboloid Advisor: Camil Muscalu First Position: Postdoctoral Researcher, Nantes Université

Brandon Shapiro Thesis: Shape Independent Category Theory Advisor: Inna Zakharevich First Position: Postdoctoral Fellow, Topos Institute

Ayah Almousa Thesis: Combinatorial characterizations of polarizations of powers of the graded maximal ideal Advisor: Irena Peeva First position: RTG Postdoctoral Fellow, University of Minnesota

Jose Bastidas Thesis: Species and hyperplane arrangements Advisor: Marcelo Aguiar First position: Postdoctoral Fellow, Université du Québec à Montréal

Zaoli Chen Thesis: Clustered Behaviors of Extreme Values Advisor: Gennady Samorodnitsky First Position: Postdoctoral Researcher, Department of and Statistics, University of Ottawa

Ivan Geffner Thesis: Implementing Mediators with Cheap Talk Advisor: Joe Halpern First Position: Postdoctoral Researcher, Technion – Israel Institute of Technology

Ryan McDermott Thesis: Phase Transitions and Near-Critical Phenomena in the Abelian Sandpile Model Advisor: Lionel Levine

Aleksandra Niepla Thesis:  Iterated Fractional Integrals and Applications to Fourier Integrals with Rational Symbol Advisor: Camil Muscalu First Position: Visiting Assistant Professor, College of the Holy Cross

Dylan Peifer Thesis: Reinforcement Learning in Buchberger's Algorithm Advisor: Michael Stillman First Position: Quantitative Researcher, Susquehanna International Group

Rakvi Thesis: A Classification of Genus 0 Modular Curves with Rational Points Advisor: David Zywina First Position: Hans Rademacher Instructor, University of Pennsylvania

Ana Smaranda Sandu Thesis: Knowledge of counterfactuals Advisor: Anil Nerode First Position: Instructor in Science Laboratory, Computer Science Department, Wellesley College

Maru Sarazola Thesis: Constructing K-theory spectra from algebraic structures with a class of acyclic objects Advisor: Inna Zakharevich First Position: J.J. Sylvester Assistant Professor, Johns Hopkins University

Abigail Turner Thesis: L2 Minimal Algorithms Advisor: Steven Strogatz

Yuwen Wang Thesis: Long-jump random walks on finite groups Advisor: Laurent Saloff-Coste First Position: Postdoc, University of Innsbruck, Austria

Beihui Yuan Thesis:  Applications of commutative algebra to spline theory and string theory Advisor: Michael Stillman First Position: Research Fellow, Swansea University

Elliot Cartee Thesis: Topics in Optimal Control and Game Theory Advisor: Alexander Vladimirsky First Position: L.E. Dickson Instructor, Department of , University of Chicago

Frederik de Keersmaeker Thesis: Displaceability in Symplectic Geometry Advisor: Tara Holm First Position: Consultant, Addestino Innovation Management

Lila Greco Thesis: Locally Markov Walks and Branching Processes Advisor: Lionel Levine First Position: Actuarial Assistant, Berkshire Hathaway Specialty Insurance

Benjamin Hoffman Thesis: Polytopes And Hamiltonian Geometry: Stacks, Toric Degenerations, And Partial Advisor: Reyer Sjamaar First Position: Teaching Associate, Department of , Cornell University

Daoji Huang Thesis: A Bruhat Atlas on the Wonderful Compactification of PS O(2 n )/ SO (2 n -1) and A Kazhdan-Lusztig Atlas on G/P Advisor: Allen Knutson First Position: Postdoctoral Associate, University of Minnesota

Pak-Hin Li Thesis: A Hopf Algebra from Preprojective Modules Advisor: Allen Knutson First position: Associate, Goldman Sachs

Anwesh Ray Thesis: Lifting Reducible Galois Representations Advisor: Ravi Ramakrishna First Position: Postdoctoral Fellowship, University of British Columbia

Avery St. Dizier Thesis: Combinatorics of Schubert Polynomials Advisor: Karola Meszaros First Position: Postdoctoral Fellowship, Department of , University of Illinois at Urbana-Champaign

Shihao Xiong Thesis: Forcing Axioms For Sigma-Closed Posets And Their Consequences Advisor: Justin Moore First Position: Algorithm Developer, Hudson River Trading

Swee Hong Chan Thesis: Nonhalting abelian networks Advisor: Lionel Levine First Position: Hedrick Adjunct Assistant Professor, UCLA

Joseph Gallagher Thesis: On conjectures related to character varieties of knots and Jones polynomials Advisor: Yuri Berest First Position: Data Scientist, Capital One

Jun Le Goh Thesis: Measuring the Relative Complexity of Mathematical Constructions and Theorems Advisor: Richard Shore First Position: Van Vleck Assistant Professor, University of Wisconsin-Madison

Qi Hou Thesis: Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces Advisor: Laurent Saloff-Coste First Position: Visiting Assistant Professor, Department of , Cornell University

Jingbo Liu Thesis: Heat kernel estimate of the Schrodinger operator in uniform domains Advisor: Laurent Saloff-Coste First Position: Data Scientist, Jet.com

Ian Pendleton Thesis:  The Fundamental Group, Homology, and Cohomology of Toric Origami 4-Manifolds Advisor: Tara Holm

Amin Saied Thesis: Stable representation theory of categories and applications to families of (bi)modules over symmetric groups Advisor: Martin Kassabov First Position: Data Scientist, Microsoft

Yujia Zhai Thesis:  Study of bi-parameter flag paraproducts and bi-parameter stopping-time algorithms Advisor: Camil Muscalu First Position: Postdoctoral Associate, Université de Nantes 

Tair Akhmejanov Thesis: Growth Diagrams from Polygons in the Affine Grassmannian Advisor: Allen Knutson First position: Arthur J. Krener Assistant Professor, University of California, Davis

James Barnes Thesis:  The Theory of the Hyperarithmetic Degrees Advisor: Richard Shore First position: Visiting Lecturer, Wellesley College

Jeffrey Bergfalk Thesis:  Dimensions of ordinals: set theory, homology theory, and the first omega alephs Advisor: Justin Moore Postdoctoral Associate, UNAM - National Autonomous University of Mexico

TaoRan Chen Thesis: The Inverse Deformation Problem Advisor: Ravi Ramakrishna

Sergio Da Silva Thesis: On the Gorensteinization of Schubert varieties via boundary divisors Advisor: Allen Knutson First position: Pacific Institute for the Mathematical Sciences (PIMS) postdoctoral fellowship, University of Manitoba

Eduard Einstein Thesis:  Hierarchies for relatively hyperbolic compact special cube complexes Advisor: Jason Manning First position: Research Assistant Professor (Postdoc), University of Illinois, Chicago (UIC)

Balázs Elek Thesis:  Toric surfaces with Kazhdan-Lusztig atlases Advisor: Allen Knutson First position: Postdoctoral Fellow, University of Toronto

Kelsey Houston-Edwards Thesis:  Discrete Heat Kernel Estimates in Inner Uniform Domains Advisor: Laurent Saloff-Coste First position: Professor of Math and Science Communication, Olin College

My Huynh Thesis:  The Gromov Width of Symplectic Cuts of Symplectic Manifolds. Advisor: Tara Holm First position: Applied Mathematician, Applied Research Associates Inc., Raleigh NC.

Hossein Lamei Ramandi Thesis: On the minimality of non-σ-scattered orders Advisor: Justin Moore First position:  Postdoctoral Associate at UFT (University Toronto)

Christine McMeekin Thesis: A density of ramified primes Advisor: Ravi Ramakrishna First position: Researcher at Max Planck Institute

Aliaksandr Patotski Thesis:  Derived characters of Lie representations and Chern-Simons forms Advisor: Yuri Berest First position: Data Scientist, Microsoft

Ahmad Rafiqi Thesis:  On dilatations of surface automorphisms Advisor: John Hubbard First position: Postdoctoral Associate, Sao Palo, Brazil

Ying-Ying Tran Thesis:  Computably enumerable boolean algebras Advisor: Anil Nerode First position: Quantitative Researcher

Drew Zemke Thesis:  Surfaces in Three- and Four-Dimensional Topology Advisor: Jason Manning First position: Preceptor in , Harvard University

Heung Shan Theodore Hui Thesis: A Radical Characterization of Abelian Varieties  Advisor: David Zywina First position: Quantitative Researcher, Eastmore Group

Daniel Miller Thesis: Counterexamples related to the Sato–Tate conjecture Advisor: Ravi Ramakrishna First position: Data Scientist, Microsoft

Lihai Qian Thesis: Rigidity on Einstein manifolds and shrinking Ricci solitons in high dimensions Advisor: Xiaodong Cao First position: Quantitative Associate, Wells Fargo

Valente Ramirez Garcia Luna Thesis: Quadratic vector fields on the complex plane: rigidity and analytic invariants Advisor: Yulij Ilyashenko First position: Lebesgue Post-doc Fellow, Institut de Recherche Mathématique de Rennes

Iian Smythe Thesis: Set theory in infinite-dimensional vector spaces Advisor: Justin Moore First position: Hill Assistant Professor at Rutgers, the State University of New Jersey

Zhexiu Tu Thesis: Topological representations of matroids and the cd-index Advisor: Edward Swartz First position: Visiting Professor - Centre College, Kentucky

Wai-kit Yeung Thesis: Representation homology and knot contact homology Advisor: Yuri Berest First position: Zorn postdoctoral fellow, Indiana University

Lucien Clavier Thesis: Non-affine horocycle orbit closures on strata of translation surfaces: new examples Advisor: John Smillie First position: Consultant in Capital Markets, Financial Risk at Deloitte Luxembourg

Voula Collins Thesis: Crystal branching for non-Levi subgroups and a puzzle formula for the equivariant cohomology of the cotangent bundle on projective space Advisor: Allen Knutson FIrst position: Postdoctoral Associate, University of Connecticut

Pok Wai Fong Thesis: Smoothness Properties of symbols, Calderón Commutators and Generalizations Advisor: Camil Muscalu First position: Quantitative researcher, Two Sigma

Tom Kern Thesis: Nonstandard models of the weak second order theory of one successor Advisor: Anil Nerode First position: Visiting Assistant Professor, Cornell University

Robert Kesler Thesis: Unbounded multilinear multipliers adapted to large subspaces and estimates for degenerate simplex operators Advisor: Camil Muscalu First position: Postdoctoral Associate, Georgia Institute of Technology

Yao Liu Thesis: Riesz Distributions Assiciated to Dunkl Operators Advisor: Yuri Berest First position: Visiting Assistant Professor, Cornell University

Scott Messick Thesis: Continuous atomata, compactness, and Young measures Advisor: Anil Nerode First position: Start-up

Aaron Palmer Thesis: Incompressibility and Global Injectivity in Second-Gradient Non-Linear Elasticity Advisor: Timothy J. Healey First position: Postdoctoral fellow, University of British Columbia 

Kristen Pueschel Thesis: On Residual Properties of Groups and Dehn Functions for Mapping Tori of Right Angled Artin Groups Advisor: Timothy Riley First position: Postdoctoral Associate, University of Arkansas

Chenxi Wu Thesis: Translation surfaces: saddle connections, triangles, and covering constructions. Advisor: John Smillie First position: Postdoctoral Associate, Max Planck Institute of

David Belanger Thesis: Sets, Models, And Proofs: Topics In The Theory Of Recursive Functions Advisor: Richard A. Shore First position: Research Fellow, National University of Singapore

Cristina Benea Thesis: Vector-Valued Extensions for Singular Bilinear Operators and Applications Advisor: Camil Muscalu First position: University of Nantes, France

Kai Fong Ernest Chong Thesis: Face Vectors and Hilbert Functions Advisor: Edward Swartz First position: Research Scientist, Agency for Science, Technology and Research, Singapore

Laura Escobar Vega Thesis: Brick Varieties and Toric Matrix Schubert Varieties Advisor: Allen Knutson First position: J. L. Doob Research Assistant Professor at UIUC

Joeun Jung Thesis: Iterated trilinear fourier integrals with arbitrary symbols Advisor: Camil Muscalu First position: Researcher, PARC (PDE and Functional Analysis Research Center) of Seoul National University

Yasemin Kara Thesis: The laplacian on hyperbolic Riemann surfaces and Maass forms Advisor: John H. Hubbard Part Time Instructor, Faculty of Engineering and Natural Sciences, Bahcesehir University

Chor Hang Lam Thesis: Homological Stability Of Diffeomorphism Groups Of 3-Manifolds Advisor: Allen Hatcher

Yash Lodha Thesis: Finiteness Properties And Piecewise Projective Homeomorphisms Advisor: Justin Moore and Timothy Riley First position: Postdoctoral fellow at Ecole Polytechnique Federale de Lausanne in Switzerland

Radoslav Zlatev Thesis: Examples of Implicitization of Hypersurfaces through Syzygies Advisor: Michael E. Stillman First position: Associate, Credit Strats, Goldman Sachs

Margarita Amchislavska Thesis: The geometry of generalized Lamplighter groups Advisor: Timothy Riley First position: Department of Defense

Hyungryul Baik Thesis: Laminations on the circle and hyperbolic geometry Advisor: John H. Hubbard First position: Postdoctoral Associate, Bonn University

Adam Bjorndahl Thesis: Language-based games Advisor: Anil Nerode and Joseph Halpern First position: Tenure Track Professor, Carnegie Mellon University Department of Philosophy

Youssef El Fassy Fihry Thesis: Graded Cherednik Algebra And Quasi-Invariant Differential Forms Advisor: Yuri Berest First position: Software Developer, Microsoft

Chikwong Fok Thesis: The Real K-theory of compact Lie groups Advisor: Reyer Sjamaar First position: Postdoctoral fellow in the National Center for Theoretical Sciences, Taiwan

Kathryn Lindsey Thesis: Families Of Dynamical Systems Associated To Translation Surfaces Advisor: John Smillie First position: Postdoctoral Associate, University of Chicago

Andrew Marshall Thesis: On configuration spaces of graphs Advisor: Allan Hatcher First position: Visiting Assistant Professor, Cornell University

Robyn Miller Thesis: Symbolic Dynamics Of Billiard Flow In Isosceles Triangles Advisor: John Smillie First position: Postdoctoral Researcher at Mind Research Network

Diana Ojeda Aristizabal Thesis: Ramsey theory and the geometry of Banach spaces Advisor: Justin Moore First position: Postdoctoral Fellow, University of Toronto

Hung Tran Thesis: Aspects of the Ricci flow Advisor: Xiaodong Cao First position: Visiting Assistant Professor, University of California at Irvine

Baris Ugurcan Thesis: LPLP-Estimates And Polyharmonic Boundary Value Problems On The Sierpinski Gasket And Gaussian Free Fields On High Dimensional Sierpinski Carpet Graphs Advisor: Robert S. Strichartz First position: Postdoctoral Fellowship, University of Western Ontario

