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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 > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2024 2024.

On Cheeger Constants of Knots , Robert Lattimer

Information Based Approach for Detecting Change Points in Inverse Gaussian Model with Applications , Alexis Anne Wallace

Theses/Projects/Dissertations from 2023 2023

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

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

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

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

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

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

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

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

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

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

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

Symmetric Generation , Ana Gonzalez

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

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

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

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

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

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

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

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

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

Permutation and Monomial Progenitors , Crystal Diaz

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

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

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

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

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

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

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

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

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

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

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

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

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

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

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

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

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

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

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

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

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

LIFE EXPECTANCY , Ali R. Hassanzadah

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

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

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

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

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

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

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

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

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

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

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

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

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

Mathematics Theses and Dissertations

Theses/dissertations from 2023 2023.

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

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

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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Guide to Graduate Studies

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

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

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

University Requirements

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

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

Qualifying Exam

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

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

Minor Thesis

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

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

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

Language Exam

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

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

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

Teaching Requirement

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

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

Final Stages

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

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

• Department of Mathematical Sciences
• Study programmes
• PhD program

Ph.D. Theses (PDF)

Bryan Advocaat, Explicit Overconvergence Rates Related to Eisenstein Series ISBN-number: 978-87-7125-228-6, 2024 TEXT: [ Bryan Advocaat TEXT PDF ].

Máté László Telek, Signed Support of Multivariate Polynomials and Applications ISBN-number: 978-87-7125-227-9, 2024 TEXT: [ Máté László Telek TEXT PDF ]. COVER: [ Máté László Telek COVER PDF ].

Jeffrey Adams, Causal Inference and Causal Discovery with Latent Variables ISBN-number: 978-87-7125-226-2, 2024 TEXT: [ Jeffrey Adams TEXT PDF ]. COVER: [ Jeffrey Adams COVER PDF ].

August Bjerg, Periodic Phenomena in the Theory of Large Atoms ISBN-number: 978-87-7125-225-5, 2024 TEXT: [ August Bjerg TEXT PDF ]. COVER: [ August Bjerg COVER PDF ].

Jacob Fronk, Multivariate Polynomials and Rational Functions of Random Matrices - A Case Study ISBN-number: 978-87-7125-224-8, 2023 TEXT: [ Jacob Fronk TEXT PDF ]. COVER: [ Jacob Fronk COVER PDF ].

Vignesh Subramanian, Categorification of Smith Theory ISBN-number: 978-87-7125-223-1, 2023 TEXT: [ Vignesh Subramanian TEXT PDF ]. COVER: [ Vignesh Subramanian COVER PDF ].

Jinwen Ye, Quantum Group Structures in Chern-Simons Theory ISBN-number: 978-87-7125-222-4, 2023 TEXT: [ Jinwen Ye TEXT PDF ]. COVER: [ Jinwen Ye COVER PDF ].

Nanna Havn Aamand, Quantum Group Structures in Chern-Simons Theory ISBN-number: 978-87-7125-221-7, 2023 TEXT: [ Nanna Havn Aamand TEXT PDF ]. COVER: [ Nanna Havn Aamand COVER PDF ].

Adriano Cordova Fedeli, Topological Hochschild homology of adic rings ISBN-number: 978-87-7125-220-0, 2023 TEXT: [ Adriano Cordova Fedeli TEXT PDF ].

Ali Muhammad, Entropy bounds for self-shrinkers with symmetries and applications ISBN-number: 978-87-7125-219-4, 2023 TEXT: [ Ali Muhammad TEXT PDF ]. COVER: [ Ali Muhammad COVER PDF ].

Alexis Aumonier, Moduli of algebraic hypersurfaces via homotopy principles ISBN-number: 978-87-7125-218-7, 2023 TEXT: [ Alexis Aumonier TEXT PDF ]. COVER: [ Alexis Aumonier COVER PDF ].

Frederik Ravn Klausen, Random Problems in Mathematical Physics ISBN-number: 978-87-7125-217-0, 2023 TEXT: [ Frederik Ravn Klausen TEXT PDF ]. COVER: [ Frederik Ravn Klausen COVER PDF ].

Paula Belzig, Quantum communication and fault-tolerance ISBN-number: 978-87-7125-216-3, 2023 TEXT: [ Paula Belzig TEXT PDF ]. COVER: [ Paula Belzig PDF ].

