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

Mathematics and Statistics Theses and Dissertations

Theses/dissertations from 2024 2024.

The Effect of Fixed Time Delays on the Synchronization Phase Transition , Shaizat Bakhytzhan

On the Subelliptic and Subparabolic Infinity Laplacian in Grushin-Type Spaces , Zachary Forrest

Utilizing Machine Learning Techniques for Accurate Diagnosis of Breast Cancer and Comprehensive Statistical Analysis of Clinical Data , Myat Ei Ei Phyo

Quandle Rings, Idempotents and Cocycle Invariants of Knots , Dipali Swain

Comparative Analysis of Time Series Models on U.S. Stock and Exchange Rates: Bayesian Estimation of Time Series Error Term Model Versus Machine Learning Approaches , Young Keun Yang

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

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Mathematics PhD theses

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

2023   2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Reham Alahmadi - Asymptotic Study of Toeplitz Determinants with Fisher-Hartwig Symbols and Their Double-Scaling Limits

Anne Sophie Rojahn –  Localised adaptive Particle Filters for large scale operational NWP model

Melanie Kobras –  Low order models of storm track variability

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

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

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

Samuel R Harrison – Stalactite Inspired Thin Film Flow

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

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

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

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

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

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

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

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

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

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

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

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

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

James Jackaman  – Finite element methods as geometric structure preserving algorithms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mark Parsons - Mathematical Modelling of Evolving Networks

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

David Gilbert - Analysis of large-scale atmospheric flows

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

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

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

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

Ashley Twigger - Boundary element methods for high frequency scattering

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

M. Henderson - Extending the edge-colouring of graphs

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

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

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

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

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

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

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

M. Johnson - Some problems on graphs and designs

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

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

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

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

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

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

J. Hudson - Numerical Techniques for Morphodynamic Modelling

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

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

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

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

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

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

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

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

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

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

I. Sciriha - On Some Aspects of Graph Spectra.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Deep Learning for Studying Materials Stability and Solving Thermodynamically Consistent PDES With Dynamic Boundary Conditions in Arbitrary Domains , Chunyan Li

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

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Assouad-type dimensions and the local geometry of fractal sets , separability properties of semigroups and algebras , groups defined by language theoretic classes , rearrangement groups of connected spaces , modern computational methods for finitely presented monoids .

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Applied and Computational Mathematics Master's Thesis - 625.803

This is the first in a two-course sequence (EN.625.803 and EN.625.804) designed for students in the master’s program who wish to work with a faculty advisor to conduct significant, original independent research in the field of applied and computational mathematics. (Each course is one semester.) A sequence may be used to fulfill two courses within the 700-level course requirements for the master’s degree; only one sequence may count toward the degree. For sequence 625.803 and 625.804, the student is to produce a bound hard-copy thesis for submission to the JHU library and an electronic version of the thesis based on standards posted at https://www.library.jhu.edu/library-services/electronic-theses-dissertations/. (The student is also encouraged to write a technical paper for publication based on the thesis.) The intent of the research is to expand the body of knowledge in the broad area of applied mathematics, with the research leading to professional-quality documentation. A full description of the guidelines (which includes the list of approved ACM research faculty) and the approval form can be found at https://ep.jhu.edu/current-students/student-forms/.

Course Prerequisite(s)

Completion of at least six courses towards the Master of Science, including EN.625.601 Real Analysis and/or EN.625.609 Matrix Theory, EN.625.603 Statistical Methods and Data Analysis, and at least one of the following three two-semester sequences: EN.625.717–EN.625.718 Advanced Differential Equations: Partial Differential Equations and Nonlinear Differential Equations and Dynamical Systems, EN.625.721– EN.625.722 Probability and Stochastic Processes I and II, or EN.625.725– EN.625.726 Theory of Statistics I and II. It is recommended that the sequence represent the final two courses of the degree.

Course Offerings

There are no sections currently offered, however you can view a sample syllabus from a prior section of this course.

Senior Theses

Available in PDF format (linked below) and in hard copy from the  Library  (through 2010, more recent ones available soon). For more information, see  Senior Thesis Guidelines.

Timothy Steven Murray,  Spectral Asymptotics of the Laplacian on Surfaces of Constant Curvature Thesis Advisor: Robert Strichartz,

January 2015

Matthew Stuart Farrell,  The Halting Program for Chip-Firing on Finite Directed Graphs Thesis Advisor: Lionel Levine,

Walter Cai,  Extremal 1-Cycle Lifespan Behavior under the Vietoris-Rips Complex Thesis Advisor: Robert Strichartz,

Sy yeu Chou,  Numerical Simulations of N Point Vortices on Bounded Domains Thesis Advisor: Robert Strichartz,

Eric Primozic, Homogeneous Groups and Covers Thesis Advisor: R. Keith Dennis,

Thomas Brooks,  Generating Sets of Mathieu Groups Thesis Advisor: R. Keith Dennis,

Weilin Li,  Boundary Value Problems on a Half Sierpinski Gasket Thesis Advisor: Robert Strichartz,

Yiran Li,  Hodge deRham Theory of Carpet Type Fractals Thesis Advisor: Robert Strichartz,

Min Lin,  Strongly Flat p-Groups Thesis Advisor: R. Keith Dennis,

Diwakar Raisingh,  Toward an Axiomatic Characterization of the Smash Sum Thesis Advisor: Lionel Levine,

Robert Ravier,  Average Values of Functions on the Sierpinski Gasket Thesis Advisor: Robert Strichartz,

Yipu Wang,  Maximal Unbalanced Families Thesis Advisors: Louis Billera, ; Justin Moore,

Sudesh Kalyanswamy,  Inverse Galois Problem for Totally Real Number Fields Thesis Advisor: Ravi Ramakrishna,

Derek Lougee,  Percolation on the Non-P.C.F. Sierpinski Gasket and Hexacarpet Thesis Advisor: Benjamin Steinhurst,

Benjamin Nachman,  Generating Sequences of the Two Dimensional Special Projective Linear Group over Fields of Prime Order , PSL(2,p) Thesis Advisor: R. Keith Dennis,