Anna Bertiger Thesis: The Combinatorics and Geometry of the Orbits of the Symplectic Group on Flags in Complex Affine Space Advisor: Allen Knutson First position: University of Waterloo, Postdoctoral Fellow

Mariya Bessonov Thesis: Probabilistic Models for Population Dynamics Advisor: Richard Durrett First position: CUNY City Tech, Assistant Professor, Tenure Track

Igors Gorbovickis Thesis: Some Problems from Complex Dynamical Systems and Combinatorial Geometry Advisor: Yulij Ilyashenko First position: Postdoctoral Fellow, University of Toronto

Marisa Hughes Thesis: Quotients of Spheres by Linear Actions of Abelian Groups Advisor: Edward Swartz First position: Visiting Professor, Hamilton College

Kristine Jones Thesis: Generic Initial Ideals of Locally Cohen-Macaulay Space Curves Advisor: Michael E. Stillman First position: Software Developer, Microsoft

Shisen Luo Thesis: Hard Lefschetz Property of Hamiltonian GKM Manifolds Advisor: Tara Holm First position: Associate, Goldman Sachs

Peter Luthy Thesis: Bi-parameter Maximal Multilinear Operators Advisor: Camil Muscalu First position: Chauvenet Postdoctoral Lecturer at Washington University in St. Louis 

Remus Radu Thesis: Topological models for hyperbolic and semi-parabolic complex Hénon maps Advisor: John H. Hubbard First position: Milnor Lecturer, Institute for Mathematical Sciences, Stony Brook University

Jenna Rajchgot Thesis: Compatibly Split Subvarieties of the Hilbert Scheme of Points in the Plane Advisor: Allen Knutson First position: Research member at the Mathematical Sciences Research Institute (fall 2012); postdoc at the University of Michigan

Raluca Tanase Thesis: Hénon maps, discrete groups and continuity of Julia sets Advisor: John H. Hubbard First position: Milnor Lecturer, Institute for Mathematical Sciences, Stony Brook University

Ka Yue Wong Thesis: Dixmier Algebras on Complex Classical Nilpotent Orbits and their Representation Theories Advisor: Dan M. Barbasch First position: Postdoctoral fellow at Hong Kong University of Science and Technology

Tianyi Zheng Thesis: Random walks on some classes of solvable groups Advisor: Laurent Saloff-Coste First position: Postdoctoral Associate, Stanford University

Juan Alonso Thesis: Graphs of Free Groups and their Measure Equivalence Advisor: Karen Vogtmann First position: Postdoc at Uruguay University

Jason Anema Thesis: Counting Spanning Trees on Fractal Graphs Advisor: Robert S. Strichartz First position: Visiting assistant professor of mathematics at Cornell University

Saúl Blanco Rodríguez Thesis: Shortest Path Poset of Bruhat Intervals and the Completecd-Index Advisor: Louis Billera First position: Visiting assistant professor of mathematics at DePaul University

Fatima Mahmood Thesis: Jacobi Structures and Differential Forms on Contact Quotients Advisor: Reyer Sjamaar First position: Visiting assistant professor at University of Rochester

Philipp Meerkamp Thesis: Singular Hopf Bifurcation Advisor: John M. Guckenheimer First position: Financial software engineer at Bloomberg LP

Milena Pabiniak Thesis: Hamiltonian Torus Actions in Equivariant Cohomology and Symplectic Topology Advisor: Tara Holm First position: Postdoctoral associate at the University of Toronto

Peter Samuelson Thesis: Kauffman Bracket Skein Modules and the Quantum Torus Advisor: Yuri Berest First position: Postdoctoral associate at the University of Toronto

Mihai Bailesteanu  Thesis: The Heat Equation under the Ricci Flow Advisor: Xiaodong Cao First position: Visiting assistant professor at the University of Rochester

Owen Baker  Thesis:  The Jacobian Map on Outer Space Advisor: Karen Vogtmann First position: Postdoctoral fellow at McMaster University

Jennifer Biermann  Thesis: Free Resolutions of Monomial Ideals Advisor: Irena Peeva First position: Postdoctoral fellow at Lakehead University

Mingzhong Cai  Thesis: Elements of Classical Recursion Theory: Degree-Theoretic Properties and Combinatorial Properties Advisor: Richard A. Shore First position: Van Vleck visiting assistant professor at the University of Wisconsin at Madison

Ri-Xiang Chen  Thesis: Hilbert Functions and Free Resolutions Advisor: Irena Peeva First position: Instructor at Shantou University in Guangdong, China

Denise Dawson  Thesis: Complete Reducibility in Euclidean Twin Buildings Advisor: Kenneth S. Brown First position: Assistant professor of mathematics at Charleston Southern University

George Khachatryan Thesis: Derived Representation Schemes and Non-commutative Geometry Advisor: Yuri Berest First position: Reasoning Mind

Samuel Kolins  Thesis: Face Vectors of Subdivision of Balls Advisor: Edward Swartz First position: Assistant professor at Lebanon Valley College

Victor Kostyuk Thesis: Outer Space for Two-Dimensional RAAGs and Fixed Point Sets of Finite Subgroups Advisor: Karen Vogtmann First position: Knowledge engineering at Reasoning Mind

Ho Hon Leung  Thesis: K-Theory of Weight Varieties and Divided Difference Operators in Equivariant KK-Theory Advisor: Reyer Sjamaar First position: Assistant professor at the Canadian University of Dubai

Benjamin Lundell  Thesis: Selmer Groups and Ranks of Hecke Rings Advisor: Ravi Ramakrishna First position: Acting assistant professor at the University of Washington

Eyvindur Ari Palsson  Thesis: Lp Estimates for a Singular Integral Operator Motivated by Calderón’s Second Commutator Advisor: Camil Muscalu First position: Visiting assistant professor at the University of Rochester

Paul Shafer  Thesis: On the Complexity of Mathematical Problems: Medvedev Degrees and Reverse Advisor: Richard A. Shore First position: Lecturer at Appalachian State University

Michelle Snider  Thesis: Affine Patches on Positroid Varieties and Affine Pipe Dreams Advisor: Allen Knutson First position: Government consulting job in Maryland

Santi Tasena Thesis: Heat Kernel Analysis on Weighted Dirichlet Spaces Advisor: Laurent Saloff-Coste First position: Lecturer professor at Chiang Mai University, Thailand

Russ Thompson  Thesis: Random Walks and Subgroup Geometry Advisor: Laurent Saloff-Coste First position: Postdoctoral fellow at the Mathematical Sciences Research Institute

Gwyneth Whieldon Thesis: Betti Numbers of Stanley-Reisner Ideals Advisor: Michael E. Stillman First position: Assistant professor of mathematics at Hood College

Andrew Cameron Thesis: Estimates for Solutions of Elliptic Partial Differential Equations with Explicit Constants and Aspects of the Finite Element Method for Second-Order Equations Advisor: Alfred H. Schatz First position: Adjunct instructor of mathematics at Tompkins Cortland Community College

Timothy Goldberg Thesis: Hamiltonian Actions in Integral Kähler and Generalized Complex Geometry Advisor: Reyer Sjamaar First position: Visiting assistant professor of mathematics at Lenoir-Rhyne University

Gregory Muller Thesis: The Projective Geometry of Differential Operators Advisor: Yuri Berest First position: Assistant professor at Louisiana State University 

Matthew Noonan Thesis: Geometric Backlund transofrmation in homogeneous spaces Advisor: John H. Hubbard

Sergio Pulido Niño Thesis: Financial Markets with Short Sales Prohibition Advisor: Philip E. Protter First position: Postdoctoral associate in applied probability and finance at Carnegie Mellon University

Home > FACULTIES > Applied Mathematics > APMATHS-ETD

Applied Mathematics Theses and Dissertations

This collection contains theses and dissertations from the Department of Applied Mathematics, collected from the Scholarship@Western Electronic Thesis and Dissertation Repository

Theses/Dissertations from 2023 2023

Visual Cortical Traveling Waves: From Spontaneous Spiking Populations to Stimulus-Evoked Models of Short-Term Prediction , Gabriel B. Benigno

Spike-Time Neural Codes and their Implication for Memory , Alexandra Busch

Study of Behaviour Change and Impact on Infectious Disease Dynamics by Mathematical Models , Tianyu Cheng

Series Expansions of Lambert W and Related Functions , Jacob Imre

Data-Driven Exploration of Coarse-Grained Equations: Harnessing Machine Learning , Elham Kianiharchegani

Pythagorean Vectors and Rational Orthonormal Matrices , Aishat Olagunju

The Magnetic Field of Protostar-Disk-Outflow Systems , Mahmoud Sharkawi

A Highly Charged Topic: Intrinsically Disordered Proteins and Protein pKa Values , Carter J. Wilson

Population Dynamics and Bifurcations in Predator-Prey Systems with Allee Effect , Yanni Zeng

Theses/Dissertations from 2022 2022

A Molecular Dynamics Study Of Polymer Chains In Shear Flows and Nanocomposites , Venkat Bala

On the Spatial Modelling of Biological Invasions , Tedi Ramaj

Complete Hopf and Bogdanov-Takens Bifurcation Analysis on Two Epidemic Models , Yuzhu Ruan

A Theoretical Perspective on Parasite-Host Coevolution with Alternative Modes of Infection , George N. Shillcock

Theses/Dissertations from 2021 2021

Mathematical Modelling & Simulation of Large and Small Scale Structures in Star Formation , Gianfranco Bino

Mathematical Modelling of Ecological Systems in Patchy Environments , Ao Li

Credit Risk Measurement and Application based on BP Neural Networks , Jingshi Luo

Coevolution of Hosts and Pathogens in the Presence of Multiple Types of Hosts , Evan J. Mitchell

SymPhas: A modular API for phase-field modeling using compile-time symbolic algebra , Steven A. Silber

Population and Evolution Dynamics in Predator-prey Systems with Anti-predation Responses , Yang Wang

Theses/Dissertations from 2020 2020

The journey of a single polymer chain to a nanopore , Navid Afrasiabian

Exploration Of Stock Price Predictability In HFT With An Application In Spoofing Detection , Andrew Day

Multi-Scale Evolution of Virulence of HIV-1 , David W. Dick

Contraction Analysis of Functional Competitive Lotka-Volterra Systems: Understanding Competition Between Modified Bacteria and Plasmodium within Mosquitoes. , Nickolas Goncharenko

Phage-Bacteria Interaction and Prophage Sequences in Bacterial Genomes , Amjad Khan

The Effect of the Initial Structure on the System Relaxation Time in Langevin Dynamics , Omid Mozafar

Mathematical modelling of prophage dynamics , Tyler Pattenden

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra , Leili Rafiee Sevyeri

Abelian Integral Method and its Application , Xianbo Sun

Theses/Dissertations from 2019 2019

Algebraic Companions and Linearizations , Eunice Y. S. Chan

Algorithms for Mappings and Symmetries of Differential Equations , Zahra Mohammadi

Algorithms for Bohemian Matrices , Steven E. Thornton

A Survey Of Numerical Quadrature Methods For Highly Oscillatory Integrals , Jeet Trivedi

Theses/Dissertations from 2018 2018

Properties and Computation of the Inverse of the Gamma function , Folitse Komla Amenyou

Optimization Studies and Applications: in Retail Gasoline Market , Daero Kim

Models of conflict and voluntary cooperation between individuals in non-egalitarian social groups , Cody Koykka

Investigation of chaos in biological systems , Navaneeth Mohan

Bifurcation Analysis of Two Biological Systems: A Tritrophic Food Chain Model and An Oscillating Networks Model , Xiangyu Wang

Ecology and Evolution of Dispersal in Metapopulations , Jingjing Xu

Selected Topics in Quantization and Renormalization of Gauge Fields , Chenguang Zhao

Three Essays on Structural Models , Xinghua Zhou

Theses/Dissertations from 2017 2017

On Honey Bee Colony Dynamics and Disease Transmission , Matthew I. Betti

Simulation of driven elastic spheres in a Newtonian fluid , Shikhar M. Dwivedi

Feasible Computation in Symbolic and Numeric Integration , Robert H.C. Moir

Modelling Walleye Population and Its Cannibalism Effect , Quan Zhou

Theses/Dissertations from 2016 2016

Dynamics of Discs in a Nematic Liquid Crystal , Alena Antipova

Modelling the Impact of Climate Change on the Polar Bear Population in Western Hudson Bay , Nicole Bastow

A comparison of solution methods for Mandelbrot-like polynomials , Eunice Y. S. Chan

A model-based test of the efficacy of a simple rule for predicting adaptive sex allocation , Joshua D. Dunn

Universal Scaling Properties After Quantum Quenches , Damian Andres Galante

Modeling the Mass Function of Stellar Clusters Using the Modified Lognormal Power-Law Probability Distribution Function , Deepakshi Madaan

Bacteria-Phage Models with a Focus on Prophage as a Genetic Reservoir , Alina Nadeem

A Sequence of Symmetric Bézout Matrix Polynomials , Leili Rafiee Sevyeri

Study of Infectious Diseases by Mathematical Models: Predictions and Controls , SM Ashrafur Rahman

The survival probability of beneficial de novo mutations in budding viruses, with an emphasis on influenza A viral dynamics , Jennifer NS Reid

Essays in Market Structure and Liquidity , Adrian J. Walton

Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods , Fei Wang

Studying Both Direct and Indirect Effects in Predator-Prey Interaction , Xiaoying Wang

Theses/Dissertations from 2015 2015

The Effect of Diversification on the Dynamics of Mobile Genetic Elements in Prokaryotes: The Birth-Death-Diversification Model , Nicole E. Drakos

Algorithms to Compute Characteristic Classes , Martin Helmer

Studies of Contingent Capital Bonds , Jingya Li

Determination of Lie superalgebras of supersymmetries of super differential equations , Xuan Liu

Edge states and quantum Hall phases in graphene , Pavlo Piatkovskyi

Evolution of Mobile Promoters in Prokaryotic Genomes. , Mahnaz Rabbani

Extensions of the Cross-Entropy Method with Applications to Diffusion Processes and Portfolio Losses , Alexandre Scott

Theses/Dissertations from 2014 2014

A Molecular Simulation Study on Micelle Fragmentation and Wetting in Nano-Confined Channels , Mona Habibi

Study of Virus Dynamics by Mathematical Models , Xiulan Lai

Applications of Stochastic Control in Energy Real Options and Market Illiquidity , Christian Maxwell

Options Pricing and Hedging in a Regime-Switching Volatility Model , Melissa A. Mielkie

Optimal Contract Design for Co-development of Companion Diagnostics , Rodney T. Tembo

Bifurcation of Limit Cycles in Smooth and Non-smooth Dynamical Systems with Normal Form Computation , Yun Tian

Understanding Recurrent Disease: A Dynamical Systems Approach , Wenjing Zhang

Theses/Dissertations from 2013 2013

Pricing and Hedging Index Options with a Dominant Constituent Stock , Helen Cheyne