Anna Kamille Nyegaard, Projections and sensitivities of life insurance liabilities ISBN-number: 978-87-7125-214-9, 2023 TEXT: [ NYEGAARD TEXT PDF ]. COVER: [ NYEGAARD COVER PDF ].

Jacob Bjerre Skov, Dynamic Term Structure Modeling and the LIBOR Transition ISBN-number: 978-87-7125-215-6, 2023 TEXT: [ SKOV TEXT PDF ]. COVER: [ SKOV COVER PDF ].

Jingxuan Zhang, Localization theory for propagation of quantum information ISBN-number: 978-87-7125-213-2, 2023 TEXT: [ ZHANG TEXT PDF ]. COVER: [ ZHANG COVER PDF ].

Johannes Agerskov, One-dimensional Dilute Quantum Gases and Their Ground State Energies ISBN-number: 978-87-7125-071-8, 2023 TEXT: [ JOAS TEXT PDF ]. COVER: [ JOAS COVER PDF ].

Mads Friis Frand-Madsen, Connections between Quantum Key Distribution and Quantum Data Hiding ISBN-number: 978-87-7125-070-1, 2023 TEXT: [ MFFM TEXT PDF ]. COVER: [ MFFM COVER PDF ].

Vincent Steffan, Tensor Decompositions, Theory and Applications in Quantum Information ISBN-number: 978-87-7125-069-5, 2023 TEXT: [ VS TEXT PDF ]. COVER: [ VS COVER PDF ].

Song Li, Mathematical tools for population genetics based on genotype data ISBN-number: 978-87-7125-068-8, 2023 TEXT: [ SL TEXT PDF ]. COVER: [ SL COVER PDF ].

Yumo Zhang, Dynamic portfolio optimization with stochastic investment opportunities ISBN-number: 978-87-7125-067-1, 2023 TEXT: [ YZ TEXT PDF ]. COVER: [ YZ COVER PDF ].

Severin Mejak, Definability of maximal discrete sets ISBN-number: 978-87-7125-063-3, 2023 TEXT: [ SM TEXT PDF ]. COVER: [ SM COVER PDF ].

Jamaal Ahmad, Matrix methods in multi-state life insurance ISBN-number: 978-87-7125-066-4, 2023 TEXT: [ JA TEXT PDF ]. COVER: [ JA COVER PDF ].

Alexander Frei, Operator algebras and nonlocal games: Optimal states and self-testing ISBN-number: 978-87-7125-065-7, 2023 TEXT: [ AF TEXT PDF ]. COVER: [ AF COVER PDF ].

Snorre Jallbjørn, Multi-Population Mortality Models and Scenario-Based Projections ISBN-number: 978-87-7125-062-6, 2023 TEXT: [ SJ TEXT PDF ]. COVER: [ SJ COVER PDF ].

Jeroen van der Meer, Higher-algebraic Picard invariants in modular representation theory ISBN-number: 978-87-7125-058-9, 2022 TEXT: [ JVDM TEXT PDF ]. COVER: [ JVDM COVER PDF ].

Nikolaj Theodor Birkmose Thams, Causality and Distribution Shift ISBN-number: 978-87-7125-061-9, 2022 TEXT: [ NT TEXT PDF ], Text in B5 format. COVER: [ NT COVER PDF ].

Phillip Bredahl Mogensen, Multiple Hypothesis Testing and Causal Discovery ISBN-number: 978-87-7125-060-2, 2022 TEXT: [ PBM TEXT PDF ], Text in B5 format. COVER: [ PBM COVER PDF ].

Kaif Hilman Tan, Norms and periodicities in genuine equivariant hermitian K–theory ISBN-number: 978-87-7125-059-6, 2022 TEXT: [ KHT TEXT PDF ], Text in B5 format. COVER: [ KHT COVER PDF ].

Debbie Kusch Falden, Projection of balances and benefits in life insurance with various dividend strategies ISBN-number: 978-87-7125-057-2, 2022 TEXT: [ DKF PDF ]; COVER: [ DKF PDF ].

Henning Olai Milhøj, Quasitraces, Tracial States, and Kaplansky's Conjecture ISBN-number: 978-87-7125-056-5, 2022 TEXT: [ HOM PDF ], NB: Text in B5 format. COVER: [ HOM PDF ].