Thomas Rudelius,  A Geometric Understanding of Ricci Curvature in the Context of Pseudo-Riemannian Manifolds Thesis Advisor: John Hubbard,

Liang Ze Wong,  Irredundant Generating Sets of Finite Nilpotent Groups Thesis Advisor: R. Keith Dennis,

Tarun Chitra,  Generalized Complex Geometry, Sasakian Manifolds and ADS/CFT Thesis Advisors: Leonard Gross, ; Liam McAllister, Physics

Matthew Guay,  Infinity-Harmonic Functions on SG Thesis Advisor: Robert Strichartz and Alexander Vladimirsky,

Richard Gustavson,  Some New Results on the Combinatorial Laplacian Thesis Advisor: Edward Swartz,

Zheng Kang Tham, Square-Free Matroids Thesis Advisor: Edward Swartz,

Divya Kirti,  An Exploration of Transmission on Random Graphs Thesis Advisor: Richard Durrett,

January 2010

Daniel Jack Collins,  Generating Sequences of Finite Groups Thesis Advisor: R. Keith Dennis,

Kai Fong Ernest Chong,  The Weak Order of Coxeter Systems and Combinatorial Properties of Descent Sets Thesis Advisor: Edward Swartz,

Steven Heilman,  Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet Thesis Advisor: Robert Strichartz,

Wei Quan Julius Poh,  Properties of Posets in Non-crossing Pairings on Bitstrings Thesis Advisor: Edward Swartz,

Jessica Lucille DeGrado,  Computing the Gradient of Laplacian Eigenfunctions on the Sierpinski Gasket (This thesis consists of a joint paper produced with Professor Strichartz and Luke Rogers following Jessica's participation in the department's 2006 REU program.) Thesis Advisor: Robert Strichartz,

Hyun Kyu Kim,  Representation Theoretic Existence Proof for Fischer Group Fi_23 Thesis Advisor: Gerhard O. Michler,

Rami Mohieddine,  Chaos in the Hodgkin-Huxley Equations: The Takens-Bodganov Cusp Bifurcation Thesis Advisor: John Guckenheimer,

Yu Tung Cheng,  Continued Fractions Thesis Advisor: Peter Kahn,

Timothy David Cramer with Farkhod Eshmatov,  Heat Kernel Coefficients for Periodic Schrödinger Operators Thesis Advisor: Yuri Berest,

Rafael Mariano Frongillo,  Topological Entropy Bounds for Hyperbolic Dynamical Systems Thesis Advisor: John Smillie,

Yan Yancy Lo,  Mathematical Models for Salmonella Transmission Dynamics Thesis Advisor: Yrjo Grohn, Cornell Vet School; John Guckenheimer,

Evan Jayson Marshak,  A Survey of Results Regarding a Computational Approach to the Zeros of Dedekind Zeta Functions Thesis Advisor: Ravi Ramakrishna,

Jeremy Kenneth Miller,  Finite Groups Acting on Surfaces as Rigid Motions of Euclidean Space Thesis Advisor: Allen Hatcher,

Thomas Franklin Church,  The Magnus Representation of the Torelli Group I_{g,1} Thesis Advisor: Karen Vogtmann,

Wai Wai Liu,  The Logarithmic Sobolev Constant of Some Finite Markov Chains Thesis Advisor: Laurent Saloff-Coste,

Philip Daniel Owrutsky,  Orthogonal Polynomials with Respect to Self-Similar Measures Thesis Advisor: Robert Strichartz,

Saúl A. Blanco Rodríguez,  Some Results on Combinatorial Game Theory Thesis Advisor: Michael Morley,

Vorrapan Chandee with Tian Tian Qiu,  Counting Kneading Sequences Thesis Advisor: Rodrigo Pérez,

Timothy D. DeVries,  The Weak Order and Flag h-Vector Inequalities Thesis Advisor: Edward Swartz,

Shawn Drenning,  Fractal Analysis: Extending the Domain Thesis Advisor: Robert Strichartz,

Tian Tian Qiu with Vorrapan Chandee,  Counting Kneading Sequences Thesis Advisor: Rodrigo Pérez,

Matthew J. Hirn,  The Refinability of Step Functions Thesis Advisor: Robert Strichartz,

Elizabeth Rach,  Power Law Distributions of Gene Family Sizes Thesis Advisor: Richard Durrett,

BYU ScholarsArchive

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

Mathematics Education Theses and Dissertations

Theses/dissertations from 2024 2024.

Rigorous Verification of Stability of Ideal Gas Layers , Damian Anderson

Documentation of Norm Negotiation in a Secondary Mathematics Classroom , Michelle R. Bagley

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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• Senior Thesis

A thesis is a more ambitious undertaking than a project. Most thesis writers within Applied Mathematics spend two semesters on their thesis work, beginning in the fall of senior year.  Students typically enroll in Applied Mathematics 91r or 99r (or Economics 985, if appropriate) during each semester of their senior year.  AM 99r is graded on a satisfactory/unsatisfactory basis.  Some concentrators will have completed their programs of study before beginning a thesis; in situations where this is necessary, students may take AM 91r for letter-graded credit, for inclusion in Breadth section (v) of the plan of study.  In the spring semester, the thesis itself may serve as the substantial paper on which the letter grade is based.  Econ 985 is also letter-graded, and may be included in the Breadth section of the plan of study in place of AM 91r.

Another, somewhat uncommon option, is that a project that meets the honors modeling requirement (either through Applied Mathematics 115 or 91r) can be extended to a thesis with about one semester of work.  Obviously the more time that is spent on the thesis, the more substantial the outcome, but students are encouraged to write a thesis in whatever time they have. It is an invaluable academic experience.

