## Fall 2024

**5010: Analysis and Geometry in Hyperbolic Space**

Instructor: Guozhen Lu

The hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is a simple example of a noncompact and complete Riemannian symmetric space of rank 1. In this course, we will begin with the intro- duction of basic models of real hyperbolic space such as the Poincare ball, the half space, the hyperboloid and the Beltrami-Klein models. We will then establish various important functional inequalities (such as the Hardy- Littlewood-Sobolev, Poincare-Sobolev and Hardy-Sobolev-Maz’ya inequal- ities) and we will investigate their best constants as well as the existence of extremal functions. The course will be self-contained and the only pre- requisite is MATH 5111 (Measure and Integral).

**Math 5020: Cluster Algebras**

Instructor: Ralf Schiffler

Cluster algebras are commutative algebras with a special combinatorial structure which are related to many different areas including Combina- torics (graphs and perfect matchings, triangulations of surfaces, positivity), Representation Theory (quivers, tilting theory, categorification, Lie theory, Coxeter groups), Number Theory (Markov numbers, continued fractions, Lagrange spectrum), Knot Theory (Alexander polynomial, Jones polyno- mial) and many more. The subject is relatively young (first paper in 2002), and it is a highly active research area. Topics covered in the course in- clude the following. Definition and examples, Laurent phenomenon and positivity, finite type classification, relation to representation theory, cat- egorification, combinatorial models, relation to knot theory and number theory.

Prerequisites: MATH 5210.

**Math 5026: Computability**

Instructor: Damir Dzhafarov

This course is an introduction to the basic concepts and techniques in computability theory. You do not need background in logic for this course, and in particular, Math 5260 is not a prerequisite even though you will need a permission number if you have not taken Math 5260. Com- putability theory is the study of which sets of natural numbers (or suitably coded algebraic objects) can be determined by a computer algorithm. Given that there are countably many computer programs but uncountable many sets of natural numbers, most sets are not computable. The focus of this course is on the non-computable sets and how they relate to each other. In particular, we will study how to measure when one set is more complicated than another and what are the natural benchmarks of complexity for non-computable sets.

**5030: Comparison Theorems in Geometry.**

Instructor: Ovidiu Munteanu

In geometry, we try to compare a certain quantity on a curved space with the corresponding quantity on a model space. The model space is often, but not always, a space of constant curvature. Some classical examples of such comparison estimates are:

Volume Comparison: Here, we analyze the volume growth of geodesic balls of arbitrary radii in a manifold with non-negative curvature. We compare this growth to the volume growth of geodesic balls in Euclidean space.

Eigenvalue Comparison: Similar to volume comparison, we compare the eigenvalues of the Laplacian on curved spaces to those on the model space (which may be spheres, Euclidean space, or hyperbolic space).

More generally, one can attempt to understand various geometric and an- alytic quantities on curved spaces by using certain model spaces as canonical examples for comparison. Well-known examples of these quantities include the heat kernel, the Green’s function of the Laplacian, and the famous Isoperimetric, Sobolev, and Minkowski inequalities.

This course will assume basic knowledge of Riemannian geometry, such as curvature and geodesics, but it will be self-contained otherwise. We will start by covering the basic ideas of the theory. Once we get a good grasp of those, we will move on to the newer and more advanced concepts and methods.

## Spring 2024

**Math 5020: Topics in Algebra: Elliptic curves**

Instructor: Alvaro Lozano-Robledo

This course will be an introduction to elliptic curves which, roughly speaking, are smooth cubic curves in the projective plane with at least one rational point (turns out they have a simple model of the form y2 = x3 + ax + b). The surprising feature of elliptic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients. Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat’s Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e., calculate its torsion and rank), and a number of more refined invariants (such as the order of the Shafarevich-Tate group).

The prerequisites for this course are the abstract algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be ”The Arithmetic of Elliptic Curves” by J. H. Silverman, which is the standard graduate-level textbook for the subject.

Prerequisites: MATH 5210 and MATH 5211 (i.e., a year of abstract algebra).