Malte Sander Leip, On the Hochschild homology of hypersurfaces as a mixed complex ISBN-number: 978-87-7125-055-8, 2022 TEXT: [ MSL PDF vol.1 ], [ MSL PDF vol.2 ] NB: Text in B5 format. COVER: [ COVER PDF vol.1 ], [ COVER PDF vol.2 ].

Sigurd Emil Rømer, Essays on rough and classical stochastic volatility ISBN-number: 978-87-7125-054-1, 2022 TEXT: [ SER PDF ],    COVER: [ SER COVER PDF ].

Manh Cuong Ngo, Modelling marine mammal reactions ISBN-number: 978-87-7125-053-4, 2022 TEXT: [ MCN PDF ],    COVER: [ MCN COVER PDF ].

Alexander Sevel Lollike, Projections in Life Insurance and the Equilibrium Approach to Utility Optimization ISBN-number: 978-87-7125-052-7, 2022 TEXT: [ ASL PDF ],    COVER: [ ASL COVER PDF ].

Marie Fialová, Aharonov-Casher theorems for manifolds with boundary ISBN-number: 978-87-7125-051-0, 2022 TEXT: [ MF PDF ],    COVER: [ MF COVER PDF ].

Luigi Pagano, The motivic zeta functions of Hilbert schemes of points on surfaces ISBN-number: 978-87-7125-050-3, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Meltem Ünel, Random trees with height dependent weights: Local limits and applications ISBN-number: 978-87-7125-049-7, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Martin Emil Jakobsen, Causality and generalizability: Identifiability and learning methods ISBN-number: 978-87-7125-048-0, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Francesco Campagna, Arithmetic and diophantine properties of elliptic curves with complex multiplication ISBN 978-87-7125-047-3, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Daria Poliakova, Homotopical algebra and combinatorics of polytopes ISBN 978-87-7125-045-9, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Sabiha Sibel Tokus, Variational methods for quantum Hamiltonians ISBN: 978-87-7125-046-6, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Maria Laura Battagliola, Quantile regression for scalar and functional clustered data and data analysis with phase-amplitude separation ISBN: 978-87-7125-044-2, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Robin Janik Sroka, Patterns in the homology of algebras: Vanishing, stability, and higher structures ISBN: 978-87-7125-043-5, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Lasse Petersen, Causal inference and machine learning ISBN: 978-87-7125-042-8, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Maxime Savoy, Combinatorial cell complexes. Duality, reconstruction and causal cobordisms ISBN: 978-87-7125-041-1, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Mikala Ørsnes Jansen, The stratified homotopy type of the reductive Borel-Serre compactification and applications to algebraic K-theory. ISBN: 978-87-7125-040-4, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Nicholas Gauguin Houghton-Larsen, A mathematical framework for causally structured dilations and its relation to quantum self-testing ISBN: 978-87-7125-039-8, 2021    TEXT: [ PDF ],    COVER: [ PDF ].

Zhipeng Duan, On equivariant Euler characteristics and spaces of trees ISBN: 978-87-7125-038-1, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Asbjørn Christian Nordentoft, On arithmetic statistics and periods of automorphic forms ISBN: 978-87-7125-037-4, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Jorge Yslas, Point process convergence of random walks and the estimation of multivariate heavy-tailed distributions ISBN: 978-87-7125-032-9, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Christian Furrer, Multi-state modeling in the mathematics of life insurance: meditations and applications ISBN: 978-87-7125-036-7, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Clemens Borys, Groups, actions, and C*-algebras ISBN: 978-87-7125-034-3, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Riccardo Pengo, Mahler measures, special values of L-functions and complex multiplication ISBN: 978-87-7125-035-0, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Philipp Schmitt, Strict quantization of certain classes of analytic functions ISBN: 978-87-7125-033-6, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Rune Christiansen, Causal inference in the presence of hidden variables: Structure learning, Effect estimation and distribution generalization ISBN: 978-87-7125-031-2, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Joshua Edward Hunt, Decompositions and obstructions for the stable module ∞-category ISBN: 978-87-7078-898-4, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Søren Wengel Mogensen, Graphical modeling in dynamical systems ISBN: 978-87-7078-897-7, 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Angélica Marcela Torres Bustos, Algebraic methods for bistability and oscillations in reaction networks ISBN: 978-87-7078-896-0. 2020    TEXT: [ PDF ],    COVER: [ PDF ].