The thesis should make substantive use of mathematical, statistical or computational modeling,  though the level of sophistication will vary as appropriate to the particular problem context.  It is expected that conscientious attention will be paid to the explanatory power of mathematical modeling of the phenomena under study, going beyond data analysis to work to elucidate questions of mechanism and causation rather than mere correlation. Models should be designed to yield both understanding and testable predictions. A thesis with a suitable modeling component will automatically satisfy the English honors modeling requirement; however a thesis won't satisfy modeling Breadth section (v) unless the student also takes AM 91r or Econ 985.

Economics 985 thesis seminars are reserved for students who are writing on an economics topic. These seminars are full courses for letter-graded credit which involve additional activities beyond preparation of a thesis. They are open to Applied Mathematics concentrators with suitable background and interests.

Students wishing to enroll in AM 99r or 91r should follow the application instructions on my.harvard.

Thesis Timeline

The timeline below is for students graduating in May. The thesis deadline for May 2025 graduates is Friday, March 28, 2025 at 2:00PM. For off-cycle students, a similar timeline applies, offset by one semester. The thesis due date for March 2025 graduates is Friday, November 22, 2024 at 2:00PM. Late theses are not accepted.

Mid to late August:

Students often find a thesis supervisor by this time, and work with their supervisor to identify a thesis problem. Students may enroll in Econ 985 (strongly recommended when relevant), AM 91r, or AM 99r to block out space in their schedule for the thesis.

Early December:

All fourth year concentrators are contacted by the Office of Academic Programs. Those planning to submit a senior thesis are requested to supply certain information. This is the first formal interaction with the concentration about the thesis.

Mid-January:

A tentative thesis title approved by the thesis supervisor is required by the concentration.

Early February:

The student should provide the name and contact information for a recommended second reader, together with assurance that this individual has agreed to serve. Thesis readers are expected to be teaching faculty members of the Faculty of Arts and Sciences or SEAS. Exceptions to this requirement must be first approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies. For AM/Economics students writing a thesis on a mathematical economics topic for the March thesis deadline, the second reader will be chosen by the Economics Department. For AM/Economics students writing for the November deadline, the student should recommend the second reader.

On the thesis due date:

Thesis due at 2pm. Late theses are not accepted. Electronic copies in PDF format should be delivered by the student to the two readers and to [email protected] (which will forward to the Directors of Undergraduate Studies, Associate and Assistant Director of Undergraduate Studies) on or before that date and time. An electronic copy should also be submitted via the SEAS  online submission tool  on or before that date. SEAS will keep this electronic copy as a non-circulating backup and will use it to print a physical copy of the thesis to be deposited in the Harvard University Archives. During this online submission process, the student will also have the option to make the electronic copy publicly available via DASH, Harvard’s open-access repository for scholarly work.

Contemporaneously, the two readers will receive a rating sheet to be returned to the Office of Academic Programs before the beginning of the Reading Period, together with their copy of the thesis and any remarks to be transmitted to the student.

The Office of Academic Programs will send readers' comments to the student in late May, after the degree meeting to decide honors recommendations.

Thesis Readers

The thesis is evaluated by two readers, whose roles are further delineated below.  The first reader is the thesis adviser.  The second and reader is recommended by the student and adviser, who should secure the agreement of the individual concerned to serve in this capacity.  The reader must be approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  The second reader is normally are teaching members of the Faculty of Arts and Sciences, but other faculty members or comparable professionals will usually be approved, after being apprised of the responsibilities they are assuming.   For theses in mathematical economics, the choice of the second reader is made in cooperation with the Economics department.  The student and thesis adviser will be notified of the designated second reader by mid-March.

The roles of the thesis adviser and of the outside reader are somewhat different.  Ideally, the adviser is a collaborator and the outside reader is an informed critics.  It is customary for the adviser's report to comment not only on the document itself but also on the background and context of the entire effort, elucidating the overall accomplishments of the student.  The supervisor may choose to comment on a draft of the thesis before the final document is submitted, time permitting.  The outside reader is being asked to evaluate the thesis actually produced, as a prospective scientific contribution — both as to content and presentation.  The reader may choose to discuss their evaluation with the student, after the fact, should that prove to be mutually convenient.

The thesis should contain an informative abstract separate from the body of the thesis.  At the degree meeting, the Committee on Undergraduate Studies in Applied Mathematics will review the thesis, the reports from the two readers and the student’s academic record. The readers (and student) are told to assume that the Committee consists of technical professionals who are not necessarily conversant with the subject matter of the thesis so their reports should reflect this audience.

The length of the thesis should be as long as it needs to be to make the arguments made, but no longer!

Thesis Examples

The most recent thesis examples across all of SEAS can be found on the Harvard DASH (Digital Access to Scholarship at Harvard) repository . Search the FAS Theses and Dissertations collection for "applied mathematics" to find dozens of examples.

Note: Additional samples of old theses can be found in McKay Library. Theses awarded Hoopes' Prizes can be found in Lamont Library.

Recent thesis titles

Theses submitted in 2024.

Theses submitted in 2023

Senior Thesis Submission Information for A.B. Programs

Senior A.B. theses are submitted to SEAS and made accessible via the Harvard University Archives and optionally via  DASH  (Digital Access to Scholarship at Harvard), Harvard's open-access repository for scholarly work.

In addition to submitting to the department and thesis advisors & readers, each SEAS senior thesis writer will use an online submission system to submit an electronic copy of their senior thesis to SEAS; this electronic copy will be kept at SEAS as a non-circulating backup. Please note that the thesis won't be published until close to or after the degree date. During this submission process, the student will also have the option to make the electronic copy publicly available via DASH.  Basic document information (e.g., author name, thesis title, degree date, abstract) will also be collected via the submission system; this document information will be available in  HOLLIS , the Harvard Library catalog, and DASH (though the thesis itself will be available in DASH only if the student opts to allow this). Students can also make code or data for senior thesis work available. They can do this by posting the data to the Harvard  Dataverse  or including the code as a supplementary file in the DASH repository when submitting their thesis in the SEAS online submission system.

