Advanced Graduate Courses 2014/15

• 1. A little review of Hilbert space
• 2. Basic definitions of linear operators, mostly on Hilbert space; examples.
• 3. Compact operators on Hilbert Spaces
• 4. Definition of a Banach algebra; examples.
• 5. Spectrum, spectral radius, holomorphic functional calculus. (The proofs for holomorphic functional calculus will be omitted.)
• 6. The weak and weak* topologies, Alaoglu’s Theorem, Krein-Milman Theorem. (Some proofs omitted, and general theory of topological vector spaces omitted.)
• 7. Maximal ideal space of a commutative Banach algebra, Gelfand transform. (Nonunital case left as an exercise.)
• 8. Definition of a C*-algebra; examples.
• 9. Foundations of the general theory of C*-algebras: continuous functional calculus, positive elements, ideals, quotients, approximate identities.
• 10. States and representations: GNS construction. Existence of irreducible representations (parts left as exercises)

Math 685

• 1. Double commutant and Kadison transitivity theorems.
• 2. Examples of simple C*-algebras: UHF algebras; crossed products by minimal homeomorphisms; Cuntz algebras; the Calkin algebra. Without proof: The reduced C*-algebra of a free group. This topic includes enough about crossed products by Z to do things like proving simplicity of specific cases.
• 3. General theory of purely infinite simple C*-algebras.
• 4. An outline of nuclearity, mostly without proofs. Tensor products of C*-algebras.

Math 686

• 1. Outlines of the proofs of Kirchberg’s stability theorems, leaving some details to be filled in.
• 2. Asymptotic morphisms and a sketch of E-theory.
• 3. Asymptotic morphisms between Kirchberg algebras: homotopy implies asymptotic unitary equivalence.
• 4. What is the UCT (Universal Coefficient Theorem)?
• 5. Sketch of the proof of the classification theorem for UCT Kirchberg algebras.

Pre-requisites: Math 616, 617, 618

Shabnam Akhtari (12:00): 607 Algebraic Number Theory

TEXTS:

• Number Fields, Daniel A. Marcus, Springer-Verlag
• Algebraic Number Fields, Gerald Janusz, AMS
• J.S. Milne Course Notes: http://www.jmilne.org/math/CourseNotes/ant.html.

This is a graduate-level course on algebraic number theory. The goal is to learn about some objects of algebraic number theory, such as number fields, function fields, local fields, class fields, and elliptic curves. We will learn the proofs of important theorems, such as unique factorization of ideals in Dedekind domains, structure of factorization of rational primes (decomposition and inertia groups), finiteness of the class group and unit group.

Pre-requisites: 500 Algebra

Alexander Ellis (12:00) 607 Categorification

Categorification is the study of “higher structures” which control more familiar algebraic objects. A basic example: singular (co)homology categorifies the Euler characteristic of a space. While categorification is an ad hoc process, decategorification usually takes the form of an Euler characteristic, Grothendieck group, or similar invariant. A categorification often has a richer structure than the original object (e.g. cup product, Poincare duality in the example of singular cohomology).

More recent instances of categorification involve higher category theory, derived and other triangulated categories, and diagrammatic algebras. In this course, our examples will come from topology and representation theory. Polynomial invariants of links such as the Jones polynomial can be categorified by link homology theories. We will start with a quick review of category theory and homological algebra, followed by a discussion of TQFTs in low dimensions. This will allow us to construct Khovanov homology, a bigraded homology theory for links which categorifies the Jones polynomials. We will then turn to examples coming from “towers of algebras”, in which induction and restriction functors categorify the multiplication and comultiplication of a Hopf algebra (work of Geissinger, Zelevinsky, Khovanov-Lauda). Finally, we will bring derived categories, higher categories, and diagrammatic algebra into the picture and study the 2-representation theory of Lie algebras and their quantized enveloping algebras (work of Chuang-Rouquier, Khovanov-Lauda, Rouquier, Cautis-Kamnitzer-Licata, and others). If time permits, we may discuss other topics as well.

By the end of this course, students should expect to have a “category number” of at least 2 (see: http://en.wikipedia.org/wiki/N-category_number).

Vadim Vologodsky (14:00) 607 Mirror Symetry

Description: The goal of the course is to understand the statement and a proof of “the mirror identity” for hypersurfaces of dimension 3. Roughly, this identity computes the number of rational curves of given degree on quintic 3-folds. Along the way we will discuss moduli spaces of algebraic curves, the quantum cohomology, and the Atiyah-Bott localization formula.

Pre-requisites: 600 topology, 600 algebra, and 2 quarters of algebraic geometry.

Weiyong He (2:00) 607 Geometric Analysis

Why does a cat curl into a ball when it is cold? Why drums are almost always round? Both questions are related to the eigenvalues of the Laplace operator. We will calculate explicitly almost all known cases. Since just a few are actually known, we will explore possible ways of estimating Laplace eigenvalues using geometric properties of the domains (e.g. area, perimeter, diameter).

The class requires basic knowledge of undergraduate analysis, matrix algebra and ode. We will try to use computers to visualize the topic.

• 1. Bochner technique (Bochner formula)
• 2. Bishop-Gromov volume comparison
• 3. Laplacian comparison
• 4. Gradient estimate of harmonic function (on manifold w/ Ricci curvature bound)-Cheng-Yau estimate
• 5. Maximum principle
• 6. Gromov-Hausdorff topology
• 7. Cheeger-Gromoll splitting theorem and almost rigidity.

Prerequisite: Riemannian geometry

Alexander Polishchuk (14:00) 607 Derived Categories in Algebraic Geometry

I’d like to present some important recent developments in algebraic geometry involving derived categories of coherent sheaves. I will focus on topics that are relevant for Kontsevich’s homological mirror conjecture (and its generalizations) in which these derived categories play a key role.

The tentative plan is to study:

• 1. Fourier-Mukai functors
• 2. Exceptional collections
• 3. Matrix factorizations (these are very useful when discussing derived categories of hypersurfaces in projective spaces).
• 4. Bridgeland stability conditions

Prerequisite: A course in Algebraic geometry.

Shabnam Akhtari (12:00) 607 Diophantine Analysis

Summary: We will introduce the concept of irrationality measure and study the approximation of irrational numbers by rational numbers. Although almost all real numbers are irrational, it is very difficult to establish the irrationality of a given number. We will study a variety of techniques which can examine the irrationality of different numbers and functions. Later we will focus on the transcendence theory.

Some specific topics to be covered:

• 1. Heights and absolute values in number fields – Diophantine approximation
• 2. Theory of continued fractions
• 3. Diophantine equations and inequalities
• 4. Theory of linear forms in logarithms
• 5. Irrationality of Riemann zeta function at special values – uniform distribution
• 6. Normal numbers
• 7. Transcendence measure

Prerequisite: 500 Algebra and 500 Real Analysis.