Francesco Chini, Some classification results for translating solitons and ancient mean curvature flows ISBN: 978-87-7078-895-3, 2019    TEXT: [ PDF ],    PDF ].-->

Mads Bonde Raad, Stability of age dependent Hawkes processes ISBN: 978-87-7078-894-6, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Kevin Aguyar Brix, Topological dynamics, groupoids and C*-algebras ISBN: 978-87-7078-893-9, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Asger Kjærulff Jensen, Tensors and the entanglement of pure quantum states ISBN: 978-87-7078-890-8, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Karen Bakke Haga, Maximal almost disjoint families, determinacy, and forcing ISBN: 978-87-7078-889-2, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Georgios Dalezios, Relative homological algebra and exact model structures ISBN: 978-87-7078-886-1, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Henrik Carøe Bylling, Bilevel optimization with applications in energy ISBN: 978-87-7078-885-4, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Vibeke Quorning, Cantor-Bendixson type ranks & Co-induction and invariant random subgroups ISBN: 978-87-7078-884-7, 2019    TEXT: [ PDF ],    COVER: [ PDF ].

Rune Harder Bak, Tensor abelian categories -- in a non-commutative setting ISBN: 978-87-7078-805-2, 2018    TEXT: [ PDF ].

Espen Auseth Nielsen, Operations on Hochschild complexes of Hopf-like algebras ISBN: 978-87-7078-883-0, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Mads Christian Hansen, Quasi-stationary distributions in stochastic reaction networks ISBN: 978-87-7078-881-6, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Martin Speirs, On the algebraic K-theory of coordinate axes and truncated polynomial algebras ISBN: 978-87-7078-916-5, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Niels Aske Lundtorp Olsen, Statistical analysis of functional data: Multivariate responses, misaligned data and local inference ISBN: 978-87-7078-915-8, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Amir Hosein Sadeghimanesh, Algebraic tools in the study of multistationarity of chemical reaction networks ISBN: 978-87-7078-914-1, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Manuel Krannich, On characteristic classes of manifold bundles ISBN: 978-87-7078-908-0, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Benjamin Böhme, Equivariant multiplications and idempotent splittings of G-spectra ISBN: 978-87-7078-913-4, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Roberto Ferrara, An information-theoretic framework for quantum repeaters ISBN: 978-87-7078-912-7, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Valerio Proietti, On K-theory, groups, and topological dynamics ISBN: 978-87-7078-906-6, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Anton Samojlow, Universality of Born-Oppenheimer curves ISBN: 978-87-7078-909-7, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Samuele Soraggi, Theory and inference on gene flow and ploidy numbers from NGS data 2018    TEXT: [ PDF ].

Jacob Østergaard, Statistical methods for neural data: Cointegration analysis of coupled neurons & generalized linear models for spike train data ISBN: 978-87-7078-903-5, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Mareile Große Ruse, Inference from stochastic processes with application to birdsongs and biomedicine ISBN: 978-87-7078-902-8, 2018    TEXT: [ PDF ],    COVER: [ PDF ].

Frederik Vissing Mikkelsen, Model selection and risk estimation with applications to nonlinear ordinary differential equation systems ISBN: 978-87-7078-901-1, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Dino Destefano, Investigating slopes of overconvergent modular forms ISBN: 978-87-7078-900-4, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Birger Brietzke, On the second order correction to the ground state energy of the dilute Bose gas ISBN: 978-87-7078-932-5, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Xiaolei Xie, Analysis of heavy-tailed time series ISBN: 978-87-7078-931-8, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Adam Lund, Spatio-temporal modeling of neuron fields ISBN: 978-87-7078-934-9, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Emil Steen Jørgensen, Diffusion models observed at high frequency and applications in finance ISBN: 978-87-7078-933-2, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Rasmus Sylvester Bryder, Boundaries, injective envelopes, and reduced crossed products ISBN: 978-87-7078-930-1, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Martin S. Christensen, Regularity of C*-algebras and central sequence algebras ISBN: 978-87-7078-929-5, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Eduardo Scarparo, Partial actions, paradoxicality and topological full groups ISBN: 978-87-7078-927-1, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Christian Majenz, Entropy in quantum information theory -- Communication and cryptography ISBN: 978-87-7078-928-8, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Matias Lolk, Dynamical systems and algebras associated with separated graphs ISBN: 978-87-7078-925-7, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Tomasz Prytula, Hyperbolic isometries of systolic complexes ISBN: 978-87-7078-923-3, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Johannes Heiny, Extreme eigenvalues of sample covariance and correlation matrices ISBN: 978-87-7078-921-9, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Clarisson Rizzie Canlubo, Non-commutative covering spaces and their symmetries ISBN: 978-87-7078-922-6, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Kang Li, Novel mathematical neural models for visual attention ISBN: 978-87-7078-920-2, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Amalie Høgenhaven, Real topological cyclic homology ISBN: 978-87-7078-919-6, 2017    TEXT: [ PDF ],    COVER: [ PDF ].

