*Multiple graduate student mentors have proposed this project; their individual proposals are listed above separately. Please note that both proposals are listed under a single project on the application form.
Prerequisites: Linear algebra (MAT 310/315), Abstract algebra (MAT 313)
Project 1:
Book: Representing Finite Groups: A Semisimple Introduction, Ambar Sengupta
Description: The idea for this project is an enrichment of abstract algebra. One goal will be to understand the p^aq^b solvability criterion of Burnside. Additionally we wish to develop the classical results of replantation theory with a view towards geometric connections.
Project 2:
Book: The Symmetric Group Representations, Combinatorial Algorithms, and Symmetric Functions, B.E Sagan
Description: This project will aim mainly to develop the representation theory of the Symmetric group. In order to do this, we will learn the general theory of finite group representations, Specht modules, combinatorial algorithms, and the theory of Symmetric functions. For those interested, we would also study an application of this theory to probability.
Prerequisites: Student should be comfortable with proof-based mathematics (MAT 200/250). Some abstract algebra (MAT 313) and/or basic topology (MAT 364, MAT 528) would be helpful.
Book: Algebraic Curves, Fulton
Description: Foundational algebraic geometry is about understanding how the properties of certain algebraic structures, most notably rings, correspond to geometric properties of shapes defined by collections of polynomials. In this project, we aim to understand some aspects of this beautiful dictionary by focusing on a particularly rich class of examples: algebraic curves. Despite their long history, these objects are still intensely studied today and have an extremely rich geometry. Depending on the student's background, it is also possible to cover more advanced aspects of the theory such as the Riemann-Roch theorem, elliptic curves, the modern scheme-theoretic perspective etc.
Prerequisites: Abstract algebra, especially basic ring theory (e.g. MAT 313)
Book: Ideals, Varieties, and Algorithms, Cox, Little, and O'Shea
Description: Algebraic geometry is the study of spaces which can be defined by polynomial equations, called varieties. The study of varieties is closely tied to the study of ideals in polynomial rings. In this project, we will learn some of the basic principles of algebraic geometry. At the same time, we will see how a powerful computational tool for ideals called a Gröbner basis can be used to work out many explicit examples. The ability to explicitly work through examples should complement the sometimes quite abstract general theory of varieties.
*Multiple graduate student mentors have proposed this project; their individual proposals are listed above separately. Please note that both proposals are listed under a single project on the application form.
Prerequisites: Topology (MAT 528), Algebra (MAT 313/524 Fall 2025)
Project 1:
Book: The Rising Sea, Ravi Vakil
Description: This project is intended to give a serious and reasonably complete introduction to algebraic geometry. Learning the subject is a demanding task, but also incredibly rewarding. The ideas that allow algebraic geometry to connect several parts of mathematics are fundamental and well-motivated. We will cover sheaves, schemes, and morphisms of schemes.
Project 2:
Book: Algebraic Geometry, Robin Hartshorne
Description: Algebraic geometry studies spaces arising as the zero sets of systems of polynomials. Hartshorne’s book provides an introduction to abstract algebraic geometry through the methods of schemes and cohomology. In this project, we will begin with several basic concepts and examples, along with a review of key ideas from commutative algebra. Our goal is to develop an understanding of the methods of schemes and cohomology presented in Chapters 2 and 3 of the book.
Prerequisites: Analysis (MAT 319/320), knowledge of real analysis and smooth manifolds may be helpful for further topics
Book: Complex Analysis, Donald Marshall
Description: Although the classical theory for functions of one complex variable has been established since the early 19th century, many open problems with deceivingly simple statements remain elusive to this day. The goal of this project is to study this classical theory. Depending on the student background and interest, further topics may be pursued, concerning complex manifolds (Hodge theory, vanishing theorems) and symplectic manifolds (pseudo-holomorphic curves).