Whether or not a student opts to make the thesis available through DASH, SEAS will provide an electronic record copy of the thesis to the Harvard University Archives. The Archives may make this record copy of the thesis accessible to researchers in the Archives reading room via a secure workstation or by providing a paper copy for use only in the reading room.  Per University policy , for a period of five years after the acceptance of a thesis, the Archives will require an author’s written permission before permitting researchers to create or request a copy of any thesis in whole or in part. Students who wish to place additional restrictions on the record copy in the Archives must contact the Archives  directly, independent of the online submission system. 

Students interested in commercializing ideas in their theses may wish to consult Dr. Fawwaz Habbal , Senior Lecturer on Applied Physics, about patent protection. See Harvard's policy for information about ownership of software written as part of academic work.

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Thesis Defenses

Julius baldauf.

Date: Thursday, March 28, 2024 | 2:10pm | Room: 2-449 | Zoom Link

Committee: Bill Minicozzi (Thesis Advisor and Examination Committee Chair), Tristan Collins, Tristan Ozuch

The Ricci Flow on Spin Manifolds

This thesis studies the Ricci flow on manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

Deeparaj Bhat

Date: Tuesday, June 25, 2024 | 3:00pm | Room: 2-136 | Zoom Link

Committee: Tomasz Mrowka (Advisor and Chair), Daniel Alvarez-Gavela and John Baldwin (Boston College)

Surgery Exact Triangles in Instanton Theory

The introduction of instanton Floer theory and Donaldson polynomial invariants in the 1980s revolutionised the study of low dimensional topology. Since then, many Floer theories have been introduced with different structural properties and qualitative features. One of these Floer theories, Heegaard Floer theory, is popular due to its computational ease and rich algebraic structure. One of the computational tools absent in other Floer theories is the integer surgery formula that computes Heegaard Floer homology of 3-manifolds obtained by surgery along knot(s) in them.

This thesis establishes a new surgery formula in instanton Floer theory. The algebraic language to express this formula is that of the derived category of chain complexes. The first part of the thesis describes this surgery formula whose statement and proof are inspired by the Atiyah-Floer conjectures. The second part then contrasts with the Heegaard Floer analogue by showing that instanton and Heegaard Floer theory cannot agree over integers.

Date: Tuesday, April 30, 2024 | 3:00pm | Room: 4-149 | Zoom Link

Committee: Alexander Rakhlin (advisor), Yury Polyanskiy, Martin Wainwright, Ankur Moitra (chair)

Smoothed Online Learning: Theory and Applications

Many of the algorithms and theoretical results surrounding modern machine learning are predicated on the assumption that data are independent and identically distributed. Motivated by the numerous applications that do not satisfy this assumption, many researchers have been interested in relaxations of this condition, with online learning being the weakest such assumption. In this setting, the learner observes data points one at a time and makes predictions, before incorporating the data into a training set with the goal of predicting new data points as well as possible. Due to the lack of assumptions on the data, this setting is both computationally and statistically challenging. In this thesis, we investigate the statistical rates and efficient algorithms achievable when the data are constrained in a natural way motivated by the smoothed analysis of algorithms. The first part covers the statistical rates achievable by an arbitrary algorithm without regard to efficiency, covering both the fully adversarial setting and the constrained setting in which improved rates are possible. The second part of the thesis focuses on efficient algorithms for this constrained setting, as well as special cases where bounds can be improved under additional structure. Finally, in the third part we investigate applications of these techniques to sequential decicions making, robotics, and differential privacy. We introduce a number of novel techniques, including a Gaussian anti-concentration inequality and a new norm comparison for dependent data.

Gonzalo Cao

Date: Monday, July 1, 2024 | 10:30am | Room: 2-361 | Zoom Link

Committee: Prof. Gigliola Staffilani (advisor and committee chair), Prof. Semyon Dyatlov and Prof. Larry Guth

Self-similar singularity formation and wellposedness theory for compressible fluids and dispersive PDE

Murilo Corato Zanarella

Date: Tuesday, April 23, 2024 | 11:00am | Room: 4-370

Committee: Wei Zhang, Zhiwei Yun and Spencer Leslie (Boston College)

First explicit reciprocity law for unitary Friedberg—Jacquet periods

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin—Selberg, with applications to the Bloch—Kato conjecture and to Iwasawa theory.

This thesis studies the case of Galois representations attached to automorphic representations on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). We prove a new "first explicit reciprocity law" in this setting, which has applications to the rank 0 case of the corresponding Bloch—Kato conjecture.

Date: Wednesday, April 24, 2024 | 3:00pm | Room: 2-142

Committee: Wei Zhang, Julee Kim, Zhiwei Yun

Local newforms and spherical characters for unitary groups

We first prove a smooth transfer statement analogous to Jacquet–Rallis’s fundamental lemma and use it to compute the special value of a local spherical character that appears in the Ichino–Ikeda conjecture at a test vector. Then we provide a uniform definition of newforms for representations of both even and odd dimensional unitary groups over p-adic fields. This definition is compatible with the one given by Atobe, Oi, and Yasuda in the odd dimensional case. Using the nonvanishing of the local spherical character at the test vector, we prove the existence of the representation containing newforms in every tempered Vogan L-packet. We also show the uniqueness of such representations in Vogan L-packets and give an explicit description of them using local Langlands correspondence.

Patrik Gerber

Date: Friday, April 26, 2024 | 9:30am | Room: 2-361 | Zoom Link

Committee: Philippe Rigollet (advisor), Yury Polyanskiy, Martin Wainwright

Likelihood-Free Hypothesis Testing and Applications of the Energy Distance

The first part of this thesis studies the problem of likelihood-free hypothesis testing: given three samples X,Y and Z with sample sizes n,n and m respectively, one must decide whether the distribution of Z is closer to that of X or that of Y. We fully characterize the problem's sample complexity for multiple distribution classes and with high probability. We uncover connections to two-sample, goodness of fit and robust testing, and show the existence of a trade-off of the form mn ~ k/ε^4, where k is an appropriate notion of complexity and ε is the total variation separation between the distributions of X and Y. We demonstrate that the family of "classifier accuracy" tests are not only popular in practice but also provably near-optimal, recovering and simplifying a multitude of classical and recent results. We generalize our problem to allow Z to come from a mixture of the distributions of X and Y, and propose a kernel-based test for its solution. Finally, we verify the existence of a trade-off between m and n on experimental data from particle physics.