Robin Reuvers. Analysis of the Bogoliubov free energy functional ISBN: 978-87-7078-917-2, 2016    TEXT: [ PDF ],    COVER: [ PDF ].

Rune Ramsdal Ernstsen. Operation, investment and hedging in electricity markets ISBN: 978-87-7078-918-9, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Ditte Mølgård Heide-Jørgensen, Operations management in short-term power markets ISBN: 978-87-7078-951-6, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Massimiliano Ungheretti, On the algebraic structure of Hochschild complexes and the free loop space ISBN: 978-87-7078-950-9, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Niek de Kleijn, Group actions on deformation quantizations and an equivariant algebraic index theorem ISBN: 978-87-7078-949-3, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Kristian Knudsen Olesen, Analytic aspects of the Thompson groups ISBN: 978-87-7078-948-6, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Martin Jönsson, Essays on quantitative finance ISBN: 978-87-7078-945-5, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Giacomo Cherubini, Studies in the hyperbolic circle problem ISBN: 978-87-7078-946-2, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Ninna Reitzel Jensen, Modern policyholder preferences and scenario-based projections ISBN: 978-87-7078-944-8, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Simon Ellersgaard Nielsen, Essays on rational portfolio theory ISBN: 978-87-7078-939-4, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Lykke Rasmussen, Computational finance - on the search for performance ISBN: 978-87-7078-943-1, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Matthias Grey, Rational homological stability for automorphisms of manifolds ISBN 978-87-7078-956-1, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Seyed Nourollah Mousavi, Analysis of functional data with focus on multinomial regression and multilevel data ISBN 978-87-7078-940-0, 2016:    TEXT: [ PDF ],    COVER: [ PDF ].

Kang Li, Property A and coarse embedding for locally compact groups ISBN 978-87-7078-936-3, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Sima Mashayekhi, Numerical methods for nonlinear PDEs in finance ISBN: 978-87-7078-935-6, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Casper Guldberg, On homotopy automorphisms of Koszul spaces ISBN 978-87-7078-959-2, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Nina Munkholt Jakobsen, Efficient estimating functions for stochastic differential equations ISBN 978-87-7078-960-8, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Kamille Sofie Tågholt Gad, Optimal stopping and policyholder behaviour in life insurance ISBN 978-87-7078-961-5, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Kristian Buchardt, Life insurance liabilities with policyholder behaviour and stochastic rates ISBN 978-87-7078-967-7, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Martha Muller, Functional data analysis applied in chemometrics ISBN 978-87-7078-968-4, 2015:    TEXT: [ PDF ],    PDF ]. -->

Mathias Makedonski, On the stability of spherically symmetric selfgravitating classical and quantum systems ISBN: 978-87-7078-962-2, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Mauricio Gomez Lopez, Spaces of piecewise linear manifolds ISBN: 978-87-7078-963-9, 2015:    TEXT: [ PDF ],    COVER: [ PDF ].

Lars Frederik Brandt Henriksen, Aspects of valuation and optimization in life insurance ISBN: 978-87-7078-965-3. 2014: TEXT: [ PDF ],    COVER: [ PDF ].

Nadim Rustom, Algebra and arithmetic of modular forms ISBN: 978-87-7078-966-0, 2014:    TEXT: [ PDF ],    COVER: [ PDF ].

Søren Knudby, Approximation properties for groups and von Neumann algebras ISBN: 978-87-7078-971-4, 2014:    TEXT: [ PDF ],    COVER: [ PDF ].

Kristian J. Moi, Equivariant homotopy theory and K-theory of exact categories with duality ISBN: 978-87-7078-970-7, 2014:    TEXT: [ PDF ],    COVER: [ PDF ].