On evolution dynamics and strategies in some host-parasite models , Liman Dai

Valuation of the Peterborough Prison Social Impact Bond , Majid Hasan

Sensitivity Analysis of Minimum Variance Portfolios , Xiaohu Ji

Eigenvalue Methods for Interpolation Bases , Piers W. Lawrence

Hybrid Lattice Boltzmann - Molecular Dynamics Simulations With Both Simple and Complex Fluids , Frances E. Mackay

Ecological Constraints and the Evolution of Cooperative Breeding , David McLeod

A single cell based model for cell divisions with spontaneous topology changes , Anna Mkrtchyan

Analysis of Re-advanceable Mortgages , Almas Naseem

Modeling leafhopper populations and their role in transmitting plant diseases. , Ji Ruan

Topological properties of modular networks, with a focus on networks of functional connections in the human brain , Estefania Ruiz Vargas

Computation Sequences for Series and Polynomials , Yiming Zhang

Theses/Dissertations from 2012 2012

A Real Options Valuation of Renewable Energy Projects , Natasha Burke

Approximate methods for dynamic portfolio allocation under transaction costs , Nabeel Butt

Optimal clustering techniques for metagenomic sequencing data , Erik T. Cameron

Phase Field Crystal Approach to the Solidification of Ferromagnetic Materials , Niloufar Faghihi

Molecular Dynamics Simulations of Peptide-Mineral Interactions , Susanna Hug

Molecular Dynamics Studies of Water Flow in Carbon Nanotubes , Alexander D. Marshall

Valuation of Multiple Exercise Options , T. James Marshall

Incomplete Market Models of Carbon Emissions Markets , Walid Mnif

Topics in Field Theory , Alexander Patrushev

Pricing and Trading American Put Options under Sub-Optimal Exercise Policies , William Wei Xing

Further applications of higher-order Markov chains and developments in regime-switching models , Xiaojing Xi

Theses/Dissertations from 2011 2011

Bifurcations and Stability in Models of Infectious Diseases , Bernard S. Chan

Real Options Models in Real Estate , Jin Won Choi

Models, Techniques, and Metrics for Managing Risk in Software Engineering , Andriy Miranskyy

Thermodynamics, Hydrodynamics and Critical Phenomena in Strongly Coupled Gauge Theories , Christopher Pagnutti

Molecular Dynamics Studies of Interactions of Phospholipid Membranes with Dehydroergosterol and Penetrating Peptides , Amir Mohsen Pourmousa Abkenar

Socially Responsible Investment in a Changing World , Desheng Wu

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Home > Physical and Mathematical Sciences > Mathematics Education > Theses and Dissertations

Mathematics Education Theses and Dissertations

Theses/dissertations from 2024 2024.

New Mathematics Teachers' Goals, Orientations, and Resources that Influence Implementation of Principles Learned in Brigham Young University's Teacher Preparation Program , Caroline S. Gneiting

Theses/Dissertations from 2023 2023

Impact of Applying Visual Design Principles to Boardwork in a Mathematics Classroom , Jennifer Rose Canizales

Practicing Mathematics Teachers' Perspectives of Public Records in Their Classrooms , Sini Nicole White Graff

Parents' Perceptions of the Importance of Teaching Mathematics: A Q-Study , Ashlynn M. Holley

Engagement in Secondary Mathematics Group Work: A Student Perspective , Rachel H. Jorgenson

Theses/Dissertations from 2022 2022

Understanding College Students' Use of Written Feedback in Mathematics , Erin Loraine Carroll

Identity Work to Teach Mathematics for Social Justice , Navy B. Dixon

Developing a Quantitative Understanding of U-Substitution in First-Semester Calculus , Leilani Camille Heaton Fonbuena

The Perception of At-Risk Students on Caring Student-Teacher Relationships and Its Impact on Their Productive Disposition , Brittany Hopper

Variational and Covariational Reasoning of Students with Disabilities , Lauren Rigby

Structural Reasoning with Rational Expressions , Dana Steinhorst

Student-Created Learning Objects for Mathematics Renewable Assignments: The Potential Value They Bring to the Broader Community , Webster Wong

Theses/Dissertations from 2021 2021

Emotional Geographies of Beginning and Veteran Reformed Teachers in Mentor/Mentee Relationships , Emily Joan Adams

You Do Math Like a Girl: How Women Reason Mathematically Outside of Formal and School Mathematics Contexts , Katelyn C. Pyfer

Developing the Definite Integral and Accumulation Function Through Adding Up Pieces: A Hypothetical Learning Trajectory , Brinley Nichole Stevens

Theses/Dissertations from 2020 2020

Mathematical Identities of Students with Mathematics Learning Dis/abilities , Emma Lynn Holdaway

Teachers' Mathematical Meanings: Decisions for Teaching Geometric Reflections and Orientation of Figures , Porter Peterson Nielsen

Student Use of Mathematical Content Knowledge During Proof Production , Chelsey Lynn Van de Merwe

Theses/Dissertations from 2019 2019

Making Sense of the Equal Sign in Middle School Mathematics , Chelsea Lynn Dickson

Developing Understanding of the Chain Rule, Implicit Differentiation, and Related Rates: Towards a Hypothetical Learning Trajectory Rooted in Nested Multivariation , Haley Paige Jeppson

Secondary Preservice Mathematics Teachers' Curricular Reasoning , Kimber Anne Mathis

“Don’t Say Gay. We Say Dumb or Stupid”: Queering ProspectiveMathematics Teachers’ Discussions , Amy Saunders Ross

Aspects of Engaging Problem Contexts From Students' Perspectives , Tamara Kay Stark

Theses/Dissertations from 2018 2018

Addressing Pre-Service Teachers' Misconceptions About Confidence Intervals , Kiya Lynn Eliason

How Teacher Questions Affect the Development of a Potential Hybrid Space in a Classroom with Latina/o Students , Casandra Helen Job

Teacher Graphing Practices for Linear Functions in a Covariation-Based College Algebra Classroom , Konda Jo Luckau

Principles of Productivity Revealed from Secondary Mathematics Teachers' Discussions Around the Productiveness of Teacher Moves in Response to Teachable Moments , Kylie Victoria Palsky

Theses/Dissertations from 2017 2017

Curriculum Decisions and Reasoning of Middle School Teachers , Anand Mikel Bernard

Teacher Response to Instances of Student Thinking During Whole Class Discussion , Rachel Marie Bernard

Kyozaikenkyu: An In-Depth Look into Japanese Educators' Daily Planning Practices , Matthew David Melville

Analysis of Differential Equations Applications from the Coordination Class Perspective , Omar Antonio Naranjo Mayorga

Theses/Dissertations from 2016 2016

The Principles of Effective Teaching Student Teachershave the Opportunity to Learn in an AlternativeStudent Teaching Structure , Danielle Rose Divis

Insight into Student Conceptions of Proof , Steven Daniel Lauzon

Theses/Dissertations from 2015 2015

Teacher Participation and Motivation inProfessional Development , Krystal A. Hill

Student Evaluation of Mathematical Explanations in anInquiry-Based Mathematics Classroom , Ashley Burgess Hulet

English Learners' Participation in Mathematical Discourse , Lindsay Marie Merrill

Mathematical Interactions between Teachers and Students in the Finnish Mathematics Classroom , Paula Jeffery Prestwich

Parents and the Common Core State Standards for Mathematics , Rebecca Anne Roberts

Examining the Effects of College Algebra on Students' Mathematical Dispositions , Kevin Lee Watson

Problems Faced by Reform Oriented Novice Mathematics Teachers Utilizing a Traditional Curriculum , Tyler Joseph Winiecke

Academic and Peer Status in the Mathematical Life Stories of Students , Carol Ann Wise

Theses/Dissertations from 2014 2014

The Effect of Students' Mathematical Beliefs on Knowledge Transfer , Kristen Adams

Language Use in Mathematics Textbooks Written in English and Spanish , Kailie Ann Bertoch

Teachers' Curricular Reasoning and MKT in the Context of Algebra and Statistics , Kolby J. Gadd

Mathematical Telling in the Context of Teacher Interventions with Collaborative Groups , Brandon Kyle Singleton

An Investigation of How Preservice Teachers Design Mathematical Tasks , Elizabeth Karen Zwahlen

Theses/Dissertations from 2013 2013

Student Understanding of Limit and Continuity at a Point: A Look into Four Potentially Problematic Conceptions , Miriam Lynne Amatangelo

Exploring the Mathematical Knowledge for Teaching of Japanese Teachers , Ratu Jared R. T. Bukarau

Comparing Two Different Student Teaching Structures by Analyzing Conversations Between Student Teachers and Their Cooperating Teachers , Niccole Suzette Franc

Professional Development as a Community of Practice and Its Associated Influence on the Induction of a Beginning Mathematics Teacher , Savannah O. Steele

Types of Questions that Comprise a Teacher's Questioning Discourse in a Conceptually-Oriented Classroom , Keilani Stolk

Theses/Dissertations from 2012 2012

Student Teachers' Interactive Decisions with Respect to Student Mathematics Thinking , Jonathan J. Call

Manipulatives and the Growth of Mathematical Understanding , Stacie Joyce Gibbons

Learning Within a Computer-Assisted Instructional Environment: Effects on Multiplication Math Fact Mastery and Self-Efficacy in Elementary-Age Students , Loraine Jones Hanson

Mathematics Teacher Time Allocation , Ashley Martin Jones

Theses/Dissertations from 2011 2011

How Student Positioning Can Lead to Failure in Inquiry-based Classrooms , Kelly Beatrice Campbell

Teachers' Decisions to Use Student Input During Class Discussion , Heather Taylor Toponce

A Conceptual Framework for Student Understanding of Logarithms , Heather Rebecca Ambler Williams

Theses/Dissertations from 2010 2010

Growth in Students' Conceptions of Mathematical Induction , John David Gruver

Contextualized Motivation Theory (CMT): Intellectual Passion, Mathematical Need, Social Responsibility, and Personal Agency in Learning Mathematics , Janelle Marie Hart

Thinking on the Brink: Facilitating Student Teachers' Learning Through In-the-Moment Interjections , Travis L. Lemon

Understanding Teachers' Change Towards a Reform-Oriented Mathematics Classroom , Linnae Denise Williams

Theses/Dissertations from 2009 2009

A Comparison of Mathematical Discourse in Online and Face-to-Face Environments , Shawn D. Broderick

The Influence of Risk Taking on Student Creation of Mathematical Meaning: Contextual Risk Theory , Erin Nicole Houghtaling

Uncovering Transformative Experiences: A Case Study of the Transformations Made by one Teacher in a Mathematics Professional Development Program , Rachelle Myler Orsak

Theses/Dissertations from 2008 2008

Student Teacher Knowledge and Its Impact on Task Design , Tenille Cannon

How Eighth-Grade Students Estimate with Fractions , Audrey Linford Hanks

Similar but Different: The Complexities of Students' Mathematical Identities , Diane Skillicorn Hill

Choose Your Words: Refining What Counts as Mathematical Discourse in Students' Negotiation of Meaning for Rate of Change of Volume , Christine Johnson

Mathematics Student Teaching in Japan: A Multi-Case Study , Allison Turley Shwalb

Theses/Dissertations from 2007 2007

Applying Toulmin's Argumentation Framework to Explanations in a Reform Oriented Mathematics Class , Jennifer Alder Brinkerhoff

What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof? , Karen Malina Duff

Probing for Reasons: Presentations, Questions, Phases , Kellyn Nicole Farlow

One Problem, Two Contexts , Danielle L. Gigger

The Main Challenges that a Teacher-in-Transition Faces When Teaching a High School Geometry Class , Greg Brough Henry

Discovering the Derivative Can Be "Invigorating:" Mark's Journey to Understanding Instantaneous Velocity , Charity Ann Gardner Hyer

Theses/Dissertations from 2006 2006

How a Master Teacher Uses Questioning Within a Mathematical Discourse Community , Omel Angel Contreras

Determining High School Geometry Students' Geometric Understanding Using van Hiele Levels: Is There a Difference Between Standards-based Curriculum Students and NonStandards-based Curriculum Students? , Rebekah Loraine Genz

The Nature and Frequency of Mathematical Discussion During Lesson Study That Implemented the CMI Framework , Andrew Ray Glaze

Second Graders' Solution Strategies and Understanding of a Combination Problem , Tiffany Marie Hessing

What Does It Mean To Preservice Mathematics Teachers To Anticipate Student Responses? , Matthew M. Webb

Theses/Dissertations from 2005 2005

Fraction Multiplication and Division Image Change in Pre-Service Elementary Teachers , Jennifer J. Cluff

An Examination of the Role of Writing in Mathematics Instruction , Amy Jeppsen

Theses/Dissertations from 2004 2004

Reasoning About Motion: A Case Study , Tiffini Lynn Glaze

Theses/Dissertations from 2003 2003

An Analysis of the Influence of Lesson Study on Preservice Secondary Mathematics Teachers' View of Self-As Mathematics Expert , Julie Stafford

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Mathematics thesis and dissertation collection

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This collection contains a selection of the latest doctoral theses completed at the School of Mathematics. Please note this is not a comprehensive record.

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Georgia Institute of Technology College of Sciences

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Dissertations

Here is the complete list of all doctoral dissertations granted by the School of Math, which dates back to 1965. Included below are also all masters theses produced by our students since 2002. A combined listing of all dissertations and theses , going back to 1934, is available at Georgia Tech's library archive. For the post PhD employment of our graduates see our  Alumni Page .

Doctoral Dissertations

Masters dissertations.