Recommended Preparation: A semester of Algebraic Number Theory, and a semester of Algebraic Geometry.

**Math 5030: Topics in Geometry and Topology: Introduction to Minimal Surfaces**

Instructor: Lan-Hsuan Huang

The question of finding a surface with the smallest area is one of the oldest problems in geometry. It gives rise to the concept of minimal surfaces. The mathe- matical theory of minimal surfaces dates back to the 18th century at the beginning of calculus of variations, and now it is still one of the most active branches of mathematics. Minimal surface theory also has developed a wide range of applica- tions in, e.g. general relativity, molecular engineering, materials science, mechanical engineering, and architecture.

The course will present mathematical properties of minimal surfaces, including the topics on the first and second variations formulas of the area, Bernstein the- orem, Douglas-Rad ́o’s resolution to the Plateau problem, curvature estimates and compactness of minimal surfaces, and the fundamental existence theory of harmonic maps by Sacks and Uhlenbeck.

**Math 5040: Stochastic control theory**

Instructor: George Yin

This is an introductory course on stochastic control theory. We will study both discrete-time and continuous-time stochastic systems. In addition to students in the mathematics department, we hope to attract students from other departments (as we did this year in the MAT 5040 class, several students from statistics and engineering departments enrolled in the class). Selected topics from the list below will be covered.

- Preliminary results (including Markov processes, diffusions, stochastic dif- ferential equations, generators and associated partial differential equations)
- calculus of variations
- controlled diffusion
- stochastic optimization
- dynamic programming
- viscosity solutions of Hamilton-Jacobi-Bellman equations
- maximum principle
- filtering
- adaptive control
- computational methods

## Fall 2023

**Math 5016: Stochastic Processes with Financial Applications**

Instructor: Oleksii Mostovyi

This course will cover topics in stochastic processes, which will lead to study- ing and solving problems arising from finance. The tentative list of topics includes Brownian motion, stochastic integration, diffusions, filtering, optimal stopping, and stochastic control. The financial applications will include pricing and hedging of European and American options and optimal investment. The focus will be on solv- ing the problems in simplified settings rather than establishing the mathematical results in full generality. No preliminary knowledge of finance is required. Working knowledge in probability will help.

**Math 5020: Local fields**

Instructor: Keith Conrad

In addition to completing the rational numbers to form the real numbers, the rationals can be completed in infinitely many other ways: one completion for each prime number p. The resulting completions are called the p-adic numbers and their finite extensions are called local fields. Many concepts from classical analysis (power series, integration, etc.) can be developed over these fields, with properties that are sometimes similar and sometimes quite different from the classical case. Local fields were originally studied and applied just within number theory, but over time they became more mainstream through connections to other areas, such as harmonic analysis, algebraic geometry, dynamical systems, and model theory. This course should be of interest to students in algebra as well as other students who want a broader sense of the scope in which analytic concepts can be developed over fields that are not just the real or complex numbers.

Prerequisites: Math 5210, either 5110 or 5310, or permission of instructor.

## Spring 2023

**Math 5010: Sobolev spaces in metric measure spaces**

Instructor: Fabrice Baudoin

Description: Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first order analysis in non-smooth settings. Based on the fundamental concept of upper gradient the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. In this course we will present that theory and study functional inequalities in that non-smooth setting.

Bibliography: Sobolev spaces on metric measure spaces by J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson.

**Math 5016: Random walks, heat kernels and applications**

Instructor: Alexander Teplyaev

Description: The topic of the first part of the course will be the relationship between random walks and the heat equation. The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation by considering the individual random particles one gains further insight into the problem. We will discuss the discrete case, random walk, and the heat equation on the integer lattice, the continuous case, Brownian motion, and the usual heat equation. Solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion can be introduced and developed from the first principles. We also will discuss martingales and fractal dimensions. The second part of the course will be devoted to a broader range of topics, selected according to the mutual interests of students.

