Book: Toric Varieties, David A. Cox, Hal Schenck, and John B. Little.
Prerequisites: Proofs. Some abstract algebra including ring theory (e.g. MAT 313) is highly recommended.
Area: Algebraic Geometry.
Description: Fundamentally, algebraic geometry is the study of the spaces formed by looking at the zero sets of systems of polynomials. Such spaces are called varieties. Toric varieties are a particularly nice class of varieties that can be represented by a combinatorial object called a fan. By examining the fan, many important computations can be carried out for toric varieties with relative easy, while they are hard to do for general varieties.
In this project, our goal will be to understand the correspondence between toric varieties and fans. We will also see how to use the fan to compute important data such as the divisor class group and resolution of singularities for a toric variety.
This project does not assume prior knowledge of algebraic geometry, so we will most likely begin by reading some key sections from a text on undergraduate algebraic geometry before starting the main text (although the main text does review all necessary concepts from algebraic geometry).
Book: Algebraic Curves and Riemann surfaces, Miranda.
Prerequisites: Basic topology (MAT 364 or MAT 530). Multivariable calculus (MAT 203 or higher). Algebra (MAT 312 or MAT 524).
Area: Algebraic geometry.
Description: Algebraic curves are the simplest and oldest objects of study in algebraic geometry, it’s very deep and still a vibrant area of research. Students can get a sense of what is algebraic geometry and know some methods to study curves. The aim of this reading project is to let students learn the Riemann-Roch theorem for algebraic curves, see how this theorem elegantly combines algebraic, complex-analytic and topological information of a compact Riemann surface (a smooth projective curve), and applications of this theorem. One possible final project is to apply this theorem and see how it could tell us the moduli space of curves has dimension 3g-3. There are two path ways to learn this, one of algebraic flavor and another of a more complex-analytic flavor, but they are all the same eventually. Students can choose how to approach it depending on their interests and what they know.
Book: Category Theory in Context, Emily Riehl.
Prerequisites: A basic knowledge of set theory is a hard prerequisite (MAT 200/250). However, the more you know, the better. In particular, some understanding of foundational concepts from abstract algebra (MAT 312), linear algebra (MAT 310), and topology (MAT 364) would be advantageous.
Area: Category Theory.
Description: Category theory offers a cross-disciplinary language that describes general phenomena in mathematics, enabling the transfer of ideas across different areas of study. It is widely utilized in various fields within mathematics, including algebraic geometry, homotopy theory, symplectic geometry, and low-dimensional topology. We shall tentatively cover selected topics from the first three chapters of Riehl's book, focusing on key concepts and applications that illustrate the relevance of category theory in these fields.
Book: Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication, David Cox.
Prerequisites: Elementary number theory (e.g. Number Theory MAT 311).
Area: Algebraic number theory.
Description: The development of number theory, at the early age, was closely associated with various Diophatine equations. Euler's proof of Fermat's theorem, which states that every prime of 4k+1 type has the form x^2+y^2, led to the discovery of quadratic reciprocity. Similar conjectures were proposed by Euler, hoping to find a general result for x^2+ny^2 with general integer n>1. Solving the cases for n=27, 64 using cubic and biquadratic reciprocity made Gauss recognized the role of higer reciprocity laws in the quadratic forms of the same "genus". Here we arrived at the fundamental idea of algebraic integers which motivated the class field theory and algebraic number theory.
Book: Fractal Geometry: Mathematical Foundations and Applications, 3rd Edition, Kenneth Falconer.
Prerequisites: A course in analysis (e.g. MAT 319/320); familiarity with measure theory (e.g. MAT324) would be helpful, but is not necessary.
Area: Analysis.
Description: Fractals arise in many areas of mathematics, including but not restricted to: dynamical systems, harmonic analysis, number theory and probability. In this project, we will first study the general theory of fractals, especially the various notions of fractal dimensions and methods for their calculation. With the general theory developed, we will study applications of it based on the student's interest. Some possible directions of the project are: analyse and construct (general) Cantor set of any given Hausdorff dimension and measure; study the fractal attractor of the logistic family and universality, etc.
Book: Iteration of Rational Functions, Alan F. Beardon.
Prerequisites: Introduction to Real Analysis (MAT 319/320 or higher); MAT 342 (or as corequisite taking MAT 536 in Spring 2025).
Area: Dynamics, Analysis.
Description: The goal of this project is to give an introduction complex dynamical systems in the plane by mainly studying polynomial dynamics. Topics include: stability of fixed points, basins of attraction, complex quadratic family, invariant sets for polynomial dynamics, conjugacies and the Mandelbrot set. One of the goals of the project will also be to generate some pictures of dynamical planes and parameter planes.
Book: Introduction to Stochastic Calculus and Applications, Fima Klebaner.
Prerequisites: Familiarity with probability, real analysis (MAT 319/320). A course in differential equations (MAT 308) and understanding the basics of measure theory (MAT 324) will be helpful.
Area: Probability, Analysis, Applied Math.