In the second part we study applications of the energy distance to minimax statistics. We propose a density estimation routine based on minimizing the generalized energy distance, targeting smooth densities and Gaussian mixtures. We interpret our results in terms of half-plane separability over these classes, and derive analogous results for discrete distributions. As a consequence we deduce that any two discrete distributions are well-separated by a half-plane, provided their support is embedded as a packing of a high-dimensional unit ball. We also scrutinize two recent applications of the energy distance in the two-sample testing literature.

Shashi Gowda

Date: Thursday, April 25, 2024 | 10:00am | Room: 32-G882 | Zoom Link

Committee: Alan Edelman, Steven Johnson, John Urschel

Symbolic-numeric programming in scientific computing

Scientific programming languages should pursue two goals: closeness to mathematical notation, and the ability to express efficient numerical algorithms. To meet these goals simultaneously, languages use imperative surface syntaxes that mimick mathematical notation. However, mimicking does not make them the same—mathematics is declarative, pliable, and caters to exploratory human nature; but algorithms are imperative and must cater to machines. Hence, there is a fundamental limit to this approach and we leave the expressive power of the symbolic representation on the table.

In this thesis, we ask the question: How can symbolic and numerical modes of computing co-exist, one informing the other? As an answer, we develop a user-extensible system that lifts numerical code into symbolic expressions and can turn symbolic expressions back into high-quality numerical code at staged compilation time, essentially providing the scientific user a way to generate programs and to treat programs as the symbolic artifacts they are. We identified siloing of symbolic software into 3 categories (one can call them “symbolic-only”, “secretly symbolic”, “secretly numerical”) which currently each reproduce similar forms of symbolic capabilities, but cannot share code between each other. Our work demonstrates that this siloing is not essential and an ecosystem of symbolic-numeric libraries can thrive in symbiosis.

Our system is adaptable to any domain: users can define 1) Symbolic variables of any type 2) the set of primitive (symbolically indivisible) functions in the domain, 3) the propa- gation of partial information, and 4) pattern-based rewrites and simplification rules. There is a tendency in scientific computing to create a “compiler for every problem” starting from scratch every time. Every such effort erects its own towers of symbolic and numerical ca- pabilites. A system like ours eliminates this redundancy and lets every scientific user be a “compiler designer” without any prior knowledge of compiler development.

Alasdair Hastewell

Date: Thursday, April 18, 2024 | 10:30am | Room: 2-449 | Zoom Link

Committee: Jörn Dunkel (chair), John Bush, Alexander Mietke

Robust spectral representations and model inference for biological dynamics

Current developments in automated experimental imaging allow for high-resolution tracking across various scales, from whole animal behavior to tissue scale single-cell trajectories during embryogenesis to spatiotemporal gene expression dynamics or neural dynamics. Transforming these high-dimensional data into effective low-dimensional models is an essential theoretical challenge that enables the characterization, comparison, and prediction of the dynamics within and across biological systems. Spectral mode representations have been used successfully across physics, from quantum mechanics to fluid dynamics, to compress and model dynamical data. However, their use in analyzing biological systems has yet to become prevalent. Here, we present a set of noise-robust, geometry-aware mathematical tools that enable spectral representations to extract quantitative measurements directly from experimental data. We demonstrate the practical utility of these methods by applying them to the extraction defect statistics in signaling fields on membranes of starfish, the inference of partial differential equations directly from videos of active particle dynamics, and the categorization of emergent patterns in spatiotemporal gene expression during bacterial swarming.

An additional challenge occurs for complex biophysical processes where the underlying dynamics are yet to be entirely determined. Therefore, we would like to use the experimental data to infer effective dynamical models directly that can elucidate the system's underlying biological and physical mechanisms. Building on spectral mode representations, we construct a generic computational framework that can incorporate prior knowledge about biological and physical constraints for inferring the dynamics of living systems through the evolution of their mode representations. We apply this framework first to single-cell imaging data during zebrafish embryogenesis, demonstrating how our framework compactly characterizes developmental symmetry breaking and reveals similarities between pan-embryo cell migration and Brownian particles on curved surfaces. Next, we apply the framework to the undulatory locomotion of worms, centipedes, robots, and snakes to distinguish between locomotion behaviors. Finally, we present an extension of the framework to the case of nonlinear dynamics when all relevant degrees of freedom are only partially observed, with applications to neuronal and chemical dynamics.

Arun Kannan

Date: Tuesday, April 23, 2024 | 1:00pm | Room: 1-273 | Zoom Link

Committee: Pavel Etingof (advisor), Roman Bezrukavnikov, Victor Kac

On Lie Theory in the Verlinde Category

A symmetric tensor category (STC) can be thought of as a “home” to do commutative algebra, algebraic geometry, and Lie theory. They are defined by axiomatizing the basic properties of a representation category of a group (or affine supergroup scheme). Are these the only examples of STCs? In characteristic zero, a famous theorem of Deligne states that, assuming a natural growth condition, representation categories of affine supergroup schemes are the only examples. However, the situation is totally different in positive characteristic, and the Verlinde category Verp is the most fundamental counterexample and appears to play a key role in generalizing the theorem of Deligne to positive characteristic. Moreover, Verp contains the category of supervector spaces. The upshot is that the study of Verp provides new algebraic structures and phenomena beyond that afforded by superalgebra and supergeometry but must also generalize what is already known.

In this thesis defense, we will first survey the theory of symmetric tensor categories. Then, we will discuss new algebraic structures that arise from the Verlinde category, including new constructions of exceptional Lie superalgebras and a generalization of Jordan algebras unique to characteristic 5. Finally, we will turn to progress made on generalizing useful algebraic techniques and machinery from the super setting to the Verp setting, like the Steinberg tensor product theorem and notions of polynomial functors.