Textbook: Random Walk and the Heat Equation by Gregory F. Lawler, University of Chicago.

## Fall 2022

**Math 5026: Generic sets and forcing in computability theory**

Instructor: Reed Solomon

Description: Forcing is a powerful technique to construct sets in computability theory. We will start by giving a number of classical constructions in computability theory using the terminology and techniques of forcing. Next we will define the forcing language and describe the connection between levels of genericity and satisfaction of formulas in the forcing language. The majority of the course will be devoted to applications of Cohen, Sacks and Mathias forcing in classical computability theory, computable combinatorics and computable model theory. There are no formal prerequisites but students should know the basic definitions in computability theory (e.g. computable and computably enumerable sets, Turing jump, arithmetic hierarchy, etc.).

**Math 5040: Stochastic Approximation and Applications**

Instructor: George Yin

Description: This course presents an introduction to stochastic approximation with various applications. Stochastic approximation stems from the goal of locating roots of a nonlinear function or finding minimizers of a function. In contrast to numerical analysis, either the precise form of the function is not known or it is too complicated to compute and only noisy measurements or observations are available. One constructs a sequence of estimates recursively to carry out the desired task. Standard procedures and their variants such as projection and truncation algorithms will be introduced. Convergence, rates of convergence, and asymptotic efficiency will be studied in connection with ordinary differential equations, stochastic differential equations, and martingale problem formulations. If time permits, large deviations will also be presented.

**Math 5121: Riemann Surfaces and Complex Manifolds**

Instructor: Damin Wu

Description: A Riemann surface is a complex manifold of complex dimension one. Every Euclidean surface admits a complex structure and hence is a Riemann surface. From the algebraic geometric viewpoint, a Riemann surface is a smooth complex algebraic curve. In this course, we shall introduce geometry of complex manifolds via the study of Riemann surfaces. Normally the techniques and concepts that look difficult and abstract in higher dimensions can be seen and understood clearly in the case of Riemann surface. We shall develop tools in differential geometry, partial differential equations, and topology including sheaf cohomology.

## Fall 2021

**Math 5010: Probabilistic techniques in analysis**

Instructor: Alexander Teplyaev

Description: This course will be based on the book by Rich Bass with the same title, and on supplementary materials. In recent years, there has been an upsurge of interest in using techniques drawn from probability to tackle problems in analysis. These applications arise in subjects such as potential theory, harmonic analysis, singular integrals, and the study of analytic functions. This book presents a modern survey of these methods at the level of a beginning Ph.D. student. Highlights of this book include the construction of the Martin boundary, probabilistic proofs of the boundary Harnack principle, Dahlberg’s theorem, a probabilistic proof of Riesz’ theorem on the Hilbert transform, and Makarov’s theorems on the support of harmonic measure.

The author assumes that a reader has some background in basic real analysis, but the book includes proofs of all the results from probability theory and advanced analysis required. Each chapter concludes with exercises ranging from the routine to the difficult. In addition, there are included discussions of open problems and further avenues of research.

**Math 5020: Commutative Algebra**

Instructor: Mihai Fulger

Description: Commutative Algebra is the study of commutative rings and modules over them. Topics include Noetherian rings and modules, ideals, rings of polynomials, Hilbert’s basis theorem, nilpotents, prime and maximal ideals and the topology on the spectrum of a ring, dimension theory, regular rings. Time permitting we will go into more advanced topics like resolutions, computational algebra, homological algebra, and Cohen-Macaulay rings.

Commutative Algebra is a precursor to the Algebraic Geometry course to be offered in Spring 2022, but will also help anyone interested in Algebraic Number Theory.

**Math 5026: Computability, Randomness and Genericity**

Instructor: David Solomon

Description: This course will start with an introduction to the basic concepts of computability theory, but the main focus will be on notions of algorithmic randomness and genericity. We will see three equivalent approaches to algorithmic randomness: incompressibility and Kolmogorov complexity; effective null sets; and effective martingales. Once we have a robust hierarchy of randomness notions, we will explore the connections between the Turing degrees of random sets and those of other classes such as PA degrees, DNR degrees and DNR2 degrees. For the last part of the course, we will turn to generic sets and study their properties and applications in the Turing degrees. This course will not assume prior knowledge of computability theory, but also will have as little overlap as possible with the topics from the computability theory course in Spring 2020.