Description: This project is an introduction to stochastic calculus, and the analysis of stochastic differential equations. Topics include introduction to probability in the language of measure theory, construction of Brownian motion as a scaling limit of random walks and its properties, stochastic integration and properties of the Ito integral- Brownian motion is nowhere differentiable, so how can we construction an integral with respect to the Weiner measure? Analysis of SDEs- we will prove and apply the beautiful Ito’s formula to obtain strong solutions to SDEs. We will then study martingales from the viewpoint of SDEs- the Ito integral is a martingale, but it turns out that the converse is also true- every martingale can be represented by an ito integral- this will be one of the biggest theorems we will study in the project. We will then look at the connection of SDEs to PDEs, in particular we will look at how the amazing properties of the heat equation and how its solutions give us a nice interpretation of hitting times. The last topic is change of measure and the Girsanov theorem which gives us a way to formulate weak solutions to SDEs.
Time permitting, and depending on interest, we can study applications to finance (black scholars model), more topics in stochastic processes like Markov chains, poisson processes, etc…
Book: Lipschitz Functions, Cobzas, Minulescu, and Nicolae.
Prerequisites: Real analysis (MAT 319/320), measure theory (MAT 324), and topology (MAT 364).
Area: Analysis.
Description: Functions are one of the main tools and objects of study in math. The two classes of functions a student typically learns about in calculus are continuous functions and differentiable functions. In between these two lies the class of Lipschitz functions. These are the functions that distort distances only by a bounded amount. We'll study how Lipschitz functions are flexible (like continuous functions), but have many of the nice properties of differentiable functions.
Book: The Symmetric Group Representations, Combinatorial Algorithms, & Symmetric Functions, by B.E Sagan.
Prerequisites: MAT 313 (Algebra I), MAT 310 (Linear Algebra), MAT 319 (Analysis) is recommended, but perhaps not necessary.
Area: Probability, Representation Theory.
Description: Many classic questions in probability can be simply stated, though often have deep theory to their study. An example of this is card shuffling; given a method of shuffling, how many times must this method be applied to adequately shuffle the cards. To answer this, a dive into the theory of representations for finite groups, specifically for the symmetric group or permutation group, can give us an answer. In this project, we would study the basics of representations, find and illustrate the correspondence between partitions and representations of the symmetric group, and see how these methods apply to card shuffling problems. Given enough time and interest, some of the theory behind Markov chains would also be of use.
Book: Introduction to Topological Manifolds, John Lee.
Prerequisites: Undergraduate topology (MAT 364).
Area: Topology.
Description: Compact surfaces are the manifolds which are most familiar from our everyday experience (image a basketball or a doughnut). How many types of surfaces are there? The classification of compact surfaces says that every compact, connected surface is topologically either a sphere, a connected sum of one or more tori, or a connected sum of one or more projective planes. In this project, we will study the classification theorem and construct compact surfaces via gluing edges of a polygon in some handy way.
Book: We can refer to any subset of these books: Bernard Schutz's A First Course in General Relativity, Robert Wald's General Relativity, Barrett O'Niell's Semi-Riemannian Geometry, Hawking and Ellis' The Large-Scale Structure of Spacetime.
Prerequisites: Required background in multivariable calculus and linear algebra (MAT 307). Helpful but not mandatory: some background in differential geometry, differential equations (MAT 308) and special relativity.
Area: Differential Geometry, Analysis, Mathematical Physics.
Description: We will study aspects of general relativity from a mathematically rigorous standpoint, based on the mathematical background and interests of the student. Potential areas of study include detailed investigation of a particular metric (like Schwarzschild/Kerr/FLRW), black hole thermodynamics, cosmology, topics in Lorentzian geometry like Penrose diagrams and the Penrose-Hawking singularity theorems, etc. While it will be tailored to the student's skills and interests, all of them will involve some differential geometry.
Book: Mathematical Methods of Classical Mechanics, V. I. Arnold.
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.
Area: Dynamics, Geometry.
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.
Book: Differential Topology, Guillemin and Pollack.
Prerequisites: Analysis MAT319/320
Area: Topology.
Description: A (smooth) manifold is a space which is locally modeled on Euclidean space. Given two manifolds which are subsets of a larger manifold (e.g., imagine drawing two circles on the surface of a donut), one can count the number of points at which they intersect.This simple idea forms the foundation of "intersection theory," which reveals deep insights into a manifold's topology. In this project, we will study the basic properties of smooth manifolds and their intersections, applying them to prove notable results in topology such as the Jordan curve and Boursak–Ulam theorems.
Book: Computational Topology: An Introduction, Herbert Edelsbrunner and John Harer.
Prerequisites: MAT211/AMS210 (Calculus I-IV). Some coding experience, a basic understanding of probability. Some Analysis (MAT 319/320) would be helpful but not necessary.
Area: Topology, Applied Mathematics.
Description: Topology, viewed by many undergraduates, is a mysterious bottle that, if opened, releases a whirlwind of abstract and dizzying geometric nonsense that has no applications to the real world. In reality, the use of topology is bountiful. The project is geared toward applying topology in a specific setting with an emphasis on building an understanding through developing solutions to real world problems. Problems and applications could be from topics such as data analysis, physics, chemistry, and biology.