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Mathematics MSc dissertations

The Department of Mathematics and Statistics was host until 2014 to the MSc course in the Mathematics of Scientific and Industrial Computation (previously known as Numerical Solution of Differential Equations) and the MSc course in Mathematical and Numerical Modelling of the Atmosphere and Oceans. A selection of dissertation titles are listed below, some of which are available online:

2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

2014: Mathematics of Scientific and Industrial Computation

Amanda Hynes - Slow and superfast diffusion of contaminant species through porous media

2014: Applicable and Numerical Mathematics

Emine Akkus - Estimating forecast error covariance matrices with ensembles

Rabindra Gurung - Numerical solution of an ODE system arising in photosynthesis

2013: Mathematics of Scientific and Industrial Computation

Zeinab Zargar - Modelling of Hot Water Flooding as an Enhanced Oil Recovery Method

Siti Mazulianawati Haji Majid - Numerical Approximation of Similarity in Nonlinear Diffusion Equations

2013: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Yu Chau Lam - Drag and Momentum Fluxes Produced by Mountain Waves

Josie Dodd - A Moving Mesh Approach to Modelling the Grounding Line in Glaciology

2012: Mathematics of Scientific and Industrial Computation

Chris Louder - Mathematical Techniques of Image Processing

Jonathan Muir - Flux Modelling of Polynyas

Naomi Withey - Computer Simulations of Dipolar Fluids Using Ewald Summations

2012: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Jean-Francois Vuillaume - Numerical prediction of flood plains using a Lagrangian approach

2011: Mathematics of Scientific and Industrial Computation

Tudor Ciochina - The Closest Point Method

Theodora Eleftheriou - Moving Mesh Methods Using Monitor Functions for the Porous Medium Equation

Melios Michael - Self-Consistent Field Calculations on a Variable Resolution Grid

2011: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Peter Barnet - Rain Drop Growth by Collision and Coalescence

Matthew Edgington - Moving Mesh Methods for Semi-Linear Problems

Samuel Groth - Light Scattering by Penetrable Convex Polygons

Charlotte Kong - Comparison of Approximate Riemann Solvers

Amy Jackson - Estimation of Parameters in Traffic Flow Models Using Data Assimilation

Bruce Main - Solving Richards' Equation Using Fixed and Moving Mesh Schemes

Justin Prince - Fast Diffusion in Porous Media

Carl Svoboda - Reynolds Averaged Radiative Transfer Model

2010: Mathematics of Scientific and Industrial Computation

Tahnia Appasawmy - Wave Reflection and Trapping in a Two Dimensional Duct

Nicholas Bird - Univariate Aspects of Covariance Modelling within Operational Atmospheric Data Assimilation

Michael Conland - Numerical Approximation of a Quenching Problem

Katy Shearer - Mathematical Modelling of the regulation and uptake of dietary fats

Peter Westwood - A Moving Mesh Finite Element Approach for the Cahn-Hilliard Equation

Kam Wong - Accuracy of a Moving Mesh Numerical Method applied to the Self-similar Solution of Nonlinear PDEs

2010: Mathematical and Numerical Modelling of the Atmosphere and Oceans

James Barlow - Computation and Analysis of Baroclinic Rossby Wave Rays in the Atlantic and Pacific Oceans

Martin Conway - Heat Transfer in a Buried Pipe

Simon Driscoll - The Earth's Atmospheric Angular Momentum Budget and its Representation in Reanalysis Observation Datasets and Climate Models

George Fitton - A Comparative Study of Computational Methods in Cosmic Gas Dynamics Continued

Fay Luxford - Skewness of Atmospheric Flow Associated with a Wobbling Jetstream

Jesse Norris - A Semi-Analytic Approach to Baroclinic Instability on the African Easterly Jet

Robert J. Smith - Minimising Time-Stepping Errors in Numerical Models of the Atmosphere and Ocean

Amandeep Virdi - The Influence of the Agulhas Leakage on the Overturning Circulation from Momentum Balances

2009: Mathematics of Scientific and Industrial Computation

Charlotta Howarth - Integral Equation Formulations for Scattering Problems

David Fairbairn - Comparison of the Ensemble Transform Kalman Filter with the Ensemble Transform Kalman Smoother

Mark Payne - Mathematical Modelling of Platelet Signalling Pathways Mesh Generation and its application to Finite Element Methods

Mary Pham - Mesh Generation and its application to Finite Element Methods

Sarah Cole - Blow-up in a Chemotaxis Model Using a Moving Mesh Method

2009: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Danila Volpi - Estimation of parameters in traffic flow models using data assimilation

Dale Partridge - Analysis and Computation of a Simple Glacier Model using Moving Grids

David MacLeod - Evaluation of precipitation over the Middle East and Mediterranean in high resolution climate models

Joanne Pocock - Ensemble Data Assimilation: How Many Members Do We Need?

Neeral Shah - Impact and implications of climate variability and change on glacier mass balance in Kenya

Tomos Roberts - Non-oscillatory interpolation for the Semi-Lagrangian scheme

Zak Kipling - Error growth in medium-range forecasting models

Zoe Gumm - Bragg Resonance by Ripple Beds

2008: Mathematics of Scientific and Industrial Computation

Muhammad Akram - Linear and Quadratic Finite Elements for a Moving Mesh Method

Andrew Ash - Examination of non-Time Harmonic Radio Waves Incident on Plasmas

Cassandra Moran - Harbour modelling and resonances

Elena Panti - Boundary Element Method for Heat Transfer in a Buried Pipe

Juri Parrinello - Modelling water uptake in rice using moving meshes

Ashley Twigger - Blow-up in the Nonlinear Schrodinger Equation Using an Adaptive Mesh Method

Chloe Ward - Numerical Evaluation of Oscillatory Integrals

Christopher Warner - Forward and Inverse Water-Wave Scattering by Topography

2008: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Fawzi Al Busaidi - Fawzi Albusaidi

Christopher Bowden - A First Step Towards the Calculation of a Connectivity Matrix for the Great Barrier Reef

Evangelia-Maria Giannakopoulou - Flood Prediction and Uncertainty

Victoria Heighton - 'Every snowflake is different'

Thomas Jordan - Does Self-Organised Criticality Occur in the Tropical Convective System?

Gillian Morrison - Numerical Modelling of Tidal Bores using a Moving Mesh

Rachel Pritchard - Evaluation of Fractional Dispersion Models

2007: Numerical solution of differential equations

Tamsin Lee - New methods for approximating acoustic wave transmission through ducts (PDF 2.5MB)

Lee Morgan - Anomalous diffusion (PDF-1.5MB)

Keith Pham - Finite element modelling of multi-asset barrier options (PDF-3MB)

Alastair Radcliffe - Finite element modelling of the atmosphere using the shallow water equations (PDF-2.5MB)

Sanita Vetra - The computation of spectral representations for evolution PDE (PDF-3.2MB)

2007: Mathematical and numerical modelling of the atmosphere and oceans

Laura Baker - Properties of the ensemble Kalman filter (PDF-3.8MB)

Alison Brass - A moving mesh method for the discontinuous Galerkin finite element technique (PDF-916KB)

Daniel Lucas - Application of the phase/amplitude method to the study of trapped waves in the atmosphere and oceans (PDF-1.1MB)

Duduzile Nhlengethwa - Petrol or diesel (PDF-1MB)

Rhiannon Roberts - Modelling glacier flow (PDF-406KB)

David Skinner - A moving mesh finite element method for the shallow water equations (PDF-4.3MB)

Jovan Stojsavljevic - Investigation of waiting times in non-linear diffusion equations using a moving mesh method (PDF-538KB)

2006: Numerical solution of differential equations

Bonhi Bhattacharya - A moving finite element method for high order nonlinear diffusion problems

Jonathan Coleman - High frequency boundary element methods for scattering by complex polygons

Rachael England - The use of numerical methods in solving pricing problems for exotic financial derivatives with a stochastic volatility

Stefan King - Best fits with adjustable nodes and scale invariance

Edmund Ridley - Analysis of integral operators from scattering problems

Nicholas Robertson - A moving Lagrangian mesh model of a lava dome volcano and talus slope

2006: Mathematical and numerical modelling of the atmosphere and oceans

Iain Davison - Scale analysis of short term forecast errors

Richard Silveira - Electromagnetic scattering by simple ice crystal shapes

Nicola Stone - Development of a simplified adaptive finite element model of the Gulf Stream

Halina Watson - The behaviour of 4-D Var for a highly nonlinear system

2005: Numerical solution of differential equations

Jonathan Aitken - Data dependent mesh generation for peicewise linear interpolation

Stephen Arden - A collocation method for high frequency scattering by convex polygons

Shaun Benbow - Numerical methods for american options

Stewart Chidlow - Approximations to linear wave scattering by topography using an integral equation approach

Philip McLaughlin - Outdoor sound propagation and the boundary element method

Antonis Neochoritis - Numerical modelling of islands and capture zone size distributions in thin film growth

Kylie Osman - Numerical schemes for a non-linear diffusion problem

Shaun Potticary - Efficient evaluation of highly oscillatory integrals

Martyn Taylor - Investigation into how the reduction of length scales affects the flow of viscoelastic fluid in parallel plate geometries

Aanand Venkatramanan - American spread option pricing

2005: Mathematical and numerical modelling of the atmosphere and oceans

Richard Fruehmann - Ageostrophic wind storms in the central Caspian sea

Gemma Furness - Using optimal estimation theory for improved rainfall rates from polarization radar

Edward Hawkins - Vorticity extremes in numerical simulations of 2-D geostrophic turbulence

Robert Horton - Two dimensional turbulence in the atmosphere and oceans

David Livings - Aspects of the ensemble Kalman filter

David Sproson - Energetics and vertical structure of the thermohaline circulation

2004: Numerical solution of differential equations

Rakhib Ahmed - Numerical schemes applied to the Burgers and Buckley-Leverett equations

James Atkinson - Embedding methods for the numerical solution of convolution equations

Catherine Campbell-Grant - A comparative study of computational methods in cosmic gas dynamics

Paresh Prema - Numerical modelling of Island ripening

Mark Webber - The point source methods in inverse acoustic scattering

2004: Mathematical and numerical modelling of the atmosphere and oceans

Oliver Browne - Improving global glacier modelling by the inclusion of parameterised subgrid hypsometry within a three-dimensional, dynamical ice sheet model

Petros Dalakakis - Radar scattering by ice crystals

Eleanor Gosling - Flow through porous media: recovering permeability data from incomplete information by function fitting .

Sarah Grintzevitch - Heat waves: their climatic and biometeorological nature in two north american reigions

Helen Mansley - Dense water overflows and cascades

Polly Smith - Application of conservation laws with source terms to the shallow water equations and crowd dynamics

Peter Taylor - Application of parameter estimation to meteorology and food processing

2003: Numerical solution of differential equations

Kate Alexander - Investigation of a new macroscopic model of traffic flow

Luke Bennetts - An application of the re-iterated Galerkin approximation in 2-dimensions

Peter Spence - The Position of the free boundary formed between an expanding plasma and an electric field in differing geometries

Daniel Vollmer - Adaptive mesh refinement using subdivision of unstructured elements for conservation laws

2003: Mathematical and numerical modelling of the atmosphere and oceans

Clare Harris - The Valuation of weather derivatives using partial differential equations

Sarah Kew - Development of a 3D fractal cirrus model and its use in investigating the impact of cirrus inhomogeneity on radiation

Emma Quaile - Rotation dominated flow over a ridge

Jemma Shipton - Gravity waves in multilayer systems

2002: Numerical solution of differential equations

Winnie Chung - A Spectral Method for the Black Scholes Equations

Penny Marno - Crowded Macroscopic and Microscopic Models for Pedestrian Dynamics

Malachy McConnell - On the numerical solution of selected integrable non-linear wave equations

Stavri Mylona - An Application of Kepler's Problem to Formation Flying using the Störmer-Verlet Method

2002: Mathematical and numerical modelling of the atmosphere and oceans

Sarah Brodie - Numerical Modelling of Stratospheric Temperature Changes and their Possible Causes

Matt Sayer - Upper Ocean Variability in the Equatorial Pacific on Diurnal to Intra-seasonal Timescales

Laura Stanton - Linearising the Kepler problem for 4D-var Data Assimilation

2001: Numerical solution of differential equations

R.B. Brad - An Implementation of the Box Scheme for use on Transcritical Problems

D. Garwood - A Comparison of two approaches for the Approximating of 2-D Scattered Data, with Applications to Geological Modelling

R. Hawkes - Mesh Movement Algorithms for Non-linear Fisher-type Equations

P. Jelfs - Conjugate Gradients with Rational and Floating Point Arithmetic

M. Maisey - Vorticity Preserving Lax-Wendroff Type Schemes

C.A. Radcliffe - Positive Schemes for the Linear Advection Equation

2000: Numerical solution of differential equations

D. Brown - Two Data Assimilation Techniques for Linear Multi-input Systems.

S. Christodoulou - Finite Differences Applied to Stochastic Problems in Pricing Derivatives.

C. Freshwater - The Muskingum-Cunge Method for Flood Routing.

S.H. Man - Galerkin Methods for Coupled Integral Equations.

A. Laird - A New Method for Solving the 2-D Advection Equation.

T. McDowall - Finite Differences Applied to Joint Boundary Layer and Eigenvalue Problems.

M. Shahrill - Explicit Schemes for Finding Soliton Solutions of the Korteweg-de Vries Equation.

B. Weston - A Marker and Cell Solution of the Incompressible Navier-Stokes Equations for Free Surface Flow.

1999: Numerical solution of differential equations

M. Ariffin - Grid Equidistribution via Various Algorithmic Approaches.

S.J. Fletcher - Numerical Approximations to Bouyancy Advection in the Eddy Model.

N.Fulcher - The Finite Element Approximation of the Natural Frequencies of a Circular Drum.

V. Green - A Financial Model and Application of the Semi-Lagrangian Time-Stepping Scheme.

D.A. Parry - Construction of Symplectic Runge-Kutta Methods and their Potential for Molecular Dynamics Application.

S.C. Smith - The Evolution of Travelling Waves in a Simple Model for an Ionic Autocatalytic System

P. Swain - Numerical Investigations of Vorticity Preserving Lax-Wendroff Type Schemes.

M. Wakefield - Variational Methods for Upscaling.

1998: Numerical solution of differential equations

C.C. Anderson - A dual-porosity model for simulating the preferential movement of water in the unsaturated zone of a chalk aquifer.

K.W. Blake - Contour zoning.

M.R. Garvie - A comparison of cell-mapping techniques for basins of attraction.

W. Gaudin - HYDRA: a 3-d MPP Eulerian hydrocode.

D. Gnandi - Alternating direction implicit method applied to stochastic problems in derivative finance.

J. Hudson - Numerical techniques for conservation laws with source terms. .

H.S. Khela - The boundary integral method.

K. Singh - A comparison of numerical schemes for pricing bond options.

1997: Numerical solution of differential equations

R.V. Egan - Chaotic response of the Duffing equation. A numerical investigation into the dynamics of the non-linear vibration equation.

R.G. Higgs - Nonlinear diffusion in reservoir simulation.

P.B. Horrocks - Fokker-Planck model of stochastic acceleration: a study of finite difference schemes.

M.A. Wlasek - Variational data assimilation: a study.

1996: Numerical solution of differential equations

A. Barnes - Reaction-diffusion waves in an isothermal chemical system with a general order of autocatalysis.

S.J. Leary - Mesh movement and mesh subdivision.

S. McAllister - First and second order complex differential equations.

R.K. Sadhra - Investigating dynamical systems using the cell-to-cell mapping.

J.P. Wilson - A refined numerical model of sediment deposition on saltmarshes.

1995: Numerical solution of differential equations

M. Bishop - The modelling and analysis of the equations of motion of floating bodies on regular waves.

J. Olwoch - Isothermal autocatalytic reactions with an immobilized autocatalyst.

S. Stoke - Eulerian methods with a Lagrangian phase in gas dynamics.

R. Coad - 1-D and 2-D simulations of open channel flows using upwinding schemes.

1994: Numerical solution of differential equations

M. Ali - Application of control techniques to solving linear systems of equations .

M.H. Brookes - An investigation of a dual-porosity model for the simulation of unsaturated flow in a porous medium .

