Prerequisites: Linear algebra (MAT 310/315), Abstract algebra (MAT 313). A hint of idea about varieties is beneficial, and the willingness to accept GIT as a blackbox.
Book: Quiver Representations and Quiver Varieties, by Alexander Kirillov Jr.
Description: Quivers serves as a graphical device visualizing a category with objects being the nodes and arrows being morphisms. Ignoring the underlying categorical background, there are rich and fruitful geometry coming from representations of quivers. Recent research work has extended this idea beyond its classical meanings to include higher arrows without needing extra knowledge than the classical theory. The goal of this project is to survey such development in the research in an exposition written by the DRP student integrating this idea to quiver theories.
Prerequisites: A solid knowledge of linear algebra (at least MAT 211)
Book: Humphreys: Introduction to Lie algebras and representation theory
Description: Lie algebras are ubiquitous in all areas of mathematics. They are some linear-algebraic objects that describe certain symmetry. In this project we study the basic theory of Lie algebras and their representations. One potential project is to learn the classification theory of simple Lie algebras.
Prerequisites: Abstract algebra I and II (MAT 313 and a second course), Linear algebra (MAT 310/315)
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: Proof-based mathematics (MAT 200/250), Abstract algebra (MAT 313), Topology (MAT 364, MAT 528). Some prior exposure to algebraic geometry would be helpful as well.
Book: Possible books include: Algebraic Curves or Toric Varieties (both by Fulton)
Description: This project is primarily supposed to be a second course in algebraic geometry. The actual topic can be decided based on the student's background and interests. Possible topics include algebraic curves or toric varieties, or, on the more advanced side, algebraic surfaces, singularity theory, or some topics in commutative algebra usually not covered in a standard class.
Prerequisites: Topology (e.g. MAT 364), Complex analysis (e.g. MAT 342)
Book: Riemann Surfaces by Simon Donaldson
Description: Riemann surfaces were introduced by Riemann to geometrically study multi-valued complex functions such as the square root or logarithm. Despite being relatively simple objects, they sit at the intersection of complex analysis, differential geometry, and algebraic geometry and thus serve as an excellent tool to introduce many foundational concepts and techniques in all of these areas. In this project, we will study the basic properties of Riemann surfaces and then have many directions for further study (e.g. elliptic integrals, Riemann-Roch formula, or uniformisation theorem) depending on the student's interest.
Prerequisites: Multivariable calculus (MAT 307), Analysis (MAT 319/320), Measure theory (MAT 324). Familiarity with basic PDE (Evans Ch. 1–5) will be useful.
Book: Note on the dispersion of waves by Sung-Jin Oh, Non-linear waves note by Jonathan Luk, Nonlinear dispersive equations: local and global analysis by Terence Tao
Description: The tentative goal of this project is to understand Sung-Jin Oh's notes presenting three distinct proofs of dispersion for the wave equation: the Fourier method, the vector field multiplier method, and wave packet decomposition. While the dispersive behavior of waves is well known, this project can serve as a useful motivation for students to learn these three complementary approaches, each of which can be adapted to a broader range of wave-type problems.
Along the way, students will also be expected to develop familiarity with some of the basic analytical tools used in the study of PDEs, including Sobolev spaces, Littlewood–Paley theory, and energy estimates.
Prerequisites: Linear algebra (MAT 310) and at least one upper-division proof-based mathematics course. Familiarity with abstract algebra (MAT 313), topology (MAT 364), or differential geometry is strongly recommended.
Book: Differential Geometry, Lie Groups, Symmetric Spaces - Sigurdur Helgason
Description: Symmetric spaces form a meeting point of differential geometry and Lie theory, where geometric symmetries can be understood through the structure of Lie groups and Lie algebras. In this project, we will begin with Lie groups, homogeneous spaces, and basic Riemannian geometry, and then develop the theory of symmetric spaces and the Cartan decomposition of semisimple Lie algebras. We will study concrete examples such as hyperbolic space and homogeneous spaces associated with classical matrix groups, as well as the duality between symmetric spaces of compact and noncompact type. Depending on the student's background and interests, later topics may include the Iwasawa decomposition, Hermitian symmetric spaces, bounded symmetric domains, and compact duals. As a motivating application, we may also discuss how these ideas appear in Hodge theory: period domains arise as homogeneous complex manifolds sitting inside compact dual flag varieties, and Hermitian symmetric spaces provide particularly important examples of this general picture. The emphasis will be on understanding the geometric meaning of the Lie-theoretic constructions through explicit examples.
Prerequisites: Differential geometry, Partial differential equations
Book: Blaine Lawson’s notes on Minimal surfaces, other resources TBD
Description: The idea of the project is to read and understand Blaine Lawson’s notes on Minimal surfaces (link: https://www.math.stonybrook.edu/~blaine/MinSubs.pdf). Additional resources might be needed but will be defined as we go through the material.
Prerequisites: Linear algebra (MAT 310/315), Abstract algebra (MAT 313), manifolds, and ideally some familiarity with physics
Book: Introduction to Supergeometry (Yuri Manin), Quantum Field Theory: Batalin-Vilkovisky Formalism and Its Applications (Pavel Mnev), Geometry of Batalin-Vilkovisky Quantization (Albert Schwarz)
Description: Supergeometry is the study of spaces ("supermanifolds") whose rings of "coordinates" are supercommutative, meaning they contain some elements that anti-commute. Any vector bundle can be canonically thought of as a supermanifold, but there is no functor going in the opposite direction because supermanifolds have bizarre automorphisms that lack a clear geometric interpretation. These automorphisms are called "supersymmetries" and are of great interest to physicists.