Sune Precht Reeh, Burnside rings of fusion systems ISBN 978-87-7078-974-5, 2014:    TEXT: [ PDF ],    COVER: [ PDF ].

Anders Christian Jensen, Statistical inference for partially observed diffusion processes ISBN: 978-87-7078-975-2, 2014:    TEXT: [ PDF ],    COVER: [ PDF ].

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Lars Winther Christensen, Functorial dimensions HCØ-tryk, ISBN 87-7834-359-3, 1999:   TEXT: [ PDF ],   COVER: [ HCØ-PDF ].

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Two Students Unravel a Widely Believed Math Conjecture

August 10, 2023

About 2,200 years ago, the Greek geometer Apollonius of Perga inquired about how circles would fit together if they all touched each other at a single point.

Samuel Velasco/ Quanta Magazine

Introduction

Summer Haag and Clyde Kertzer had high hopes for their summer research project. Blindsiding an entire subfield of mathematics was not one of them.

In May, Haag was finishing her first year of graduate school at the University of Colorado, Boulder, where Kertzer was an undergraduate. Both looked forward to a break from classes. Haag planned to explore new hikes and climbing routes. Kertzer, a Boulder native, wanted to play soccer and prepare his grad school application. But as aspiring research mathematicians, they had also applied for a half-time summer research program in the group of the mathematician Katherine Stange .

Stange is a number theorist who describes herself as a mathematical “ frog ” — someone who delves deep into one problem’s intricacies before hopping to another. She is interested in “simple-seeming questions that lead to a richness of structure,” she said. Her projects often poke at number theory’s elusive open problems by using computers to generate large data sets.

Haag and Kertzer began the program on Haag’s 23rd birthday with a weeklong primer on Apollonian circle packings — the ancient study of how circles can harmoniously squeeze into one larger circle.

Imagine arranging three coins so that each one touches the others. You can always draw a circle around them that touches all three from the outside. Then you can start to ask questions: How does the size of that bigger circle relate to those of the three coins? What size circle will fit into the gap between the three coins? And if you start to draw circles that fill in progressively smaller and smaller gaps between circles — creating a fractal pattern known as a packing — how do the sizes of those circles relate to one another?

Rather than think about the diameter of these circles, mathematicians use a measure called curvature — the inverse of the radius. So a circle with radius 2 has curvature 1/2, and a circle with radius 1/3 has curvature 3. The smaller the circle, the bigger the curvature.

Renaissance mathematicians proved that if the first four circles have a curvature that’s an integer, the curvatures of all the subsequent circles in the packing are guaranteed to be whole numbers. That’s remarkable on its own. But mathematicians have taken the problem a step further by asking questions about which integers show up as the circles get smaller and smaller and the curvatures get larger and larger.

In 2010, Elena Fuchs , a number theorist now at the University of California, Davis, proved that curvatures follow a particular relationship that forces them into certain numerical buckets. Shortly afterward, mathematicians became convinced that not only must the curvatures fall into one bucket or another, but also that every possible number in each bucket must be used. The idea came to be known as the local-global conjecture.

“Lots of works referenced it as though it were already fact,” Kertzer said. “We discussed it as though it was going to be proven at some point in the near future.”

James Rickards , a mathematician at Boulder who works with Stange and the students, had written code to examine any desired arrangement of circle packings. So when Haag and Kertzer joined the group on May 15, they thought they’d create cool plots of the reliable local-to-global rule kicking in.

At the start of their project, Summer Haag (left) and Clyde Kertzer used data plots to discover that something was amiss.

Ilyas Salhi (left); Courtesy of Clyde Kertzer

Stange flew to France for a conference in early June. When she returned on June 12, the team huddled around charts that demonstrated how a few buckets seemed to be missing certain numbers.

“We weren’t investigating this phenomenon,” Rickards said. “I wasn’t trying to test that it’s true. I knew it was true — I just assumed it was true. And then suddenly, we’re faced with data that says it’s not.”

By the end of the week, the team was confident the conjecture was false. Numbers they expected to appear never did. They worked out a proof, and on July 6 they posted their work to the scientific preprint site arxiv.org.