Daniil Kliuev

Date: Tuesday, April 16, 2024 | 2:30pm | Room: 2-131

Committee: Pavel Etingof, Roman Bezrukavnikov and Ivan Loseu (Yale)

Positive traces and analytic Langlands correspondence

I will describe my results with co-authors in two directions.

The first direction is the problem of classification of positive traces on quantized Coulomb branches. In the most general setting, this problem generalizes the classical problem of describing irreducible unitary representations of real reductive Lie groups. We consider the case of Kleinian singularities of type $A$ and provide a complete classification of positive traces.

The second direction is analytic Langlands correspondence, which is the following. Let $X$ be a smooth irreducible projective curve over $\mathbb{C}$, $G$ be a complex reductive group. On one side of this conjectural correspondence there are $G^{\vee}$-opers on $X$ satisfying a certain topological condition ({\it real} opers), where $G^{\vee}$ is Langlands dual group. On the other side there is joint spectrum of certain operators on $L^2(Bun_G)$, called Hecke operators, where $Bun_G$ is the variety of stable parabolic $G$-bundles on $X$ and $L^2(Bun_G)$ is a Hilbert space of square-integrable half-densities. We prove most of the main conjectures of analytic Langlands correspondence in the case when $G=\operatorname{PGL}_2(\mathbb{C})$ and $X$ either a genus one curve with points or $X$ is $\mathbb{P}^1$ with higher structures at points.

Vasily Krylov

Date: Monday, April 29, 2024 | 9:30am | Room: 2-143

Committee: Roman Bezrukaunikov (advisor), Zhiwei Yun, and Ivan Loseu (Yale)

Geometry and representation theory of symplectic singularities in the context of symplectic duality

This thesis studies the geometry and representation theory of various symplectic resolutions of singularities from different perspectives. Specifically, we establish a general approach to attack the Hikita-Nakajima conjecture and illustrate this approach in the example of ADHM spaces. We also study minimally supported representations of the quantizations of ADHM spaces and provide explicit formulas for their characters. Lastly, we describe the monodromy of eigenvalues of quantum multiplication operators for type A Nakajima quiver varieties by examining Bethe subalgebras in Yangians and linking their spectrum with Kirillov-Reshetikhin crystals.

Jae Hee Lee

Date: Monday, April 1, 2024 | 3:00pm | Room: 2-361 | Zoom Link

Committee: Prof. Paul Seidel (thesis advisor), Prof. Pavel Etingof, Prof. Denis Auroux (External, Harvard)

Equivariant quantum connections in positive characteristic

Date: Tuesday, April 23, 2024 | 1:30pm | Room: 13-1143

Committee: Davesh Maulik, Michael Hopkins, Haynes Miller, and Jeremy Hahn

The algebraic K-theory of the chromatic filtration and the telescope conjecture

Chromatic homotopy theory gives a conceptual framework with which to understand the stable homotopy theory, by decomposing the stable homotopy category into monochromatic pieces. There are two variants of these monochromatic pieces, the K(n) and T(n)-local categories, the former of which is often quite understandable in terms of formal groups of height n, and the latter of which detects the so-called v_n-periodic part of the stable homotopy groups of spheres. I will explain how algebraic K-theory has refined our understanding of these monochromatic pieces. On one hand, algebraic K-theory is an important structural invariant of these categories that 'stably' classifies objects and their automorphisms, and I will explain some tools we have to computationally access the K-theory of these categories. On the other hand, the algebraic K-theory of such categories are interesting as spectra: they detect a lot of information about the stable homotopy groups of spheres and have helped us understand the difference between the T(n) and K(n)-local categories.

Nicholas Liu

Date: Tuesday, July 2, 2024 | 11:00am | Room: 2-142

Committee: John Bush, Matthew Durey, and Rodolfo Ruben Rosales

Orbital stability in a classical pilot-wave system

The hydrodynamic bouncing droplet system, consisting of millimetric droplets bouncing on a vibrating fluid bath, displays many quantum mechanical phenomena on a macroscopic scale. These phenomena include tunnelling, diffraction and wave-like statistics. This thesis focuses on the features responsible for the quantisation of orbital radii, and rationalises this quantisation in terms of the stability of circular orbits arising in the presence of a rotating frame and a central force, and are most pronounced when the waves generated by each bounce decay slowly. The faster the waves decay, the less past impacts affect the droplet’s future dynamics, corresponding to lower path memory. We conduct an analytical investigation into the stability of circular orbits using a generalized theoretical framework that allows for an exploration of classical pilot-wave dynamics both inside and outside the experimentally accessible parameter regime. The exploration of parameter regimes beyond those accessible with the hydrodynamic system reveals much richer orbital dynamics. Our novel mathematical approach allows for evaluation of the integrals appearing in the stability problem in terms of Bessel functions of complex order, and thus facilitates asymptotic expansions of the stability problem in various limits. Within the experimental parameter regime, we demonstrate that in a rotating frame, circular orbits destabilise only via resonant instabilities, for which the growing perturbations oscillate at a frequency that is an integer multiple of the orbital frequency. Conversely, in a central force, non-resonant instabilities arise, for reasons discussed herein. Outside the experimental parameter regime, we show how the non-resonant instability leads to counter-intuitive scenarios; for example, where circular orbits are stabilised by increasing memory. In the limit of vanishing particle inertia, infinite path memory and a linear spring force, we demonstrate the intriguing possibility of infinitely many sharply quantised orbital states, where the allowed orbital radii exist in vanishingly thin intervals, and are stabilised by the combined influence of the time-averaged wave field and spring force. We demonstrate that these sharply quantised orbital states are only stable for higher memory. We then consider the effect of weak external forces on spin states, circular orbits arising in the absence of external forces, and show that the destabilisation of spin states depends in a complex manner on the type of external force applied. Finally, we show that the instability of large circular orbits is related to the in-line speed oscillations of free walking droplets in a manner that is independent of the external force.