A.J. Crossley - Application of Roe's scheme to the shallow water equations on the sphere .

D.A. Kirkland - Huge singular values and the distance to instability. .

B.M. Neil - An investigation of the dynamics of several equidistribution schemes .

1993: Numerical solution of differential equations

P.A. Burton - Re-iterative methods for integral equations .

J.M. Hobbs - A moving finite element approach to semiconductor process modelling in 1-D. .

L.M. Whitfield - The application of optimal control theory to life cycle strategies .

S.J. Woolnough - A numerical model of sediment deposition on saltmarshes .

1992: Numerical solution of differential equations

I. MacDonald - The numerical solution of free surface/pressurized flow in pipes. .

A.D. Pollard - Preconditioned conjugate gradient methods for serial and parallel computers. .

C.J. Smith - Adaptive finite difference solutions for convection and convection-diffusion problems .

1991: Numerical solution of differential equations

K.J. Neylon - Block iterative methods for three-dimensional groundwater flow models .

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Home > A&S > Math > Math Undergraduate Theses

Mathematics Undergraduate Theses

Theses from 2019 2019.

The Name Tag Problem , Christian Carley

The Hyperreals: Do You Prefer Non-Standard Analysis Over Standard Analysis? , Chloe Munroe

Theses from 2018 2018

A Convolutional Neural Network Model for Species Classification of Camera Trap Images , Annie Casey

Pythagorean Theorem Area Proofs , Rachel Morley

Euclidian Geometry: Proposed Lesson Plans to Teach Throughout a One Semester Course , Joseph Willert

Theses from 2017 2017

An Exploration of the Chromatic Polynomial , Amanda Aydelotte

Complementary Coffee Cups , Brandon Sams

Theses from 2016 2016

Nonlinear Integral Equations and Their Solutions , Caleb Richards

Principles and Analysis of Approximation Techniques , Evan Smith

Theses from 2014 2014

An Introductory Look at Deterministic Chaos , Kenneth Coiteux

A Brief Encounter with Linear Codes , Brent El-Bakri

Axioms of Set Theory and Equivalents of Axiom of Choice , Farighon Abdul Rahim

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Department of Mathematics

This page contains details for the topics available for final year dissertations for MMath students, and for projects for BSc students. For full information on the BSc and MMath Final Year Projects, please see this page.

These topics are also offered to students in MSc Mathematics.

For more information on any of these projects, please contact the project supervisor.

For more information, please email Dr Miroslav Chlebík or visit his staff profile

A continuous real-valued function !$u$! defined on a domain !$U\subseteq \mathbb{R}^n$! (!$n\geq 2$!) is called absolutely minimizing , if for any open set !$V\subset U$! and any Lipschitz function !$v$! on !$\overline{V}$! !$$ v\bigm|_{\partial V}=u\bigm|_{\partial V} \qquad \implies \qquad \|\nabla u\|_{L^\infty(V)}\leq \|\nabla v\|_{L^\infty(V)}.$$! It is well-known that !$u$! is absolutely minimizing if and only if it is the solution of the infinity Laplacian, which is the (highly degenerate) Euler-Lagrange equation for the prototypical problem in the calculus of variations in !$L^\infty$!. The problem of regularity of these functions is widely open, at this time it is unknown whether they are differentiable everywhere if !$n>2$!. We examine various techniques to study pointwise behaviour of these functions.

Miroslav Chlebik Presentation [PDF 309.98KB]

Key words: Lipschitz mappings, optimal Lipschitz extension,degenerate elliptic PDEs, infinity harmonic functions.

Recommended modules: Functional Analysis, Partial Differential Equations

References:

!$[1]$! Aronsson, G., Crandall, M. G. and Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41(2004), no. 4, 439--505

!$[2]$! Crandall, M. G., Evans, L. C. and Gariepy, R. F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Diff. Equations 13(2001), no. 2, 123--139

Hausdorff dimension is the principal notion of dimension in the context of fractal sets in !$\mathbb{R}^n$!, or even for general metric spaces. However, other definitions are in widespread use, for example, packing dimension, upper and lower box-counting dimension, upper and lower Minkowski dimension, ... We will examine some of these and their inter-relationship.

Key words: Hausdorff dimension, Lipschitz mappings, rectifiable sets, fractals

Recommended modules: Measure and Integration, Functional Analysis

!$[1]$! Falconer, K., Fractal geometry: Mathematical Foundations and Applications, John Wiley & Sons Ltd., 1990

A curve !$C$! in the plane has the increasing chord property if !$\|x_2-x_3\|\leq \|x_1-x_4\|$! whenever !$x_1$!, !$x_2$!, !$x_3$! and !$x_4$! lie in that order on !$C$!. Larman & Mc Mullen showed that !$$ L\leq 2\sqrt 3|a-b|, $$! where !$C$! is a plane curve with the increasing chord property with length !$L$! and endpoints !$a$! and !$b$!. We will examine how to improve the above constant "!$2\sqrt 3$!". (It is conjectured that !$L\leq \frac23\pi|a-b|$!, with equality if !$C$! consists of two sides of a Reuleaux triangle.)

Key words: curve length, Lipschitz curve, calculus of variations

!$[1]$! Larman, D. G. and McMullen P., Arcs with increasing chords, Proc. Cambridge Philos. Soc. 72(1972), 205--207

For more information, please email Marianna Cerasuolo

This project will focus on understanding, through a strong mathematical approach, the dynamics of tumour cells. From Britton: “Biological processes such as cell proliferation are normally extremely tightly controlled through feedback processes that are mainly chemically mediated... There are cell populations that escape from such controls through mutations that allow them to manipulate their local environment. Some mutations that cancer cells undergo may be sufficient to allow the immune system to recognise them as foreign, and hence to mount a defence against them.” However, such defence is not always effective. The use of reaction diffusion equation models for the description of the dynamics of tumour growth and the processes involved is explored. The project will focus on the effect of environmental inhomogeneity on the mutations (evolution) of a tumour growing cell-population.

[1] Burbanks, A., Cerasuolo, M., Ronca, R., & Turner, L. (2023). A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies. Mathematical Biosciences, 355, 108940.

[2] Krause, A. L., Gaffney, E. A., & Walker, B. J. (2023). Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems. Bulletin of Mathematical Biology, 85(2), 14.

Physiological signals such as ECG and blood pressure contain a high degree of chaos. Often the absence of chaos and the consequent regularization of the signal implies a deterioration of the patient’s health status. In this project baseline and LPS data of blood pressure over 24 hours will be represented and interpreted using various mathematical techniques, for example:

• Heart rate variability (HRV) measure

• SPAR waveform variability measures

• Fractal dimension measures All different methods used to extract the deterioration information will be compared with the aim of finding the optimal technique for the data of interest.

[1] Aston P.J., Nandi M., Christie M., & Huang Y., (2014),“Comparison of Attractor Reconstruction and HRV Methods for Analysing Blood Pressure Data”, Computing in Cardiology. Volume 41

[2] Steven H. Strogatz,(1994), “Nonlinear Dynamics and Chaos: With applications to physics Biology, Chemistry and Engineering”,Perseus Books Publishing.

For more information, please email Dr Antoine Dahlqvist or visit his staff profile

See PDF for full description

Antoine Dahlqvist - Random matrices and Free Probability [PDF 345.10KB]

Antoine Dahlqvist - Brownian queues [PDF 151.46KB]

For more information, please email Dr Masoumeh Dashti or visit her staff profile

Studying the convergence properties of sequences of probability measures comes up in many applications (for example in the study of approximations of probability measures and stochastic inverse problems). In such problems, it is of course important to choose an appropriate metric on the space of the probability measures. This project consists of learning about some of the important metrics on the space of probability measures (for example: Hellinger, Prokhorov and Wasserstein), and studying the relationship between them. We also look at convergence properties of some sampling techniques.

Key words: probability metrics, rates of convergence, Bayesian inverse problems

Recommended modules: Introduction to Probability, Measure and Integration.

!$[1]$! Gibbs A. L. and Su F. E. (2002) On choosing and bounding probability metrics.

!$[2]$! Robert, C. P. and Casella, G. (2004) Monte Carlo statistical methods. Second edition. Springer Texts in Statistics. Springer-Verlag, New York.

Consider the problem of finding the initial temperature field of a one dimensional heat equation from (noisy) measurements of the temperature function at a positive time. This is an example of an inverse problem (considering the underlying heat equation, given initial temperature field, as the direct problem). Such problems where the function of interest cannot be observed directly, and has to be obtained from other observable quantities and through the mathematical model relating them, appears in many practical situations. Inverse problems in general do not satisfy Hadamard's conditions of well-posedness: for example in the case of the above inverse heat problem, the solution (here the initial field) does not depends continuously on the temperature function at a positive time. We can, however, obtain a reasonable approximation of the solution in a stable way by regularizing the problem using a priori information about the solution. In this project, we will study classical regularization methods, and also the Bayesian approach to regularization in the case of statistical noise.

Key words: Inverse problems, Tikhonov regularization, Bayesian regularization

Recommended modules: Partial differential equations, Functional analysis, Probability and statistics, Measure and Integration.

!$[1]$! Engl H. W., Hanke M. and Neubauer A. (2000) Regularization of inverse problems, Kluwer Academic Publishers.

!$[2]$! Stuart A. (2010) Inverse problems: a Bayesian perspective, 19 , 451--559.

We start by studying Leray-Hopf weak solutions of the three dimensional Navier-Stokes equations which are known to exit globally (for all positive times). The strong solutions are only known to exist locally. There are, however, results which show the global existence of strong solutions under extra conditions on the velocity field or pressure (conditional regularity results). In this direction, we will study Serrin's conditional regularity result and then examine similar conditions in terms of the pressure field.

Key words: Navier-Stokes equations, Regularity theory

Recommended modules: Partial differential equations, Functional analysis, Measure and Integration.

!$[1]$! Chae L. and Lee J. (2001) Regularity criterion in terms of pressure for the NavierStokes equations, Nonlinear Analysis 46 . 727-735

!$[2]$! Serrin J. (1962) On the interior regularity of weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis , 9 , 187-195.

!$[3]$! Temam R. (2001) Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society.

For more information, please email Dr Nicos Georgiou or visit his staff profile

\begin{equation} \Psi(x,y) =\left\{ \begin{array}{lll} x, & \textrm {if } x < py \\ \displaystyle \frac{2\sqrt{pxy}-p(x+y)}{q}, & \textrm {if } p^{-1}y\geq x\geq py \\ y, &\textrm {if } y < px \end {array} \right. \end{equation}

There is a vast literature in statistical physics that studies this model as a simplified alternative to the hard longest common subsequence (LCS) model (see other projects).

Key words: Longest increasing path, Hammersley process, totally asymmetric simple exclusion process, corner growth model, last passage percolation, subadditive ergodic theorem

The goal of this project is three-fold. First there is the theoretical component of understanding the mathematics behind the hydrodynamic limits of the particle system and find the limiting PDE. Second, we will use free traffic data and develop statistical tests to identify and estimate relevant parameters that appear in the hydrodynamic limit above. The third is to develop Monte Carlo algorithms that take the estimated parameters, build the stochastic model, and show us the traffic progress in a given road network.

Supervisor: Dr. Nicos Georgiou

Helpful mathematical background: Random processes, Monte Carlo Simulations, Statistical Inference.

Some Bibliography:

[1] N. Georgiou, R. Kumar and T. Seppäläinen TASEP with discontinuous jump rates https://arxiv.org/pdf/1003.3218.pdf

[2] H.J. Hilhorst and C. Appert-Rolland, A multi-lane TASEP model for crossing pedestrian traffic flows https://arxiv.org/pdf/1205.1653.pdf

[3] J.G. Brankov, N.C. Pesheva and N. Zh. Bunzarova, One-dimensional traffic flow models: Theory and computer simulations. Proceedings of the X Jubilee National Congress on Theoretical and Applied Mechanics, Varna, 13-16 September, 2005(1), 442–456.

For more information, please email Dr Peter Giesl or visit his staff profile

  Peter Giesl - Computational analysis of periodic orbits in nonsmooth differential [PDF 11.57KB]

  Peter Giesl - Calculation of Contraction Metrics [PDF 16.77KB]

  Peter Giesl Project 3 [PDF 92.74KB]

For more information, please email Chris Hadjichrysanthou

Novel models will be developed to describe the dynamical changes of different respiratory viruses, like SARS-CoV and influenza, at different levels, from the cellular level to the individual and population level. The models will be extended to incorporate the impact of a range of prophylactic and therapeutic interventions to control i) viral replication within an individual host, ii) transmission of viral infections between individuals. Following an analytical investigation of the models and the derivation of important quantities, such as the basic reproductive number, generation times and area under the viral load and epidemic curves, we will solve them numerically and fit them to real data from clinical and epidemiological studies. This will enable the improvement of the models based on a number of selection criteria and identifiability analysis techniques. Depending on the interests and skills, stochastic algorithms will be developed to simulate the stochastic processes and quantify uncertainty in the model outputs. Some of the questions that you will be able to answer by the end of the project are:

- What are the most appropriate mathematical models to describe within- and between-host viral infection dynamics given the infection, the available data and the quantities we want to consider?

- What is the time window for prophylactic and therapeutic interventions to prevent an infection, or the development of mild/severe symptoms? What should be the optimal treatment efficacy?

- What should be the optimal vaccination and/or treatment strategy to prevent an outbreak, or reduce severe cases/hospitalisations/deaths below a certain threshold?

- Who should be prioritised for vaccination/treatment in a highly heterogenous population? An old, isolated person or a highly connected child?

The various components of this project can be extended in different ways and could constitute individual projects. During the project, you may have the opportunity to meet with leading researchers in the area of infectious disease epidemiology, and attend meetings with pharmaceutical companies, so you see how theory is linked with practice and real-world problems.

We will describe complex evolutionary and/or epidemic processes in non-homogeneous populations, characterised by high heterogeneity in demographic factors and contacts between individuals. Starting from the master equations we will introduce approximations that can reduce the number of system’s states while maintaining the accuracy of the prediction of the stochastic process. Both deterministic and stochastic systems will be tested and compared on a range of real-world networks, using data from epidemiological studies. The importance of the properties of the contact structure in the evolution of different systems will be studied.

Alzheimer’s disease is a progressive neurodegenerative disease which is rapidly becoming one of the leading causes of disability and mortality. We aim to develop mathematical, statistical and computational tools that will generate insights into the development and progression of Alzheimer’s disease, address the therapeutic challenges and accelerate the development of much-needed treatments. Clinical data from thousands of individuals will be analysed to try to identify changes in biological and clinical markers that indicate disease progression. Statistical, mathematical and computational techniques will be employed to describe the long-term changes of potential biomarkers, and indicators of cognitive and functional abilities, using short-term data. The focus will be on the stage prior to the clinical presentation of the disease.