Fuchs remembers talking to Stange soon after the proof clicked into place. “How much do you believe the local-to-global conjecture?” Stange asked. Fuchs responded that of course she believed it. “Then she showed me all these data and I said, ‘Oh my goodness, that’s amazing,’” Fuchs said. “I mean, I really believed that the local-to-global conjecture was true.”

“Once you see it, you just say ‘Aha! Of course!’” said Peter Sarnak , a mathematician at the Institute for Advanced Study and Princeton University whose early observations helped fuel the local-global conjecture.

“It’s a fantastic insight,” added Alex Kontorovich of Rutgers University. “We’re all kicking ourselves that we didn’t find it 20 years ago, when people first started playing with this.”

Amid the rubble left by the result, the work has exposed a crack in the foundation of other conjectures in number theory. Mathematicians have been left to wonder what widely held belief might be the next to fall.

Roundabout History

Apollonian circle packings get their name from their probable originator, Apollonius of Perga. About 2,200 years ago, the Greek geometer wrote a book called Tangencies about how to construct a circle that’s tangent to any three others. The book has been lost to time. But about 500 years later, the Greek mathematician Pappus of Alexandria put together a compendium that would survive the collapse of the Byzantine empire.

Apollonius of Perga is considered one of the great mathematicians of antiquity. His work, seen here in a 9th-century Arabic translation, continued to develop the geometric ideas of Euclid, who lived about a century earlier.

Bodleian Libraries/Oxford University

Using only Pappus’ description of Tangencies , Renaissance mathematicians attempted to retrace the original work. By 1643, René Descartes had discovered a simple relationship between the curvatures of any four circles that are tangent to each other. Descartes asserted that the sum of all the squared curvatures equals half the square of the sum of the curvatures. This means that, given three circles, it’s possible to calculate the radius of a fourth tangent circle. For example, if you have three circles with curvatures of 11, 14 and 15, you can plug those numbers into Descartes’ equation and calculate the curvature of the circle that would fit inside them: 86.

In 1936, the Nobel Prize-winning radiochemist Frederick Soddy noticed something odd as he built packings with Descartes’ relation. As the circles got smaller and curvatures bigger, he expected to get gnarly numbers with square roots or infinite decimals. Instead, all the curvatures were integers. This was a fairly straightforward consequence of Descartes’ equation, but nobody had noticed for hundreds of years. It inspired Soddy to publish a poem in the scientific journal Nature , which began:

For pairs of lips to kiss maybe Involves no trigonometry. ’Tis not so when four circles kiss Each one the other three.

The Possible and the Inevitable

Once it was established that there are packings full of integers, mathematicians tried to find patterns in those integers.

In 2010, Fuchs and Katherine Sanden set out to build on a paper from 2003 . The duo observed that if you divided each curvature in a given packing by 24, a rule emerged. Some packings only have curvatures with remainders of 0, 1, 4, 9, 12 or 16, for example. Others only leave remainders of 3, 6, 7, 10, 15, 18, 19 or 22. There were six different possible groups.

As mathematicians examined the different categories of packings, they began to notice that for small enough circles — those with large curvatures — it seemed that every possible number within each category appeared for packings of that type. This idea came to be called the local-global conjecture. Proving it became “one of these little mathematicians’ dreams of mine,” Fuchs said. “Like, maybe at some point many years from now I’ll be able to solve it.”

In 2012, Kontorovich and Jean Bourgain (who died in 2018 ) proved that virtually every number predicted by the conjecture does occur. But “virtually all” does not mean “all.” For example, perfect squares are rare enough that, mathematically, “virtually all” integers are not perfect squares, even though, for instance, 25 and 49 are. Mathematicians thought the rare counterexamples that remained possible after Kontorovich and Bourgain’s paper didn’t actually exist, mostly because the two or three most well-studied circle packings seemed to follow the local-global conjecture so well, Kontorovich said.

Cranking Up That Dial

When Haag and Kertzer started this summer in Boulder, Rickards scribbled ideas on a blackboard in Stange’s office. “We had a whole list,” Rickards said. They had four or five starting points to experiment with. “Things you can just play with and see what happens.”

One idea was to compute all the possible circle packings that contain two arbitrary curvatures A and B. Rickards wrote a program that outputs a sort of ledger that reports which integers show up to the party when A is hosting.

Based on this program, Haag rustled together a Python script that plotted tons of simulations at once. It was like a multiplication table: Haag chose which rows and columns to include based on their remainders when they were divided by 24. Pairs of numbers that appear in an Apollonian packing together got white pixels; those that don’t got black pixels.