**Math 5030: Topics in Geometric Partial Differential Equations**

Instructor: Lan-Hsuan Huang

Description: This course is an introduction to analytical methods, specifically partial differential equations (PDEs), on the study of geometry and topology. The course will begin with an introduction to some basics of elliptic PDEs, functional analysis, and notions of curvatures. Then I will present more advanced topics, with an emphasis on isometric embedding of Riemannian manifolds and curvature problems in mathematical relativity. In particular, I plan to discuss the following topics:

1. Nirenberg’s resolution to Weyl problem for surfaces of positive Gauss

curvature

2. Nash’s isometric embedding theorem

3. Schoen-Yau’s positive mass theorem in mathematical relativity

I will try to make the course self-contained. While it is useful to have prior knowledge in either differential geometry or PDEs, it is not required to take this course.

**Math 5040: Padé approximation and its applications**

Instructor: Maksym Derevyagin

In this course we are going to study basics of the theory:

2. Various ways of computations of Padé approximants.

3. Connections with some convergence acceleration methods.

4. Connections with continued fractions.

5. Convergence theory.

We will mainly follow the book by G. Baker and P. Graves-Morris entitled Padé approximants, Parts I and II but will also use modern papers when discussing some applications and new insightful ideas.

Prerequisites: familiarity with basics of complex analysis and linear algebra.

## Spring 2021

**Math 5010: Singular integrals and applications**

Instructor: Vasileios Chousionis

Description: This course will focus on the modern theory of singular integrals and its connections to geometric measure theory, complex analysis and potential theory. We will cover topics such as:

- Overview of Calderon-Zygmund theory on spaces of homogeneous type
- Non-homogeneous Calderon-Zygmund theory
- The Cauchy transform and Analytic capacity
- The Riesz transform and removability for Lipschitz harmonic functions
- Singular Integrals on Lipschitz graphs

Good understanding of measure theory is required. Some knowledge of Fourier Analysis might be useful but 5140 is not a prerequisite.

**MATH 5020: The Arithmetic of Elliptic Curves**

Instructor: Alvaro Lozano-Robledo

Description: This course will be an introduction to elliptic curves, which roughly speaking are smooth cubic curves in the projective plane (turns out they have a simple model of the form $y^2=x^3+ax+b$). The surprising feature of elliptic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients.

Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat’s Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e., calculate its torsion and rank), and a number of more refined invariants (such as the order of the Shafarevich-Tate group).

The prerequisites for this course are the Abstract Algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be “The Arithmetic of Elliptic Curves,” by Silverman, which is the standard graduate-level textbook for the subject.

**MATH 5026: Introduction to Reverse Mathematics**

Instructor: Reed Solomon

Description: In mathematics, we sometimes encounter theorems that “are equivalent to the axioms of choice” or “require the parallel postulate to prove”. Reverse mathematics is the study of which axioms are required to prove specific theorems. In general, the axioms of set theory are too powerful for a reasonable analysis of this type, so the setting of reverse mathematics is second order arithmetic.

This course will start with an introduction to second order arithmetic and its most prominent subsystems. For each of the subsystems, we will analyze theorems from a variety of branches of mathematics that are equivalent to the subsystem.

The end of the course will focus on the existence of special models of the subsystems that give rise to conservation results.

The only prerequisite for this course is a general introduction to logic such as Math 5260.

**Math 5030: Geometric Analysis on Manifolds**

Instructor: Ovidiu Munteanu

Description: This course is an introduction to the linear theory of partial differential equations on open manifolds. Assuming some appropriate information on curvature, we will study properties of solutions to the Laplace and heat equations, which in turn will give us more insight about the geometry and topology of the underlying manifold.