Prerequisites: Analysis (MAT 319/320)
Book: Fourier Analysis: An Introduction, Stein & Shakarchi
Description: Ideas from Fourier analysis permeate much of the present-day analysis. Stein & Shakarchi's Fourier Analysis: An Introduction elucidates applications to other sciences and links to topics such partial differential equations and number theory. The goal of this project is to read the first four chapters, which cover Fourier series and some applications. Time permitting, we will also examine the Fourier Transform, finite Fourier analysis, and Dirichlet's theorem.
Prerequisites: Multivariable calculus (MAT 307), Analysis (MAT 319/320), Measure theory (MAT 324)
Book: The Incompressible Euler and Navier-Stokes Equations, Jacob Bedrossian, and Vlad Vicol; Partial Differential Equations, Evans
Description: The tentative goal of this project is to prove the short time existence and uniqueness of solution of fundamental fluid equations such as Euler equation or Navier-Stokes equation. Along the way, we aim to develop basic theoretical background, in particular Fourier Analysis, Functional Analysis, as well as the rigorous meaning of 'the existence of solution' for a partial differential equation.
*Multiple graduate student mentors have proposed this project; their individual proposals are listed above separately. Please note that both proposals are listed under a single project on the application form.
Prerequisites: Linear algebra (MAT 310/315), Analysis (MAT 319/320), ordinary differential equations (MAT 303/308)
Project 1:
Book: Partial Differential Equations, Evans; Partial Differential Equations: An Introduction, Strauss
Description: In this project, students will begin by studying the classical theory of the Laplace equation and harmonic functions. This will include the mean value property, maximum principle, and the fundamental solution of the Laplace equation. Depending on interest/background, we will study the weak formulation in order to develop ideas related to Sobolev spaces and functional analysis techniques.
Project 2:
Book: Partial Differential Equations, Evans
Description: PDEs are the basis of a mathematical study of many processes in the natural and social sciences. This project is intended to be a mathematically rigorous introduction to aspects of PDE theory. The field is very broad in scope, and the subject matter can be tailored to the interests of the student: an overview of the Laplace, heat and wave equations; a deeper dive into elliptic, parabolic, or hyperbolic PDEs (of which the previous three are the respective prototypes); weak formulation of PDEs (i.e., why it is useful to look for badly behaved "solutions" of PDEs); or even a deeper study of particular PDEs that show up in other fields of math/science (e.g. the curve-shortening flow in differential geometry).
Prerequisites: Two years of calculus (e.g., MAT 203). Having seen real analysis and some measure theory would be optimal, but is not required.
Book: Fractals Everywhere, Michael Barnsley
Description: In this project, we will be concerned with discovering and analyzing sets to which the general descriptive term "fractal" applies. We will learn how to construct fractal sets by way of the Banach contraction mapping theorem. We will learn how to find quantitative characteristics of fractals such as dimension (which need not be an integer!). More advanced topics may include the description of fractals in the parameter plane (the Mandelbrot set), the construction of measures on fractals, and applications to computer graphics (time permitting).
Prerequisites: A course in analysis (e.g. MAT 319/320); basic knowledge of measure theory (e.g. MAT 324) might be needed, but could be learned throughout the project.
Book: One-Dimensional Dynamics: From Poincaré to renormalization, Yiheng Dong, Marco Martens, and Liviana Palmisano
Description: In mathematics, a dynamical system is a "state space" X together with a map f from X to itself. We are interested in studying the iterations of the points in X under the map f, which we call the orbits. Given two dynamical systems, the behavior of the orbits may be the same combinatorially, topologically or geometrically, so a natural goal is to classify a family of dynamical systems based on their orbits behavior.
One of the simplest family of dynamical systems is the family of circle homeomorphisms. However, there is a surprisingly rich theory of their dynamics. In this project, we aim to understand the conditions under which two circle homeomorphisms are equivalent combinatorially or topologically. These are based on the classical theory of Poincaré and Denjoy. If time permits and if the student already has a solid background in dynamics, we could further study the modern theory of renormalization, which is an important tool for the so-called rigidity problem, i.e., the geometrical equivalence of dynamical systems.
