Graduate Courses, Summer 2022 – Spring 2023

Other good textbooks:

1. Hartshorne, Algebraic Geometry
2. Mumford, The Red Book of Varieties and Schemes
3. Mumford-Oda, Algebraic Geometry, II
4. Görtz-Wedhorn, Algebraic Geometry, I

1. Sheaf cohomology
2. Schemes
3. Morphisms (affine, proper, flat, smooth, étale, etc. )
4. Cohomology of projective schemes.

This course is designed to give an introduction to representation theory, with an emphasis on Lie algebras and algebraic groups. The class is designed to be suitable both for students planning to specialize in representation theory and for those who need it for applications. It will start with an outline of the representation theory of finite groups over the complex numbers. We will then introduce complex algebraic groups and their Lie algebras. After discussing the basic theory of nilpotent, solvable, and semisimple Lie algebras, we will describe the classification of semisimple Lie algebras. We will continue with the universal enveloping algebra of a Lie algebra and the Poincaré-Birkhoff-Witt theorem. We will then cover highest-weight representations of Lie algebras, including Verma modules and finite-dimensional irreducible representations. We will also discuss the relationship between the representations of a semisimple algebraic group and the representations of its Lie algebra.

We use our own lecture notes. But those are very close to the book by Folland. There are several other very good books on analysis and measure theory:

1. R. G. Bartle: The Elements of Integration and Lebesgue Measure
2. A. Friedman: Foundations of Modern Analysis.
3. P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
4. F. Jones: Lebesgue Integration on Euclidean Space.
5. L. Richardson: Measure and Integral: An Introduction to Real Analysis.
6. H. L. Royden: Real Analysis.

Math 7365 is a course on stochastic processes. A stochastic process can be thought as a random function of time which originates in modeling temporal dynamics of many systems. Detailed modeling of such systems requires incorporation of their inherent randomness which deterministic methods fail to capture. Examples of such systems are numerous - from biological networks to financial markets. Markov processes, in particular, form one of the most important classes of stochastic processes that are ubiquitous in probabilistic modeling. They also lead to probabilistic interpretations of a large class of PDEs. For example, Brownian motion is the underlying Markov process whose probability distribution satisfies the heat equation.

This course is a continuation/concurrence of Math 7360. Together they give a comprehensive introduction to measure theoretic probability which should be ideal for those wishing to study probability, or use it as a tool in analysis, statistics, mathematical biology, economics, finance or applied mathematics. The course also serves as a gateway to Math 7366 on stochastic analysis.

The course will cover theory of martingales and Markov processes in discrete-time. Some specific topics include Doob's decomposition theorem, Doob's inequalities, Burkholder-Davis-Gundy inequality, Kolmogorov's equations, generators, stationary measures, some elementary stability theory, ergodic theory and Poisson processes. If time allowed, we will also briefly discuss about the continuous-time case, in particularly, the Brownian motion and some stochastic algorithms like Markov Chain Monte Carlo.

Additional optional references:

• Elliptic partial differential equations by Qing Han and Fanghua Lin

1. Introduction of Laplace's equation, the heat equation, and the wave equation
2. Introduction of Sobolev spaces on weak derivatives, traces and Sobolev embedding
3. Elliptic equations on existence, regularity and maximum principle
4. Introduction of Calculus of Variations on Euler-Lagrange equations
5. Existence of minimizers and eigenvalue of self-adjoint elliptic operators

A Classical Introduction to Modern Number Theory (Springer GTM 84), by Kenneth Ireland and Michael Rosen. (We will mainly follow chapters 6, 8-14, 16-19 of the textbook. )

Some useful references:

1. Gauss and Jacobi Sums, by Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams
2. Number Fields, by Daniel A. Marcus
3. Rational Points on Elliptic Curves, by Joseph H. Silverman, and John Tate.

Character sums and zeta functions are closely related to the L-functions and Galois representations attached to a given algebraic variety. We will look at some examples arising from Diophantine equations and also certain special elliptic curves.

The beginning of this course will focus on the combinatorial structure of Coxeter groups, and then we will move to studying their geometry and topology. The main topic comes from the 1988 Ph.D thesis of Moussong, who showed that any infinite Coxeter group acts by isometries on a contractible cell complex admitting a singular, nonpositively curved metric; a coarse geometric analogue of Euclidean or hyperbolic space. Throughout the course, there will also be a focus on using Coxeter groups to construct exotic examples of manifolds and other complexes.

Possible topics for the end of the semester are buildings, various flavors of cohomological invariants, applications to combinatorics, and Artin groups.