- What is the expected probability and time required for an individual to develop Alzheimer’s dementia given its demographic and genetic characteristics, as well as levels of certain biological and cognitive markers?

- How factors like education could affect the clinical progression of the disease?

- If hypothetical treatments that reduce the accumulation of certain proteins in the brain lead to the decrease of the rate of cognitive decline, what is the time window for intervention to delay the occurrence of Alzheimer’s clinical symptoms for x years, given a certain treatment efficacy?

The project requires advanced statistical analysis skills.

For more information, please email Philip Herbert

Many processes may be modelled by partial differential equations (PDE), some of these may take place in a thin region. In the limit of the thinness tending to zero, one might justify modelling the process by a PDE on a surface. To begin with, this project would seek to describe surfaces and various quantities upon that surface, for example the normal vector. With geometric notions in mind, one may define a surface gradient and pose surface PDEs. Finally, one might be able to provide well-posedness for a simple surface PDE. Computational results would accompany this project well.

In this project, we wish to understand some of the mathematical background for optimisation under constraints. Frequently constraints will take the form of a partial differential equation (PDE), and the optimisation may be related to quantities of interest from that PDE. A prototype example is: where should I heat (or cool) the room in order to ensure that the room has a temperature profile which suits the task at hand. Here the quantity of interest is the deviation from the desired temperature profile and the PDE constraint is Laplace's equation. This project will investigate the applications of functional analytic theorems to show well-posedness of a variety of optimisation problems. Tjis project would be well complimented by computational results.

For more information, please email Prof. James Hirschfeld or visit his staff profile

Given one or more polynomials in several indeterminates, what do their set of common zeros look like? Curves and surfaces are typical examples. This topic examines the basic theory of such objects. It can be done both at an elementary level and at a more sophisticated level. The material of the Term 7 course on Ring Theory would be handy.

James Hirschfeld Presentation 1 [PDF 36.89KB]

Key words: polynomial, algebraic geometry

Recommended modules: Coding Theory

!$[1]$! Reid, M. Undergraduate Algebraic Geometry, University Press, 1988.

!$[2]$! Semple, J. G. and Roth, L. Introduction to Algebraic Geometry, Oxford University Press, 1949

Cubic curves in the plane may have a singular point or be non-singular. The non-singular points on a cubic form an abelian group, which leads to many remarkable properties such as the theory of the nine associated points, from which many other results can be deduced. A non-singular (elliptic) cubic is one of the most beautiful structures in mathematics.

James Hirschfeld Presentation 2 [PDF 25.58KB]

Key words: algebraic curve, cubic, group

!$[1]$! Seidenberg, A. Elements of the Theory of Algebraic Curves Addison-Welsley 1968

!$[2]$! Clemens, C.H. A scrapbook for Complex Curve Theory Plenum Press 1980

In defining a vector space, the scalars belong to a field, which can also be finite, such as the integers modulo a prime. Many combinatorial problems reduce to the study of geometrical configurations, which in turn can be analysed in a geometry over a finite field.

James Hirschfeld Presentation 3 [PDF 26.96KB]

Key words: geometry, projective plane, finite field

!$[1]$! Dembowski, P. Finite Geometries, Springer Verlag, 1968

!$[2]$! Hirschfeld, J.W.P. Projective Geometries over a Finite Field Oxford University Press, 1998.

Error correction codes are used to correct errors when messages are transmitted through a noisy communication channel. Here is the basic idea.

To send just the two messages YES and NO, the following encoding suffices: YES = 1, NO = 0:

If there is an error, say 1 is sent and 0 arrives, this will go undetected. So, add some redundancy: YES = 11, NO = 00:

Now, if 11 is sent and 01 arrives, then an error has been detected, but not corrected, since the original messages 11 and 00 are equally plausible. So, add further redundancy: YES = 111, NO = 000:

Now, if 010 arrives, and it is supposed that there was at most one error, we know that 000 was sent: the original message was NO. Most of the theory depends on vector spaces over a finnite field.

References 1. R. Hill, A First Course on Coding Theory, Oxford, 1986; QE 1302 Hil. The course is mostly based on this book. 2. V.S. Pless, Introduction to the Theory of Error-Correcting Codes, Wiley, 1982, 1989; QE 1302 Ple. 3. S. Ling and C.P. Xing, Coding Theory, a First Course, Cambridge, 2004; QE 1302 Lin. 4. https://www.maths.sussex.ac.uk/Staff/JWPH/TEACH/CODING21/index.html

For more information, please email Dr Konstantinos Koumatos or visit his staff profile

From the prototypical example of steel to modern day shape-memory alloys, materials undergoing martensitic transformations exhibit remarkable properties and are used in a wide range of applications, e.g. as thermal actuators, in medical devices, in automotive engineering and robotics.

The properties of these materials, such as the toughness of steel or Nitinol being able to remember its original shape, are related to what happens at small length scales and the ability of these materials to form complex microstructures. Hence, understanding how microstructures form and how they give rise to these properties is key, not only to find new applications, but also to design new materials.

A mathematical model, developed primarily in the last 30 years [1,2,3], views microstructures as minimizers of an energy associated to the material and has been very successful in explaining many observables. In fact, it has been successful even in contributing to the design of new smart materials which exhibit enhanced reversibility and low hysteresis, properties which are crucial in applications.

In this project, we will review the mathematical theory - based on nonlinear elasticity and the calculus of variations - and how it has been able to give rise to new materials with improved properties. Depending upon preferences, the project can be more or less technical.

Key words: microstructure, energy minimisation, elasticity, calculus of variations, non-convex variational problems

Recommended modules: Continuum Mechanics, Partial Differential Equations, Functional Analysis, Measure and Integration

!$[1]$! J. M. Ball, Mathematical models of martensitic microstructure, Materials Science and Engineering A 378, 61--69, 2004

!$[2]$! J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Archive for Rational Mechanics and Analysis 100 (1), 13--52, 1987

!$[3]$! K. Bhattacharya, Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect, Oxford University Press, 2003

!$[4]$! X. Chen, V. Srivastava, V. Dabade R. D. James, Study of the cofactor conditions: conditions of supercompatibility between phases, Journal of the Mechanics and Physics of Solids 61 (12), 2566--2587, 2013

!$[5]$! S. Muller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems, 85--210, 1999

The equilibrium problem of nonlinear elasticity can be formulated as that of minimising an energy functional of the form !$$ \mathcal E(u) = \int_\Omega W(\nabla u(x))\,dx, $$! subject to appropriate boundary conditions on !$\partial\Omega$!, where !$\Omega\subset \mathbb{R}^n$! represents the elastic body at its reference configuration and !$u:\Omega\to \mathbb{R}^n$! is a deformation of the body mapping a material point !$x\in \Omega$! to its deformed configuration !$u(x)\in \mathbb{R}^n$!. The function !$W$! is the energy density associated to the material and physical requirements force one to assume that !$$ W(F) \to \infty, \mbox{ as }\det F\to0^+ \mbox{ and } W(F) = \infty, \,\det F \leq 0. \tag{$\ast$} $$! As the determinant of the gradient expresses local change of volume, the conditions above translate to the requirement of infinite energy to compress a body to zero volume as well as the requirement that admissible deformations be orientation-preserving. It turns out that (!$\ast$!) is incompatible with standard conditions required on !$W$! to establish the existence of minimisers in the vectorial calculus of variations. In this project, we will review classical existence theorems as well as the seminal work of J. Ball [1] proving existence of minimisers for !$\mathcal E$! and energy densities !$W$! that are !${\it polyconvex}$! and fulfil condition (!$\ast$!). Such energies cover many of the standard models used in elasticity.

Key words: nonlinear elasticity, polyconvexity, quasiconvexity, existence theories, determinant constraints

Recommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration

!$[1]$! J. M. Ball, Convexity conditions and existence theorems in elasticity, Archive for Rational Mechanics and Analysis 63 (4), 337--403, 1977

!$[2]$! B. Dacorogna, Direct methods in the calculus of variations, volume 78, Springer, 2007

Existence of solutions to nonlinear PDEs often relies in the following strategy: construct a suitable sequence of approximate solutions and prove that, up to a subsequence, the approximations converge to an appropriate solution of the PDE. A priori estimates coming from the PDE itself typically allow for convergence of the approximation to be established in some weak topology which, however, does not suffice to pass to the limit under a nonlinear quantity. This loss of continuity with respect to the weak topology is a great obstacle in nonlinear problems. In a series of papers in the 1970's, L. Tartar and F. Murat (see [3] for a review) introduced a remarkable method, referred to as compensated compactness, which gives conditions on nonlinearities !$Q$! that allow one to establish the implication: !$$ V_j \rightharpoonup V \Longrightarrow Q(V_j) \rightharpoonup Q(V)\tag{$\ast$} $$! under the additional information that the sequence !$V_j$! satisfies some differential constraint, e.g. the !$V_j$!'s could be gradients, thus satisfying the constraint !${\rm curl}\, V_j = 0$!. Note that (!$\ast$!) is not true in general and it is the additional information on !$V_j$! that ``compensates'' for the loss of compactness. In this project, we will review the compensated compactness theory and investigate its consequences on the existence theory for scalar conservation laws in dimension 1 via the vanishing viscosity method. In particular, we will use the so-called div-curl lemma to prove that a sequence !$u^\varepsilon$! verifying \begin{align*} \partial_t u^\varepsilon + \partial_x f(u^\varepsilon) & = \varepsilon \partial_{xx} u^\varepsilon\\ u(\cdot,t = 0) & = u_0 \end{align*} converges in an appropriate sense as $\varepsilon\to0$ to a function $u$ solving the conservation law \begin{align*} \partial_t u + \partial_x f(u) & = 0\\ u(\cdot,t = 0) & = u_0. \end{align*}

Key words: compensated compactness, div-curl lemma, weak convergence, oscillations, convexity, wave cone, conservation laws, vanishing viscosity limit

Recommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration (essential)

!$[1]$! C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2010

!$[1]$! L. C. Evans, Weak convergence methods for nonlinear partial differential equations, American Mathematical Society, 1990

!$[1]$! L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt symposium, 136--212, 1979

For !$t\in \mathbb{R}$!, consider the system of ordinary differential equations !$$ \frac{d}{dt}X(t) = b(X(t)),\quad X(0) = x\in \mathbb{R}^n. \tag{!$\ast$!} $$! The classical Cauchy-Lipschitz theorem (aka Picard-Lindel\"of or Picard's existence theorem) provides global existence and uniqueness results for (!$\ast$!) under the assumption that the vector field !$b$! is Lipschitz. However, in many cases (e.g. fluid mechanics, kinetic theory) the Lipschitz condition on !$b$! cannot be assumed as a mere Sobolev regularity seems to be available.

In pioneering work, Di Perna and Lions [2] established existence and uniqueness of appropriate solutions to (!$\ast$!) under the assumption that !$b\in W^{1,1}_{{\tiny\rm loc}}$!, a control on its divergence is given and some additional integrability holds. In this project, we will review the elegant work of Di Perna and Lions.

Remarkably, their proof of a statement concerning ODEs is based on the transport equation (a partial differential equation) !$$ \partial_t u(x,t) + b(x)\cdot {\rm div}\, u(x,t) = 0, \quad u(x,0) = u_0(x) $$! and the concept of renormalised solutions introduced by the same authors. The relation between (!$\ast$!) and the transport equation lies in the method of characteristics which states that smooth solutions of the transport equation are constant along solutions of the ODE, i.e. !$$ u(X(t),t) = u(X(0),0) = u_0(X(0)) = u_0(x). $$!

Key words: ODEs with Sobolev coefficients, DiPerna-Lions, transport equation, renormalised solutions, continuity equation

Recommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration (desirable)

!$[1]$! C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Seminaire Bourbaki 972, 2007

!$[2]$! R. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, 511--517, 1989

!$[3]$! L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998

For more information, please email [email protected] or visit her staff profile

This project aims to identify and analyse models of coupled elements, which are connected with time-delays. These types of systems arise in various different disciplines, such as engineering, physics, biology etc. The interesting feature where the current state of the system depends on the state of the system some time ago makes such models much more realistic and leads to various potential scenarios of dynamical behaviour. The models in this project will be analysed analytically to understand their stability properties and find critical time delays as well as numerically using MATLAB.

For more information, please email Dr Omar Lakkis or visit his staff profile

Geometric constructs such as curves, surfaces, and more generally (immersed) manifolds, are traditionally thought as static objects lying in a surrounding space. In this project we view them instead as moving within the surrounding space. While Differential Geometry, which on of the basis of Geometric Motions, is a mature theory, the study of Geometric Motions themselves has only really picked-up in the late seventies of the past century. This is quite surprising given the huge importance that geometric motions play in applications which range from phase transition to crystal growth and from fluid dynamics to image processing. Here, following the so-called classical approach, we learn first about some basic differential geometric tools such as the mean and Gaussian curvature of surfaces in usual 3-dimensional space. We then use these tools to explore a fundamental model of geometric motions: the Mean Curvature Flow. We review the properties of this motion and some of its generalisations. We look at the use of this motion in applications such as phase transition. This project has the potential to extend into a research direction, depending on the students will and ability to pursue this. Extra references will be given in that case. One way of performing this extension would be to implement computer code simulating geometric motions and analysing the algorithms.

Omar Lakkis Presentations [PDF 358.53KB]

Key words: Parabolic Partial Differential Equations, Surface Tension, Geometric Measure Theory, Fluid-dynamics, Growth Processes, Mean Curvature Flow, Ricci Flow, Differential Geometry, Phase-field, Level-set, Numerical Analysis

Recommended modules: Finite Element Methods, Measure and Integration, Numerical Linear Algebra, Numerical Differential Equations, Intro to Math Bio, Applied Whatever Modelling.

!$[1]$! Gurtin, Morton E., Thermomechanics of evolving phase boundaries in the plane. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993. ISBN 0-19-853694-1

!$[2]$! Huisken, Gerhard, Evolution Equations in Geometry, in Mathematics unlimited-2001 and beyond, 593-604, Springer, Berlin, 2001.

!$[3]$! Spivak, Michael, A Comprehensive Introduction to Differential Geometry. Vol. III. Second edition. Publish or Perish, 1979. ISBN 0-914098-83-7

!$[4]$! Struwe, Michael, Geometric Evolution Problems. Nonlinear Partial Differential Equations in Differential Geometry (Park City, UT, 1992), 257-339, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.

Stochastic Differential Equations (SDEs) have become a fundamental tool in many applications ranging from environmental risk management to mechanical failure control and from neurobiology to financial analysis. While the need for effective numerical solutions of SDEs, which are differential equations with a probabilistic (uncertain) data, closed form solutions are seldom available.

This project can be specialised, according to the student's tastes and skills into 3 different flavours: (1) Analysis/Theory, (2) Analysis/Computation, (3) Computational/Modelling.