Prerequisites: Basic topology (MAT 364), Analysis in several dimensions (MAT 322), Linear algebra (MAT 310/315), or equivalents. Familiarity with smooth manifolds is helpful.
Book: Topological Methods in Hydrodynamics, V. I. Arnold and Boris Khesin; An Introduction to the Geometry and Topology of Fluid Flows, edited by Renzo L. Ricca
Description: There are hidden geometric and topological structures in fluid problems that reveal information about the dynamics of the fluid that would otherwise be tucked away by the complexity and individuality of the problem. This project explores these structures. We will follow one of two texts. One textbook focuses on the more grounded study of fluid dynamics but from a topological viewpoint, covering at least: classification of 3d steady flows, topological obstructions to energy relaxation, asymptotic linking number. The other text focuses on a survey of many topological techniques and ideas used in fluid dynamics covering at least: knots, braids, topological aspects of vortex motion. Independent of the text, we will explore fundamental ideas in fluids and topological dynamics such as: topologically conjugate dynamical systems, topological entropy, Euler’s equations and Navier-Stokes equations.
Prerequisites: Analysis in several dimensions (e.g. MAT 322), topology (e.g. MAT 364), linear algebra (e.g. MAT 310/315). Some understanding of foundational concepts from manifold theory and group theory is strongly recommended.
Book: Gauge Fields, Knots and Gravity, John Baez and Javier Muniain
Description: In modern mathematics, gauge theory plays an important role quite independently of its physical interpretation. It can be used to define invariants in differential geometry, notably for 3- and 4-manifolds. The central goal of this project is to understand vector bundles and connections, two key ingredients in gauge theory. Based on student backgrounds and interests, different topics can be covered, including Yang-Mills theory, Chern-Simons theory, and link invariants for knots.
*Multiple graduate student mentors have proposed this project; their individual proposals are listed below separately. Please note that both proposals are listed under a single project on the application form.
Prerequisites: Multivariable calculus and linear algebra (MAT 307). Background in differential geometry and differential equations (MAT 308) is helpful but not necessary.
Project 1:
Book: Harvey Reall's Part III lecture notes on general relativity and black holes; General Relativity, Robert Wald
Description: In this project, we propose an introductory study of mathematical general relativity. Our broad goal is to explore how physical concepts/theories can be made mathematically rigorous. Through the project, we will explore various concepts in differential geometry, differential equations, and analysis and investigate how these concepts arise in general relativity. In particular, we can examine specific topics in general relativity, like the Penrose diagrams, singularities, and the cosmic censorship conjectures which form the foundations of modern research in the field.
Project 2:
Book: We can refer to any subset of these books: A First Course in General Relativity, Bernard Schutz; General Relativity, Robert Wald; Semi-Riemannian Geometry, Barrett O'Niell; The Large-Scale Structure of Spacetime, Hawking and Ellis
Description: We will study aspects of general relativity from a mathematically rigorous standpoint, tailored to the mathematical background and interests of the student (although all of them will involve some differential geometry). Some potential projects (by no means exhaustive) are listed below.
Intro to the Mathematics of GR: The aim is to build up a background in differential geometry, as well as special relativity and other notions of physics as needed, to understand the Einstein Field Equations which are at the heart of GR. Time permitting, we can study a little bit about some special solutions (black holes/cosmology) or gravitational waves.
From Geometry to Physics: GR allows us to connect differential geometry to physical phenomena. We can make a detailed study of particular black hole solutions (Schwarzschild/Kerr/Reissner–Nordström etc.) and cosmological models (FLRW), including mathematical theorems about their validity. We can also study phenomena like gravitational waves, redshift, gravitational lensing, precession of orbits, etc.