We will work our way up to the definition of odd-symplectic supermanifolds and the proof of the Odd-Symplectic Stokes' Theorem. This theorem is the core of the Batalin-Vilkovisky (BV) formalism, a very powerful method which can be used to treat any quantum field theory (QFT), and which can be taken as the starting point for a mathematically rigorous definition of perturbative QFT.
Prerequisites: Real analysis (at least MAT 319), or clear experience with point-set topology or real analysis at a higher level
Book: Problems and Theorems in Classical Set Theory, Peter Komjath & Vilmos Totik
Description: This project will be focused on giving the student a learning-by-discovery approach to advanced set theory and its applications to real analysis, algebra, and combinatorics. Topics covered could be notions of cardinals, ordinals, ordinal arithmetic, transfinite enumeration, pathological sets and functions in R^n, Paradoxical sets and the Banach-Tarski paradox, infinitary combinatorics, and more. This project will be ideal for a student who wants to adopt the "learning by discovery" tradition of Hungarian mathematicians.
Prerequisites: Geometry and topology (MAT 548). Background in algebraic topology (MAT 541) and differential topology (MAT 566) would be helpful.
Book: Lectures on Symplectic Geometry by Ana Cannas da Silva; Zinger's notes on mirror symmetry (available at https://www.math.stonybrook.edu/~azinger/mat566-spr25/MirrSym.pdf)
Description: Integrations over manifolds usually contain interesting geometric information, but they can be hard to compute. The localization theorem by Atiayh and Bott reduces the integral to the one over fixed locus if there is a Lie group acting on the manifold. We will first explore the concept of equivariant cohomology in which the theory of localization lives. Then we will learn the statement of the localization theorem and prove it. If time permits, we will apply it on some examples.
Prerequisites: Point-set topology/metric spaces (MAT 528), Abstract algebra (MAT 313); these are the only essential prerequisites. It's preferable that the student is familiar with fiber/vector bundles and homotopy/homology groups, otherwise, they should be willing to learn the basics quickly. Knowledge of more advanced topics (such as algebraic topology or differential geometry) is helpful but not required. The more you know, the better.
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 algebraic topology and geometry. This project is ideal for students interested in topological, algebraic, and categorical, rather than analytical, aspects of geometry.
Prerequisites: Linear algebra (MAT 310/315), Multivariable calculus (MAT 307), Point-set topology (MAT 528)
Book: Topology from the Differentiable Viewpoint Milnor; Differential Topology Guillemin and Pollack
Description: This project should serve as a fairly comprehensive introduction to Differential Topology. We will cover topics such as smooth manifolds, transversality, vector fields, the Pontryagin-Thom construction, etc. We develop these tools with the aim of proving the correspondence between framed cobordism classes of manifolds and the stable homotopy groups of spheres.
Prerequisites: Morse theory, Homotopy theory, Homology, basic knowledge of smooth manifolds
Book: Lectures on the h-cobordism theorem Milnor
Description: The mid to late 1900's saw an explosive growth in our understanding of high-dimensional manifolds through the study of cobordisms. This culminated in the classification of high-dimensional manifolds via the use of surgery theory. In this project we will work through one of the early triumphs from this period: Smale's h-cobordism theorem. We will work through Milnor's elegant treatise on this theorem using Morse theory. If time permits, we will also take a look at the notes on surgery theory by Ranicki.
Prerequisites: Proof-based mathematics (MAT 200), Algebra (MAT 312/313), Analysis (MAT 319/320)
Book: Topology by J. Munkres or Topology Now! by R. Messer and P. Straffin
Description: How many holes does a straw have? How many does a shirt have? These recurring questions have an answer, and math can provide it!
In this project we aim to introduce the notion of a topology and its main properties. On this base, with the help of some group theory, we can build the notion of a fundamental group of a topological space, which is able to detect holes of the given space. We can then proceed to compute some examples of fundamental groups.
Prerequisites: Analysis (MAT 319/20), Multivariable calculus (MAT 307), preferrably Topology (MAT 528)
Book: Milnor, Morse theory; Milnor, Topology from the differentiable viewpoint
Description: Morse theory connects the topology of a manifold with the critical points of certain smooth functions on it and plays an important role in many areas of mathematics. This project will begin with the basics of smooth manifolds and then develop the foundations of Morse theory. Finally, we will explore some applications to the calculus of variations, particularly the study of geodesics on Riemannian manifolds.
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 very helpful.
Book: Milnor, Lectures on the h-cobordism theorem, and Cieliebak-Eliashberg-Mishachev, Introduction to the h-Principle
Description: Recent AI breakthroughs in math have demonstrated that human mathematicians are not that good at coming up with explicit constructions of (counter-)examples. This makes it all the more miraculous that, on the other hand, there are several instances where mathematicians have discovered systematic (often, in some sense, implicit) procedures of constructions of objects that (a priori) seem impossible to directly/explicitly construct, and sometimes that are even counterintuitive. Some famous examples are: Nash's C^1-isometric embedding theorem, Smale's sphere eversion, Whitney trick and the h-cobordism theorem... In this reading project, we will study one or several of these examples, and explore general principles (e.g. Gromov's h-principle) of such "flexibility" phenomena in differential topology.