Calder Morton-Ferguson

Date: Friday, April 26, 2024 | 1:30pm | Room: 2-449 | Zoom Link

Committee: Roman Bezrukavnikov (advisor), Zhiwei Yun, Ivan Loseu

Kazhdan-Laumon categories and representations

In 1988, D. Kazhdan and G. Laumon constructed the \emph{Kazhdan-Laumon category}, an abelian category $\mathcal{A}$ associated to a reductive group $G$ over a finite field, with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample for $G = SL_3$.

Since the early 2000s, there has been little activity in the study of Kazhdan-Laumon categories, despite them being beautiful objects with many interesting properties related to the representation theory of $G$ and the geometry of the basic affine space $G/U$. In the first part of this thesis, we conduct an in-depth study of Kazhdan-Laumon categories from a modern perspective. We first define and study an analogue of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$ for Kazhdan-Laumon categories and study its combinatorics, establishing connections to Braverman-Kazhdan's Schwartz space on the basic affine space and the semi-infinite flag variety. We then study the braid group action on $D^b(G/U)$ (the main ingredient in Kazhdan and Laumon's construction) and show that it categorifies the \emph{algebra of braids and ties}, an algebra previously studied in knot theory; we then use this to provide conceptual and geometric proofs of new results concerning this algebra.

After Bezrukavnikov and Polishchuk's counterexample to Kazhdan and Laumon's original conjecture, Polishchuk made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension. He suggested that this conjecture could lead toward a proof that Kazhdan and Laumon's construction is well-defined, and he proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the final chapter of this thesis, we prove Polishchuk's conjecture for all types, and prove that Kazhdan and Laumon's construction is indeed well-defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

Matthew Nicoletti

Date: Monday, April 29, 2024 | 2:30pm | Room: 2-361 | Zoom Link

Committee: Alexei Borodin (Advisor, chair), Scott Sheffield, Lauren Williams (Harvard)

Title: Stochastic Dynamics on Integrable Lattice Models

The purpose of this thesis is to present some new results related to the six-vertex and dimer model. One theme is the construction and analysis of Markov processes which are naturally associated to these lattice models. Certain integrability properties of the six-vertex and dimer model, often related to the Yang--Baxter equation, allow for the construction of associated Markov chains. In some cases, these are measure preserving Markov chains on configurations of the lattice model. In other cases, they arise via transfer matrices, after choosing a distinguished time coordinate, as a continuous time degeneration of the "time evolution" of the lattice model itself. It is often the case that the integrability of the underlying lattice model provides powerful tools to study the associated Markov chains or their marginals, which are sometimes Markov chains themselves.

In particular together with coauthors, we construct and analyze Markov chains on six-vertex configurations and on dimer model configurations, both of which are models for surface growth in the (2+1)-dimensional "Anisotropic KPZ" (or "AKPZ") universality class; we construct a Markov chain generalizing "domino shuffling" which samples exactly from a recently introduced probability measure on tuples of interacting dimer configurations; using a version of the usual domino shuffling algorithm, we construct and analyze deterministic "t-embeddings" of certain dimer graphs, which discretize minimal surfaces carrying the conformal structure of the limiting Gaussian free field; we construct stationary measures for several colored interacting particle systems using the Yang—Baxter equation.

Alexander Ortiz

Date: Wednesday, April 24, 2024 | 1:15pm | Room: 2-449 | Zoom Link

Committee: Larry Guth (advisor), David Jerison, Gigliola Staffilani

Sparse Fourier restriction for the cone

If the Fourier transform of F(x) is supported near a segment of the light-cone in R^3, what is the shape of the level sets U(N) = {x in R^3 : |F(x)| > N} for large values of N? In 2000, Thomas Wolff had a creative idea to study a related question based on the method of point-circle duality, and used it in a pivotal way to prove the first sharp L^p-decoupling estimates for the cone in R^3 for large values of p.

I will discuss new weighted L^2 estimates of F(x) which give us insight into the shape of level sets. I will explain how we use some of the same key ideas introduced by Wolff, together with a few new ones in the same spirit. By Wolff's method, our main theorem will partly be an application of a recent circular maximal function estimate due to Pramanik—Yang—Zahl in 2022 from their study of Kaufman-type restricted projection problems.

Date: Wednesday, April 3, 2024 | 3:30pm | Room: 2-449

Committee: Prof. Yufei Zhao (advisor and chair), Prof. Dor Minzer, and Prof. Philippe Rigollet

Random and exact structures in combinatorics

We aim to show various developments related to notions of randomness and structure in combinatorics and probability. One central notion, that of the pseudorandomness-structure dichotomy, has played a key role in additive combinatorics and extremal graph theory. In a broader view, randomness (and the pseudorandomness notions which resemble it along various axes) can be viewed as a type of structure in and of itself which has certain typical and global properties that may be exploited to exhibit or constrain combinatorial and probabilistic behavior.

These broader ideas often come in concert to allow the construction or extraction of exact behavior. We look at three particular directions: the singularity of discrete random matrices, thresholds for Steiner triple systems, and improved bounds for Szemerédi's theorem. These concern central questions of the areas of random matrices, combinatorial designs, and additive combinatorics.

Mehtaab Sawhney

Date: Wednesday, April 17, 2024 | 2:00pm | Room: 2-449

Committee: Yufei Zhao, Dor Minzer, and Philippe Rigollet

Probabilistic and Analytic Methods in Combinatorics

The defense will center on fast algorithms for discrepancy theory. Discrepancy theory is broadly concerned with the following problem; given a set of objects, we aim to partition them into pieces which are “roughly equal”. We will focus specifically on vector balancing: given a set of vectors, one seeks to divide them into two parts with approximately equal sum.

Important results in this area, including Spencer’s six standard deviations suffice and Banaszczyk's results towards the Komlós conjecture, were originally purely existential. However, since work of Bansal from 2010, it has become clear that such existential results can often be made algorithmic. We will explain a pair of such results. The first concerns bounds for online vector balancing obtained via a certain Gaussian fixed point random walk. The second gives an algorithmic form of Spencer's theorem that runs in near input sparsity time.