Haag plowed through dozens of plots — one for each pair of remainders in each of the six groups.

Katherine Stange, a mathematician at the University of Colorado, Boulder, would use the students’ insight to prove that the local-global conjecture doesn’t hold.

Nicholas Christoffersen

They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. “We expected the black dots to peter out,” Stange said. Rickards added, “I thought maybe it would even be possible to prove they peter out.” He speculated that by looking at charts that synthesized many packings together, the team would be able to prove results that weren’t possible when they looked at any one packing on its own.

While Stange was away, Haag wound up plotting every pair of remainders — about 120. No surprises there. Then she went big.

Haag had been plotting how 1,000 integers interact. (The graph is bigger than it sounds, since it involves 1 million possible pairs.) Then she cranked the dial up to 10,000 times 10,000. In one graph, regular rows and columns of black specks refused to dissolve. It looked nothing like what the local-global conjecture would predict.

The team met on a Monday after Stange returned. Haag presented her graphs, and they all focused on the one with the weird dots. “It was just a continual pattern,” Haag said. “And that was when Kate said, ‘What if the local-global conjecture isn’t true?’”

“This looks like a pattern. It has to continue. So the local-global conjecture must be false,” Stange recalled thinking. “James was more skeptical.”

“My first thought was there must be a bug in my code,” Rickards said. “I mean, that was the only reasonable thing I could think of.”

Within half a day, Rickards came around. The pattern ruled out all pairs where the first number is of the form 8 × (3 n ± 1) 2 and the second is 24 times any square. This means 24 and 8 never appear in the same packing. Numbers you’d expect to occur don’t.

“I was kind of giddy. It’s not very often that something really surprises you,” Stange said. “But that’s the magic of playing with data.”

The July paper outlines a rigorous proof that the pattern they observed continues indefinitely, disproving the conjecture. The proof hinges on a centuries-old principle called quadratic reciprocity that involves the squares of two prime numbers. Stange’s team discovered how reciprocity applies to circle packings. It explains why certain curvatures can’t be tangent to each other. The rule, called an obstruction, propagates throughout the whole packing. “It’s just an entirely new thing,” said Jeffrey Lagarias , a mathematician at the University of Michigan who was a co-author on the 2003 circle-packing paper. “They’ve found it ingeniously,” Sarnak said. “If these numbers did appear, they would violate reciprocity.”

The Fallout

A number of other conjectures in number theory may now be in doubt. Like the local-global conjecture, they are hard to prove but have already been shown to hold for virtually all cases and are generally assumed to be true.

For example, Fuchs studies Markov triples, sets of numbers that satisfy the equation x 2 + y 2 + z 2 = 3 xyz . She and others have shown that certain types of solutions are connected for prime numbers greater than 10 392 . Everyone believes the pattern should continue to infinity. But in light of the new result, Fuchs has allowed herself to feel a twinge of doubt. “Maybe I’m missing something,” she said. “Maybe everyone’s missing something.”

“Now that we have a single example where it’s false, the question is: Is it false for these other examples too?” Rickards said.

There’s also Zaremba’s conjecture. It says that a fraction with any denominator can be expressed as a continued fraction that uses only the numbers between 1 and 5. In 2014, Kontorovich and Bourgain showed that Zaremba’s conjecture holds for almost all numbers. But the surprise about circle packing has undermined confidence in Zaremba’s conjecture.

If the packing problem is a harbinger of things to come, computational data may be the tool of its undoing.

“I always find it fascinating when new mathematics is born out of just purely looking at data,” Fuchs said. “Without it, it’s really hard to imagine that [they] would have stumbled upon this.”

Stange added that none of this would have happened without the low-stakes summer project. “Serendipity and an attitude of playful exploration both have such a huge role in discovery,” she said.

“It was pure coincidence,” Haag said. “If I didn’t go big enough, we wouldn’t have noticed it.” The work bodes well for the future of number theory. “You can glean understanding of mathematics through your intuition, through proofs,” Stange said. “And you trust that a lot because you spent a lot of time thinking about it. But you can’t argue with the data.”

Editor’s note : Alex Kontorovich is a member of Quanta Magazine ’s scientific advisory board. He was interviewed for this story but did not otherwise contribute to its production.

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