Mass in General Relativity: GR has several models for isolated massive systems, but it is much harder to come up with a general notion of mass (and other useful quantities like angular momentum, etc.) that makes sense from the viewpoint of both geometry and physics. We can study the work that goes into developing such a notion, and various related results like the positive mass theorem and Penrose inequality.
Causality and Singularities: Are gravitational singularities (Big Bang, black holes, etc.) even real? When do they have to have event horizons? These questions took a long time for mathematicians and physicists working in GR to answer (the key breakthrough being the Penrose-Hawking singularity theorems, which are purely geometric results!), and several aspects are still open. We can study some of this journey, which will also allow us to talk about things like gravitational collapse and black hole thermodynamics.
Prerequisites: Analysis (e.g. MAT 319/320), topology (e.g. MAT 364) and linear algebra (e.g. MAT 310). Some familiarity with smooth manifolds will be useful.
Book: Mathematical Methods of Classical Mechanics, V. I. Arnold
Description: The goal of this project is to study the formulation of classical mechanics using modern geometric language. The aim is to understand the origins of various contemporary mathematical concepts—such as smooth manifolds, Lie theory, and symplectic geometry—in the context of physics. Based on student backgrounds and interests, different topics can be covered, including variational principles and Lagrangian mechanics, the symplectic formulation of Hamiltonian mechanics, perturbation theory, Kolmogorov-Arnold-Moser (KAM) theory, and geometric methods in ordinary differential equations.
Prerequisites: Multivariable calculus and linear algebra (MAT 307)
Book: Visual Differential Geometry and Forms, Tristan Needham
Description: The goal is to learn about metrics and curvature, mostly of surfaces, which are ways to describe the shape of a space. The chosen book has many pictures to accompany the mathematics and presents many examples to gain a good understanding. A good ultimate goal could be to understand the Gauss-Bonnet theorem from multiple perspectives. It is one of the major results in geometry which states that while surfaces can be curved in many ways, the "total curvature" on a surface depends only on the topology.
Other directions can also potentially be pursued. For example, other tools such as differential forms and vector bundles, which help with understanding, especially in higher dimensions, may be discussed. Alternatively, we can also work towards the development of Einstein's equations and General Relativity, which is a major application of abstract Differential Geometry.
Prerequisites: Complex analysis (e.g. MAT342), multivariable calculus (e.g. MAT 322), topology (e.g. MAT 364) and linear algebra (e.g. MAT 310). Some understanding of foundational concepts from manifold theory and group theory is strongly recommended.
Book: Riemann surfaces, Simon Donaldson
Description: The theory of Riemann surfaces provides a platform for a large number of mathematical theories including low-dimensional topology, algebraic curve, hyperbolic geometry. Based on student backgrounds and interests, different topics can be covered, including Riemann-Roch theorem, Jacobians of Riemann surfaces, uniformization theorem.
*Multiple graduate student mentors have proposed this project; their individual proposals are listed above separately. Please note that both proposals are listed under a single project on the application form.
Prerequisites: Point-set topology (MAT 528), Abstract algebra (MAT 313)
Project 1:
Book: Category Theory in Context, Emily Riehl; Topology: A Categorical Approach, Terilla et al.
Description: Since its advent in the 1940s, "category theory" has become increasingly useful as a language with which to describe mathematical ideas-- in particular, letting one easily transfer concepts from one context to another. For instance, when we talk about a "product" in mathematics, we could mean a product of: sets, vector spaces, rings, groups, topological spaces, etc. Despite this sheer breadth of definitions however, category theory tells us that these are all just examples of one type of product: the "categorical product" of two objects. If one wishes to learn, say, algebraic geometry, algebraic number theory, or even functional programming, categorical concepts are essential!