George Stepaniants

Date: Thursday, April 25, 2024 | 2:30pm | Room: 4-149 | Zoom Link

Committee: Philippe Rigollet, Jörn Dunkel, Sasha Rakhlin

Inference from Limited Observations in Statistical, Dynamical, and Functional Problems

Observational data in physics and the life sciences comes in many varieties. Broadly, we can divide datasets into cross-sectional data which record a set of observations at a given time, dynamical data which follow how observations change in time, and functional data which observe data points over a space (and possibly time) domain. In each setting, prior knowledge of statistical, dynamical systems, and physical theory allow us to constrain the inferences and predictions we make from observational data. This domain knowledge becomes of paramount importance when the data we observe is limited: due to missing labels, small sample sizes, unobserved variables, and noise corruption.

This thesis explores several problems in physics and the life sciences, where the interplay of domain knowledge with statistical theory and machine learning allows us to make inferences from such limited data. We begin in Part I by studying the problem of feature matching or dataset alignment which arises frequently when combining untargeted (unlabeled) biological datasets with low sample sizes. Leveraging the fast numerical methods of optimal transport, we develop an algorithm that gives a state-of-the-art solution to this alignment problem with optimal statistical guarantees. In Part II we study the problem of interpolating the dynamics of points clouds (e.g., cells, particles) given only a few sparse snapshot recordings. We show how tools from spline interpolation coupled with optimal transport give efficient algorithms returning smooth dynamically plausible interpolations. Part III of our thesis studies how dynamical equations of motion can be learned from time series recordings of dynamical systems when only partial observations of these systems are captured in time. Here we develop fast routines for gradient optimization and novel tools for model comparison to learn such physically interpretable models from incomplete time series data. Finally, in Part IV we address the problem of surrogate modeling, translating expensive solvers of partial differential equations for physics simulations into fast and easily-trainable machine learning algorithms. For linear PDEs, our prior knowledge of PDE theory and the statistical theory of kernel methods allows us to learn the Green's functions of various linear PDEs, offering more efficient ways to simulate physical systems.

Date: Wednesday, April 3, 2024 | 2:00pm | Room: 2-255

Committee: Scott Sheffield (advisor), Alexei Borodin, Nike Sun

Conformal welding of random surfaces from Liouville theory

Liouville quantum gravity (LQG) is a natural model describing random surfaces, which arises as the scaling limit for random planar maps. Liouville conformal field theory (LCFT) is the underlying 2D CFT that governs LQG. Schramm-Loewner evolution (SLE) is a random planar curve, which describes the scaling limits of interfaces in many statistical physics models. As discovered by Sheffield (2010), one of the deepest results in random geometry is that SLE curves arises as the interfaces under conformal welding of LQG surfaces.

In this thesis, we present some new results on conformal welding of LQG surfaces as well as their applications towards the theory of SLE. We first define a three-parameter family of random surfaces in LQG which can be viewed as the quantum version of triangles. Then we prove the conformal welding result of a quantum triangle and a two-pointed quantum disk, and deduce integrability results for chordal SLE with three force points.

The second main result is regarding the conformal welding of a multiple number of LQG surfaces, where under several scenarios, we prove that the output surfaces can be described in terms of LCFT, and the random moduli of the surface is encoded in terms of the partition functions for the SLE curves.

The third part is about the conformal welding of the quantum disks with forested boundary, where we prove that this conformal welding gives a two-pointed quantum disk with an independent SLE$_\kappa$ for $\kappa\in(4,8)$. We further extend to the conformal welding of a multiple number of forested quantum disks, where as an application, for $\kappa\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for $\kappa \in (6,8)$ and $N\ge 3$ prior to our work.

The conformal loop ensemble (CLE) is a random collection of planar loops which locally look like SLE. For $\kappa \in (4,8)$, the loops are non-simple and may touch each other and the boundary. As a second application, we derive the probability that the loop surrounding a given point in the non-simple conformal loop ensemble touches the domain boundary.

Danielle Wang

Date: Tuesday, April 23, 2024 | 1:00pm | Room: 4-265

Committee: Wei Zhang (advisor/chair), Julee Kim, Spencer Leslie (Boston College)

Twisted Gan-Gross-Prasad conjecture for unramified quadratic extensions

The global twisted GGP conjecture is a variant of the Gan-Gross-Prasad conjecture for unitary groups, which considers the restriction of an automorphic representation of GL(V) to its subgroup U(V), for a skew-Hermitian space V. It relates the nonvanishing of a certain period integral to the central value of an L-function attached to the representation.

Catherine Wolfram

Date: Tuesday, April 30, 2024 | 3:30pm | Room: 2-255 | Zoom Link

Committee: Scott Sheffield (thesis advisor), Alexei Borodin, Curtis McMullen

Random geometry in two and three dimensions

A central theme in random geometry is the interplay between discrete models and continuum ones that appear in scaling limits. Surprising structure and symmetry often arises in these scaling limits, leading to an interplay between combinatorics, probability, complex analysis, and geometry.

The dimer model is one of the classical lattice models of statistical mechanics and can be defined in any dimension. In the first half of this thesis, we prove a large deviation principle for dimer tilings in three dimensions. This generalizes a two-dimensional result of Cohn, Kenyon, and Propp, and is one of the first results for dimers in any dimension $d>2$. Many ideas and constructions used to study dimers are specific to two dimensions, so our arguments start from a smaller set of tools including Hall's matching theorem, the qualitative description of the Gibbs property, and a double dimer swapping operation.

In the second half of this thesis, we study discrete, geometrically-motivated coordinates called shears on the space of circle homeomorphisms up to M\"obius transformations. The Weil-Petersson Teichm\"uller space is a subspace of this which has been of long-term interest in geometry and string theory and has recent connections to SLE curves in probability. We introduce and study natural $\ell^2$ spaces in terms of shears, and obtain sharp results comparing them to H\"older classes of circle homeomorphisms and the Weil-Petersson class. We also give some preliminary results about i.i.d. Gaussian random shears.

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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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