For example, we will use harmonic functions to count the number of ends of open manifolds, and see applications to rigidity results for manifolds with more than one end.

The techniques developed in this theory are essential to many other problems in geometric analysis, such as in the study of geometric flows on manifolds.

The course follows Peter Li’s book Geometric Analysis, Cambridge Studies in Advanced Mathematics

Prerequisites: Basic knowledge of Riemannian geometry will be assumed. The course is self contained on the PDE’s side.

## Fall 2020

**Math 5010: Yang-Mills Theory**

Instructor: Maria Gordina

Description: We will discuss mathematical foundations of the standard model of elementary particle theory.

We will begin with classical physics equations of Newtonian mechanics, then will move to the Lagrangian mechanics, Hamiltonian mechanics, quantum mechanics (including Heisenberg versus Schroedinger picture), and finally arrive to the quantum field theory. In parallel we will need to understand two classical theories, namely, Maxwell’s equations and Yang-Mills equations. The needed differential geometrical notions will be covered as well.

**Math 5020: Enumerative Combinatorics**

Instructor: Thomas Roby

Description: The course will give an introduction to enumerative combinatorics at the graduate level, focusing on techniques useful for those specializing in other research areas as well as combinatorics. Topics will include: basic enumeration, multiset permutations and statistics, generating functions, bijective proofs, q-analogues, sieve methods, exponential formula, Lagrange inversion.

Text: Enumerative Combinatorics 1 (2nd edition) and 2, by Richard Stanley (selections from chapters 1,2,4,5, and 6).

**Math 5031: Einstein Manifolds**

Instructor: Fabrice Baudoin

Recordings of Lectures: https://sites.google.com/site/fabricebaudoinwebpage/einstein-manifolds?authuser=0

Description: In this course we will study some topics in Riemannian and pseudo- Riemannian geometry. We will mostly focus on Ricci curvature and its applications. The course will start with basics about Riemannian and pseudo-Riemannian geometry. We will assume familiarity with differential manifolds and basic calculus on them.

We will cover the following topics:

• Linear connections on vector bundles: Torsion, Curvature, Bianchi identities

• Riemannian and pseudo-Riemannian manifolds

• Get the feel of Ricci curvature: Volume comparison theorems, Bonnet-Myers theorem

• Ricci curvature as a PDE

• Einstein manifolds and topology

• Homogeneous Riemannian manifolds

• Kahler and Calabi-Yau manifolds

• Quaternion-Kahler manifolds

Text: Einstein manifolds, by A.L. Besse, Springer, 1987.

**Math 5040: Topics in PDE**

Instructor: Xiaodong Yan

Description: This topic class will cover two different topics in PDE. In 1970s, De Giorgi made a conjecture regarding level sets for solutions of semilinear PDE’s. Over the years, many mathematicians have made contributions to the problem. For the first topic, we will discuss results and technique for classical De Giorgi conjecture and its extension to fractional PDEs and (possibly) models related to thin film equations.

First introduced by Skyrme in 1960s, skyrmions are topological solitons that emerge in many physical contexts such as superfluid, Bos-Einstein condensates with ferromagnetic order, liquid crystals and magnetism. Magnetic skyrmions have attracted a lot of attentions in recent years due to their topological structure which hold promises for future information technologies. In the second part of the class, we will discuss some recent results on magnetic skyrmions.

**Math 5121: Conformal Dynamics: from the complex plane to Carnot groups**

Instructor: Vasileios Chousionis

Description: The course will cover a variety of topics whose common underlying theme is the iteration of conformal maps. Such as:

- Basics of complex dynamics including iteration of meromorphic functions
- Conformal maps in Euclidean spaces and conformal fractals
- Rigidity of conformal maps in Carnot groups
- Conformal graph directed Markov systems on Carnot groups
- Dynamics of real, complex and Iwasawa continued fractions

On the way we are going to introduce and employ ideas from geometric measure theory, ergodic theory, geometric function theory and sub-Riemannian geometry.