(1) We explore the rich theory of stochastic processes, stochastic integration and theory (existence, uniqueness, stability) of stochastic differential equations and their relationship to other fields such as the Kac-Feynman Formula (related to quantum mechanics and particle physics), or Partial Differential Equations and Potential Theory (related to the work of Einstein on Brownian Motion), stochastic dynamical systems (large deviation) or Kolmogorov's approach to turbulence in fluid-dynamics. Prerequisites for this direction are some knowledge of probability, stochastic processes, partial differential equations, measure and integration and functional analysis.

(2) We review the basics of SDEs and then look at a practical way of implementing algorithms, using any one of Octave/Matlab/C/C++, that give us a numerical solution. In particular, we learn about pseudorandom numbers, Monte-Carlo methods, filtering and the interpretation of those numbers that our computer produces. Although not a strict prerequisite, some knowledge of probability, ordinary differential equations and their numerical solution will be useful.

(3) We look at practical models in environmental sciences, medicine or engineering involving uncertainty (for example, the ideal installation of solar panels in a region where weather variability can affect their performance). We study these models both from a theoretical point of view (connecting to their Physics) and we run simulations using computational techniques for stochastic differential equations. The application field will be emphasised and must be clearly to the student's liking. (Although very interesting as a topic, I prefer not to deal with financial applications.) The prerequisites are probability, random processes, numerical differential equations and some of the applied/modelling courses.

Key words: Stochastic Differential Equations, Scientific Computing, Random Processes, Probability, Numerical Differential Equations, Environmental Modelling, Stochastic Modelling, Feynman-Kac Formula, Ito's Integral, Stratonovich's Integral, Stochastic Calculus, Malliavin Calculus, Filtering.

Recommended modules: Probability Models, Random Processes, Numerical Differential Equations, Partial Differential Equations, Introduction to Math Biology, Fluid-dynamics, Statistics.

!$[1]$! L.C. Evans, An Introduction to stochastic differential equations. Lecture notes on authors website (google: Lawrence C Evans). University of California Berkley.

!$[2]$! C. W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences. 3rd ed., Springer Series in Synergetics, vol. 13, Springer-Verlag, Berlin, 2004. ISBN 3-540-20882-8

!$[3]$! P.E. Kloeden; E. Platen; H. Schurz, Numerical solution of SDE through computer experiments. Universitext. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN 3-540-57074-8

!$[4]$! A. Beskos and A. Stuart, MCMC methods for sampling function space, ICIAM2007 Invited Lectures (R. Jeltsch and G. Wanner, eds.), 2008.

!$[5]$! Joseph L. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1984 edition. ISBN 3-540-41206-9

For more information, please email Prof. Michael Melgaard or visit his staff profile

Quantum Operator Theory concerns the analytic properties of mathematical models of quantum systems. Its achievements are among the most profound and most fascinating in Quantum Theory, e.g., the calculation of the energy levels of atoms and molecules which lies at the core of Computational Quantum Chemistry.

Among the many problems one can study, we give a short list:

• The atomic Schrödinger operator (Kato's theorem and all that);
• The periodic Schrödinger operator (describing crystals);
• Scattering properties of Schrödinger operators (describing collisions etc);
• Spectral and scattering properties of mesoscopic systems (quantum wires, dots etc);
• Phase space bounds (say, upper bounds on the number of energy levels) with applications, e.g., the Stability of Matter or Turbulence.

Key words: differential operators, spectral theory, scattering theory.

Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations.

!$[1]$! M. Melgaard, G. Rozenblum, Schrödinger operators with singular potentials, in: Stationary partial differential equations Vol. II, 407--517, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.

!$[2]$! Reed, M., Simon, B., Methods of modern mathematical physics. Vol. I-IV . Academic Press, Inc., New York, 1975, 1978,1979,1980.

Quantum Mechanics (QM) has its origin in an effort to understand the properties of atoms and molecules. Its first achievement was to establish the Schrödinger equation by explaining the stability of the hydrogen atom; but hydrogen is special because it is exactly solvable. When we proceed to a molecule, however, the QM problem cannot be solved in its full generality. In particular, we cannot determine the solution (i.e., the ground state ) to !$HΨ=EΨ$!, where !$H$! denotes the Hamiltonian of the molecular system, !$Ψ$! is the wavefunction of the system, and !$E$! is the lowest possible energy. This problem corresponds to finding the minimum of the spectrum of !$H$! or, equivalently, !$$E= \inf \{ \, \mathcal{E}^{\rm QM}(Ψ) \, : \, Ψ \in \mathcal{H}, \:\: \| Ψ \|_{L^{2}} =1 \, \}, where \ \mathcal{E}^{\rm QM}(Ψ):= \langle Ψ, H Ψ \rangle_{L^{2}}$$! and !$\mathcal{H}$! is the variational (Hilbert) space. For systems involving a few (say today six or seven) electrons, a direct Galerkin discretization is possible, which is known as Full CI in Computational Chemistry. For larger systems, with !$N$! electrons, say, this direct approach is out of reach due to the excessive dimension of the space !$ℜ^{3N}$! on which the wavefunctions are defined and the problem has to be approximated. Quantum Chemistry (QC), as pioneered by Fermi, Hartree, Löwdin, Slater, and Thomas, emerged in an attempt to develop various ab initio approximations to the full QM problem. The approximations can be divided into wavefunction methods and density functional theory (DFT). For both, the fundamental questions include minimizing configuration, divided into Question I (i) necessary and sufficient conditions for existence of a ground state (=a minimizer), and Question I (ii) uniqueness of a minimizer, and Question II, necessary and sufficient conditions for multiple (nonminimal) solutions (i.e., excited states ).

A magnetic field has two effects on a system of electrons: (i) it tends to align their spins, and (ii) it alters their translational motion. The first effect appears when one adds a term of the form !$-eħm^{-1} {s} \cdot \mathcal{B}$! to the Hamiltonian, while the second, diamagnetic effect arises from the usual kinetic energy !$(2m)^{-1} | {\mathbf p} |^{2}$! being replaced by !$(2m)^{-1} | \mathbf {p} -(e/c) \mathcal{A}|^{2}$!. Here !${\mathbf p}$! is the momentum operator, !$\mathcal{A}$! is the vector potential, !$\mathcal{B}$! is the magnetic field associated with !$\mathcal{A}$!, and !${s}$! is the angular momentum vector. Within the numerical practice, one approach is to apply a perturbation method to compute the variations of the characteristic parameters of, say, a molecule, with respect to the outside perturbation. It is interesting to go beyond and consider the full minimization problem of the perturbed energy. In Hartree-Fock theory, one only takes into account the effect (ii), whereas in nonrelativistic DFT it is common to include the spin-dependent term and to ignore (ii) and to study the minimization of the resulting nonlinear functional, which depends upon two densities , one for spin "up" electrons and the other for spin "down" electrons. Each density satisfies a normalisation constraint which can be interpreted as the total number of spin "up" or "down" electrons.

The proposed project concerns the above-mentioned problems within the context of DFT in the presence of an external magnetic field. More specifically, molecular Kohn-Sham (KS) models, which turned DFT into a useful tool for doing calculations, are studied for the following settings:

Recent results on rigorous QC are found in the references.

• As a first step towards systems subject to a magnetic field, Question I(i) is addressed for the unrestricted KS model, which is suited for the study of open shell molecular systems (i.e., systems with a odd number of electrons such as radicals, and systems with an even number of electrons whose ground state is not a spin singlet state). The aim is to consider the (standard and extended) local density approximation (LDA) to DFT.
• The spin-polarized KS models in the presence of an external magnetic field with constant direction are studied while taking into account a realistic local spin-density approximation, in short LSDA.

Resonances play an important role in Chemistry and Molecular Physics. They appear in many dynamical processes, e.g. in reactive scattering, state-to-state transition probabilities and photo-dissociation, and give rise to long-lived states well above scattering thresholds. The aim of the project is carry out a rigorous mathematical study on the use of Complex Absorbing Potentials (CAP) to compute resonances in Quantum Dynamics.

In a typical quantum scattering scenario particles with positive energy arrive from infinity, interact with a localized potential !$V(x)$! whereafter they leave to infinity. The absolutely continuous spectrum of the the corresponding Schrödinger operator !$T(\hbar)=-\hbar^{2}D+V(x)$! coincides with the positive semi-axis. Nevertheless, the Green function !$G(x,x'; z)= \langle x | (T(\hbar)-z)^{-1}| x \rangle$! admits a meromorphic continuation from the upper half-plane !$\{ \, {\rm Im}\, z >0 \,\}$! to (some part of) the lower half-plane !$\{ \, {\rm Im}\, z < 0 \,\}$!. Generally, this continuation has poles !$z_{k} =E_{k}-i Γ_{k}/2$!, !$Γ_{k}>0$!, which are called resonances of the scattering system. The probability density of the corresponding "eigenfunction" !$Ψ_{k}(x)$! decays in time like !$e^{-t Γ_{k}/ \hbar}$!, thus physically !$Ψ_{k}$! represents a metastable state with a decay rate !$Γ_{k}/ \hbar$! or, re-phrased, a lifetime !$\tau_{k}=\hbar / Γ_{k}$!. In the semi-classical limit !$\hbar \to 0$!, resonances !$z_{k}$! satisfying !$Γ_{k}=\mathcal{O}(\hbar)$! (equivalently, with lifetimes bounded away from zero) are called "long-lived".

Physically, the eigenfunction !$Ψ_{k}(x)$! only make sense near the interaction region, whereas its behaviour away from that region is evidently nonphysical (Outgoing waves of exponential growth). As a consequence, a much used approach to compute resonances approximately is to perturb the operator !$T(\hbar)$! by a smooth absorbing potential !$-iW(x)$! which is supposed to vanish in the interaction region and to be positive outside. The resulting Hamiltonian !$T_{W}(\hbar):=T(\hbar)-iW(x)$! is a non-selfadjoint operator and the effect of the potential !$W(x)$! is to absorb outgoing waves; on the contrary, a real-valued positive potential would reflect the waves back into the interaction region. In some neighborhood of the positive axis, the spectrum of !$T_{W}(\hbar)$! consists of discrete eigenvalues !$\tilde{z}_{k}$! corresponding to !$L^{2}$!-eigenfunctions !$\widetilde{Ψ}_{k}$!.

As mentioned above, the CAP method has been widely used in Quantum Chemistry and numerical results obtained by CAP are very good. The drawback with the use of CAP is that there are no proof that the correct resonances are obtained. (This is in stark contrast to the mathematically rigorous method of complex scaling). In applications it is assumed implicitly that the eigenvalues !$\tilde{z}_{k}$! near to the real axis are small perturbations of the resonances !$z_{k}$! and, likewise, the associated eigenfunctions !$\widetilde{Ψ}_{k}$!, !$Ψ_{k}(x)$! are close to each other in the interaction region. Stefanov (2005) proved that very close to the real axis (namely, for !$| {\rm Im}\, \tilde{z}_{k}| =\mathcal{O}(\hbar^{n})$! provided !$n$! is large enough), this is in fact true. Stefanov's proof relies on a series of ingenious developments by several people, most notably Helffer (1986), Sjöstrand (1986, 1991, 1997, 2001, 2002), and Zworski (1991, 2001).

The first part of the project would be to understand in details Stefanov's work [2] and, subsequently, several open problems await.

Key words: operator and spectral theory, semiclassical analysis, micro local analysis.

!$[1]$! J. Kungsman, M. Melgaard, Complex absorbing potential method for Dirac operators. Clusters of resonances, J. Ope. Th., to appear.

!$[2]$! P. Stefanov, Approximating resonances with the complex absorbing potential method, Comm. Part. Diff. Eq. 30 (2005), 1843--1862.

The Choquard equation in three dimensions reads:

!$$\begin{equation} \tag*{(0.1)} -Δ u - \left( \int_{ℜ^{3}} u^{2}(y) W(x-y) \, dy \right) u(x) = -l u , \end{equation}$$! where !$W$! is a positive function. It comes from the functional:

!$$\mathcal{E}^{\rm NR}(u) = \int_{ℜ^{3}} | \nabla u |^{2} \, dx -\int \int | u(x) |^{2} W(x-y) |u(y)|^{2} \, dx dy,$$!

which, in turn, arises from an approximation to the Hartree-Fock theory of a one-component plasma when !$W(y) =1/ | y | $! (Coulomb case). Lieb (1977) proved that there exists a unique minimizer to the constrained problem !$E^{\rm NR}(\nu) = \inf \{ \, \mathcal{E}(u) \, : \, u \in \mathcal{H}^{1}(ℜ^{3}), \| u \|_{L^{2}} \leq \nu \, \}$!.

The mathematical difficulty of the functional is caused by the minus sign in !$\mathcal{E}^{\rm NR}$!, which makes it impossible to apply standard arguments for convex functionals. Lieb overcame the lack of convexity by using the theory of symmetric decreasing functions. Later Lions (1980) proved that the unconstrained problem (0.1) possesses infinitely many solutions. For the constrained problem, seeking radially symmetric, normalized functions !$\| u \|_{L^{2}} =+1$!, or more generally, seeking solutions belonging to:

!$$\mathcal{C}_{N}= \{ \, φ \in \mathcal{H}_{\rm r}^{1} (ℜ^{3}) \, : \, \| φ \|_{L^{2}} =N \, \} ,$$! the situation is much more complicated and conditions on !$W$! are necessary. In the Coulomb case, Lions proves that there exists a sequence !$(l_{j}, u_{j})$!, with !$l_{j} > 0$!, and !$u_{j}$! satisfies !${(0.1)}$! (with !$l=l_{j}$!) and belongs to !$\mathcal{C}_{1}$!

We may replace the negative Laplace operator by the so-called quasi-relativistic operator, i.e., the pseudodifferential operator !$\sqrt{ -δ +m^{2} } -m$!; this is the kinetic energy operator of a relativistic particle of mass !$m \geq 0$!. It is defined via multiplication in the Fourier space with the symbol !$\sqrt{k^{2} +m^{2}} -m$!, which is frequently used in relativistic quantum physics models as a suitable replacement of the full (matrix valued) Dirac operator. The associated time-dependent equation arises as an effective dynamical description for an !$N$!-body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity, as recently shown by Elgart and Schlein (2007). This system models a Boson star .

Several questions arise for the quasi-relativistic Choquard equation (existence, uniqueness, positive solutions etc) and the first part of the project would be to get acquainted with recent (related) results, e.g., [1] and [2].

!$[1]$! S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift fr Angewandte Mathematik und Physik (ZAMP) , vol. 63 (2012), 233-248.

!$[2]$! M. Melgaard, F. D. Zongo, Multiple solutions of the quasi relativistic Choquard equation, J. Math. Phys. !${53}$!(2012), 033709 (12 pp).