Unfortunately, however, the same generality that makes category theory so useful is also what lends it a reputation for being unnecessarily abstract and opaque. Emily Riehl's book Category Theory in Context attempts to demystify the language of category theory and provides many examples for a reader to sink their teeth into; John Terilla et. al's Topology: A Categorical Approach does similarly, but instead focuses solely on taking familiar point-set topological constructions and reintroducing them via easily generalizable categorical methods (serving as a relatively gentle way to gain some familiarity with category theory). Both books are available freely online, and can be read in whatever ratio is most agreeable/approachable to a given student.
Project 2:
Book: The main reference will be the notes provided by and discussions with the mentor. We will also occasionally discuss sections from Basic Category Theory by Tom Leinster and Stable Homotopy and Generalised Homology by J. F. Adams.
Description: Although category theory is now a central tool across many areas of mathematics, including but not limited to algebraic geometry, algebraic topology, and low-dimensional topology, it has often been dismissed as “abstract nonsense” lacking geometric intuition. This project aims to challenge that view by exploring the geometric foundations of key categorical ideas, leading to a modern perspective on some of the most celebrated results in 20th-century topology and geometry by Milnor, Smale, Kervaire, Novikov, Adams, Sullivan, and others. We will start with the basics of category theory up to the notion of representability, which will motivate our transition to stable homotopy theory and to various forms of cobordism theories, such as complex, oriented, and spin cobordism. This project is ideal for students interested in topological, algebraic, and categorical, rather than analytical, aspects of geometry.
Prerequisites: MAT 548 (prev. MAT 530), MAT 531, MAT 541
Book: Characteristic classes, Milnor & Stasheff
Description: Characteristic classes are certain cohomology classes attached to real or complex vector bundles. Although they can be used to tell vector bundles apart, their real importance lies in how they make difficult geometric problems computable. They give us a way to translate geometric questions about manifolds into algebraic ones about cohomology. Much of modern geometry and topology can be seen as exploring the consequences of these simple but powerful ideas.
*Multiple graduate student mentors have proposed this project; their individual proposals are listed above separately. Please note that all three proposals are listed under a single project on the application form.
Prerequisites: Analysis (MAT 319/320), Multivariable calculus (MAT 307), preferably point-set topology (MAT 528)
Project 1:
Book: Topology from a Differential Viewpoint, Milnor
Description: Topology from a Differentiable Viewpoint by Milnor is a short book which covers the essentials of differential topology. Classical definitions and results such as the Brouwer degree of a map, Sard's theorem, the Poincaré-Hopf theorem, and the Pontryagin-Thom construction are discussed. This book is wonderful to read and is worth hanging onto every word.
Project 2:
Book: Morse Theory, Milnor
Description: Milnor's Morse Theory covers the essentials of Morse theory and provides a primer on Riemannian geometry. Morse theory is a lens through which one can see how (smooth) manifolds are decomposed into pieces called "handles." The applications of this theory in modern mathematics are pervasive and knowing about this topic behooves any mathematician.
Project 3:
Book: Morse Theory, Milnor; Lectures on the h-cobordism theorem, Milnor
Description: In the 20th century, much of the efforts of topologists was focused on the study of manifolds. During this time, Marston Morse introduced what is now dubbed Morse theory as a technique in studying manifolds. Using this technique, Stephen Smale proved the Poincare conjecture in dimensions 7 and higher (later resolving the cases of 5 and 6). Soon after, Smale proved the celebrated h-cobordism theorem, which became a powerful tool in the classification of high dimensional manifolds. We will study these techniques through two beautiful expositions written by John Milnor.
Prerequisites: Linear algebra (MAT 310/315)
Book: Toric topology, Buchstaber & Panov
Description: We want to understand one or more of the following topics: 1. Geometry of combinatorics of polytopes, 2. Combinatorial structures, 3. Toric varieties, 4. Half-dim torus actions, 5. Applications to Unitary bordism. Our ultimate goal will be to understand the relationship between Toric and Quasitoric manifolds, and models of generators for the Unitary bordism ring, highlighting the combinatorial advantages of using such models.