Topics courses

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

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

Description: Padé approximants are a frequently used tool for the solution of mathematical and physical problems: solution of nonlinear equations, acceleration of convergence, numerical integration by means of nonlinear techniques, solution of ordinary and partial differential equations.
In this course we are going to study basics of the theory:
1. Definitions and fundamental properties.

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:

  1.  Overview of Calderon-Zygmund theory on spaces of homogeneous type
  2.  Non-homogeneous Calderon-Zygmund theory
  3.  The Cauchy transform and Analytic capacity
  4.  The Riesz transform and removability for Lipschitz harmonic functions
  5.  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:

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.