The purpose is to study neural networks and deep learning, applied to a specific real-world problem. Projects include applications to Finance (deep hedging, calibration, option pricing, etc.), Quantum Physics/Chemistry (deep variational Monte Carlo simulations, solving (generalized) eigenvalue problem for the Schrödinger equation), and other applied topics where we need to solve Partial Differential Equations (PDEs).

Bibliogrpahy

[1] J. Berner, P. Grohs, G. Kutyniok, P. Petersen, The modern mathematics of deep learning. Mathematical aspects of deep learning, 1–111, Cambridge Univ. Press, Cambridge, 2023.

[2] M. López de Prado, Advances in Financial Machine Learning, J. Wiley and Sons, Ltd, 2018.

[3] E. O. Pyzer-Knapp, M. Benatan, Deep Learning for Physical Scientists: Accelerating Research with Machine Learning, John Wiley and Sons Ltd, 2021.

Tensor methods are increasingly finding significant applications in deep learning, computer vision, and scientific computing. Possible projects include image classification, image reconstruction, noise filtering, sensor measurements, low memory optimization, solving PDEs, supervised/unsupervised learning, grid-search and/or DMRG-type algorithms, hidden Markov models, convolutional rectifier networks, neuroscience (neural data, medical images etc), biology (genomic signal processing, low-rank tensor model for gene–gene interactions) or multichannel EEG signals.

Quantum physics (grid-based electronic structure calculations using tensor decomposition approach etc.), Latent Variable Models (community detection through tensor methods, topic models (say, co-occurrence of words in a document), latent trees etc),

Bibliography

[1] W. Hackbusch, Tensor spaces and numerical tensor calculus. Springer, Cham, 2019.

[2] T. G. Kolda, B, W. Bader, Tensor decompositions and applications. SIAM Rev. 51 (2009), no. 3, 455--500.

[3] Y. Panagakis, J. Kossaifi et al, Tensor Methods in Computer Vision and Deep Learning, Proc. IEEE 109 (2021), no. 5, 863-890.

For more information, please email Prof Veronica Sanz or visit her staff profile

PLEASE NOTE THAT PROF SANZ IS NOT AVIALABLE FOR PROJECT SUPERVISION IN 19/20

In High Energy Particle Physics we contrast data with new theories of Nature. Those theories are proposed to solve mysteries such as 1.) what is the Dark Universe made of, 2.) why there is so much more matter than antimatter in the Universe, and 3.) how can a light Higgs particle exist.

To answer these questions, we propose mathematical models and compare with observations. Sources of data are quite varied and include complex measurements from the Large Hadron Collider, underground Dark Matter detection experiments and satellite information on the Cosmic Microwave Background. We need to incorporate all this data in a framework which allows us to test hypotheses, and this is usually done via a statistical analysis, e.g. Bayesian, which provides a measure of how well a hypothesis can explain current observations. Alas, this approach has so far been unfruitful and is driving the field of Particle Physics to an impasse.

In this project, we will take a different and novel approach to search for new physics. We will assume that our inability to discover new physics stems from strong theoretical biases which have so far guided analyses. We will instead develop unsupervised searching techniques, mining on data for new phenomena, avoiding as much theoretical prejudices as possible. The project has a strong theoretical component, as the candidate will learn the mathematical/physical basis of new physics theories including Dark Matter, the Higgs particle and Inflation. The candidate will also learn about current unsupervised-learning techniques and the interpretation of data in High-Energy Physics.

The strategy adopted for this project holds the potential to open a new avenue of research in High Energy Physics. We are convinced that this departure from conventional statistical analyses mentioned above is the most effective way to discover new physics from the huge amount of data produced in the Large Hadron Collider and other experiments of similar scale.

Reaching the scientific goals outlined here would require modelling huge amounts of data at different levels of purity (raw measurements, pseudo-observables, re-interpreted data), and finding patterns which had not been detected due to a focus on smaller sets of information. Hence, we believe that research into unsupervised learning in this context will have far reaching applications beyond academic pursuits. As the world becomes increasingly data-orientated, so does our reliance on novel algorithms to make sense of the information we have in our possession. To give some examples, we can easily expect the development of unsupervised learning integrated into facial recognition software and assist in the discovery of new drugs, which provides a boost in the security and medical sector respectively.

For more information, please email Dr Nick Simm or visit his staff profile

Simm: Random matrix theory and the Riemann zeta function [PDF 156.59KB]

Simm: Asymptotic analysis of integrals and applications [PDF 124.34KB]

For more information, please email Dr Ali Taheri or visit his staff profile

The study of boundary behaviour of holomorphic functions in the unit disc is a classical subject which has been revived and generalised to higher dimensions as well as other geometries due to recent developments in the theory of ellipic PDEs, e.g., one such development being the H1 and BMO duality.

The aim of this project is more modest and lies in understanding the interplay between holomorphic functions in the disc on the one hand and the Poisson integral of Borel measures on the boundary circle. The results here lead to surprising qualitative properties of holomorphic functions.

Key words: Poisson integrals, Nevanlinna class, Non-tangential convergence, M&F Riesz theorem

Recommended modules: Complex Analysis, Functional Analysis, Measure Theory

!$[1]$! Real and Complex Analysis by Rudin

!$[2]$! Introduction to !$H^p$! spaces by Koosis

!$[3]$! Theory of !$H^p$! spaces by Duren

!$[4]$! Bounded Analytic Functions by Garnett.

Fourier analysis has been one of the major sources of interesting and fundamental problems in analysis. It alone plays one of the most significant roles in the development of mathematical analysis in the past 2 centuries.

The aim of this project is to study Fourier series, specifically in the context of: !$L^2$! -- the Hilbert space approach, continuous functions, and !$L^p$! with !$1 < p < ∞$!.

Particular emphasis goes towards the convergence/divergence properties using Functional analytic tools, Baire category arguments, singular integrals.

Key words: !$L^p$! spaces, Summability kernels, Baire category, Singular integrals, Hilbert transform

Recommended modules: Complex Analysis, Functional Analysis, Measure and Integration

!$[1]$! Fourier Analysis, T.W. Koner, Cambridge University Press, 1986

!$[2]$! Real and Complex Analysis, W. Rudin, McGraw Hill, 1987

!$[3]$! Real Variable Methods in Harmonic Analysis, A. Torchinsky, Dover, 1986.

In the theory of nonlinear partial differential equations the study of the oscillation and concentration phenomenon plays a key role in settling the question of the existence of solutions. Here the aim is to understand the basics of weak versus strong convergence for sequences of functions and to introduce a tool known as Young measures for detecting the mechanisms that could prevent strong convergence.

Key words: Young measures, Weak convergence, Div-Curl lemma

Recommended modules: Partial Differential Equations, Functional Analysis, Measure Theory

!$[1]$! Parameterised Measures and Variational Principles, P.Pedregal, Birkhäuser, 1997.

!$[2]$! Partial Differential Equations, L.C. Evans, AMS, 2010.

!$[3]$! Weak Convergence Methods in PDEs, L.C. Evans, AMS, 1988.

Harmonic maps between manifolds are extremals of the Dirichlet energy. It is well-known that depending on the topology and global geometry of the domain and target manifolds these harmonic maps can develop singularities in all forms and shapes. The aim of this project is to introduce the student to the theory and some of the basic ideas and important tools involved.

Key words: Harmonic maps, Dirichlet energy, Minimal connections, Singular cones.

Recommended modules: Partial Differential Equations, Introduction to Topology, Algebraic Topology, Functional Analysis

!$[1]$! Infinite dimensional Morse theory by Chang

!$[2]$! Two reports on Harmonic maps by Eells and Lemaire

!$[3]$! Cartesian Currents in the Calculus of Variations by Giaquinta, Modica and Soucek.

For more information, please email James Van Yperen or visit his staff profile

Mathematical models found in nature are typically stochastic in nature, and thus difficult to calibrate and analyse. Birth-death processes (a model to describe population evolution) are no different, however they are Markovian – that is, they satisfy the Markov property. Under some assumptions about the size of the population, one can take expectation of the process and derive ODEs for the mean of the population over time. In this project, we will develop a parameter estimation framework to calibrate the ODE to some given data. Dependent on the student’s interest, we can look at nonlinear birth-death processes, parameter identifiability issues and analysis, or deriving an ODE for the variance and improving the calibration method.

Recommended Modules:

Advanced Numerical Analysis, Programming through Python, Statistical Inference

Other helpful modules:

Introduction to Mathematical Biology, Probability Models, Random Processes

[1] : Raol JR, Girija G, Singh J. Modelling and parameter estimation of dynamic systems. Iet; 2004 Aug 13.

[2] : Jones DS, Plank M, Sleeman BD. Differential equations and mathematical biology. CRC press; 2009 Nov 9.

[3] : Stortelder WJ. Parameter estimation in dynamic systems. Mathematics and Computers in Simulation. 1996 Oct 1;42(2-3):135-42.

[4] : Allen L. An introduction to mathematical biology. Prentice Hall, 2007.

Curve shortening flow, mean curvature flow for curves, is a mathematical phenomenon where a curve is moving in a direction and velocity proportional to its own curvature. Formally, it is a type of geometric partial differential equation. By parametrising the curve, one derives a nonlinear first order in time and second order in arc-length partial differential equation for the coordinates of the curve as it moves over time. In this project we will be looking at the numerical simulation of curve shortening flow for different curves using the finite element method, which will involve the derivation programming of a finite element scheme. Depending on the student’s interest, we can look into using linear and quadratic elements, different types of parametrisations, or conduct some finite element analysis on a simpler parabolic PDE.

Advanced Numerical Analysis, Numerical Solution to Partial Differential Equations, Partial Differential Equations, Programming through Python

[1] Deckelnick K, Dziuk G, Elliott CM. Computation of geometric partial differential equations and mean curvature flow. Acta numerica. 2005 May;14:139-232.

[2] Brenner SC. The mathematical theory of finite element methods. Springer; 2008.

[3] Thomée V. Galerkin finite element methods for parabolic problems. Springer Science & Business Media; 2007 Jun 25.

[4] Barrett JW, Garcke H, Nürnberg R. Parametric finite element approximations of curvature-driven interface evolutions. InHandbook of numerical analysis 2020 Jan 1 (Vol. 21, pp. 275-423). Elsevier.

For more information, please email Dr Chandrasekhar Venkataraman or visit his staff profile

The formation of structure or patterns from homogeneity is ubiquitous in biological systems such as the intricate markings on sea shells, pigment patterns on the wings of butterflies and the regular structures made by populations of cells. Their is a rich theory for mathematical modelling of these phenomena that typically involves systems of PDEs. In this project we will understand and analyse some classical models for pattern formation and then extend them to take into account phenomena such as non-local interactions or growth and curvature. Dependent on the interests of the student we will either focus on the approximation of the models or their analysis.

Recommended modules: Introduction to Mathematical Biology, Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++

!$[1]$! Turing, A. M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B

!$[2]$! Murray JD (2013) Mathematical Biology II: Spatial Models and Biomedical Applications. Springer New York

!$[3]$! Kondo, S.,and Miura, T. (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science.

!$[4]$! Plaza, R. G., Sanchez-Garduno, F., Padilla, P., Barrio, R. A., & Maini, P. K. (2004). The effect of growth and curvature on pattern formation. Journal of Dynamics and Differential Equations

Mathematical modelling, analysis and simulation can help us understand a number of cell biological questions such as, How do cells move? How do they interact with their environment and each other? How do cell scale interactions influence tissue level phenomena? In this project we will review and extend models for either cell migration, receptor-ligand interactions or cell signalling. The models typically involve geometric PDE with coupled systems of equations posed in different domains, cell interior, cell-surface, extracellular space. Dependent on the interests of the student we will either focus on the derivation, the approximation, or the analysis of the models.

!$[1]$! Elliott, C. M., Stinner, B., and Venkataraman, C. (2012). Modelling cell motility and chemotaxis with evolving surface finite elements. Journal of The Royal Society Interface

!$[2]$! Croft, W., Elliott, C. M., Ladds, G., Stinner, B., Venkataraman, C., and Weston, C. (2015). Parameter identification problems in the modelling of cell motility. Journal of mathematical biology

!$[3]$! Elliott, C. M., Ranner, T., and Venkataraman, C. (2017). Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics. SIAM Journal on Mathematical Analysis

!$[4]$! Ptashnyk, M., and Venkataraman, C. (2018). Multiscale analysis and simulation of a signalling process with surface diffusion. arXiv preprint

For more information, please email Vladislav Vysotskiy >

The topic of this project is at the intersection of probability, ergodic theory, number theory, and dynamical systems. It is well-known that any real number can be represented by its decimal, binary, ternary, etc. expansion. But what if we try to expand in a non-integer basis? Such expansions are known as beta-expansions. Do they have the same properties as the usual ones? For example, what can we say about frequencies of digits? Are all patterns of digits possible? Does every real number have a unique beta-expansion? The project aims to address questions of such type to study basic properties of beta-expansions.

[1] A. Renyi. Representations for real numbers and their ergodic properties (1957)

[2] W. Parry. Representations for real numbers (1964)

The topic of this project is at the intersection of probability and measure theory. The Bernoulli convolution is the distribution of a power series in x whose coefficients are independent identically distributed Bernoulli(1/2) random variables. These distributions have surprising different properties depending on the value of x, e.g. they are singular for all 0<x<1/2 and have density for almost all (but not all!) 1/2<x

What are the properties that make certain values of x special? What happens at such x? Is there any connection with the famous Cantor function (aka Devil’s staircase)? The project aims to address questions of such type to study basic properties of Bernoulli convolutions.

[1] B. Solomyak. Notes on Bernoulli convolutions (2017)

For more information, please email Dr Minmin Wang or visit her staff profile

Minmin Wang - Probabilistic and combinatorial analysis of coalescence [PDF 64.35KB]

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MSc Mathematical & Computational Finance: sample dissertations

Below are some examples of MSc dissertations from previous years, which received high marks:

• Optimal Strategies from forward versus classical utilities
• Robust Pricing of Derivatives on Realised Variance
• Log Mean-Variance Portfolio Theory and Time Inconsistency
• Multilayer network valuation under bail-in
• Topological Persistence in Market Micro Structure
• Volatility is Rough
• Deep learning approach to hedging (2019 prize for best Master’s Thesis in Quantitative Finance by Natixis Foundation for Research & Innovation.)
• Risk Management with Generative Adversarial Networks.pdf  (Won award for best Masters Thesis in Quantitative Finance in Europe in 2020)
• Hawkes Process-Driven Models for Limit Order Book Dynamics.pdf   (Won award for best Masters Thesis in Quantitative Finance in Europe in 2021)
• Evaluating Credit Portfolios under IFRS 9 in the UK Economy

• Our Mission

11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes. 

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!” 

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

HELP OTHER TEACHERS OUT!

We’d love this article to be an evolving document of lesson ideas that make math relevant to kids. So, teachers, please tell us about your go-to activities that connect math to kids’ real world experiences.