# Advanced Graduate Courses 2012/13

See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2012 |
WINTER 2013 |
SPRING 2013 |

511 Intro to complex analysis I | 512 Intro to complex analysis II | |

J. Isenberg | J. Isenberg | |

513 Intro to analysis I | 514 Intro to analysis II | 515 Intro to analysis III |

C. Sinclair | C. Sinclair | P. Gilkey |

520 Differential equations I | 521 Differential equations II | 522 Differential equations III |

M. Yattselev | M. Bownik | M. Bownik |

531 Intro to topology I | 532 Intro to topology II | 533 Intro to diff’l geometry |

V. Vologodsky | V. Vologodsky | J. Isenberg |

544 Intro to algebra I | 545 Intro to algebra II | 546 Intro to algebra III |

N. Proudfoot | A. Vaintrob | A. Vaintrob |

556 Networks and combinatorics | 557 Discrete dynamical systems | |

A. Vaintrob | L. Santiago-Moreno | |

567 Stochastic processes | 558 Cryptography | |

D. Levin | B. Young | |

607 Algebraic number theory | 607 Analytic number theory | 607 Hodge theory |

C. Sinclair | C. Sinclair | W. He |

607 Potential theory/Brownian motion | 607 Simple rings | |

B. Siudeja | A. Berenstein | |

607 Group Cohomology | ||

D. Sinha | ||

616 Real analysis I | 617 Real analysis II | 618 Real analysis III |

H. Lin | H. Lin | C. Phillips |

634 Algebraic topology I | 635 Algebraic topology II | 636 Algebraic topology III |

B. Botvinnik | B. Botvinnik | B. Botvinnik |

637 Differential geometry I | 638 Differential geometry II | 639 Differential geometry III |

W. He | P. Gilkey | P. Gilkey |

647 Abstract algebra I | 648 Abstract algebra II | 649 Abstract algebra III |

J. Brundan | J. Brundan | J. Brundan |

672 Probability theory I/II | 673 Probability theory II/III | |

B. Siudeja | B. Siudeja | |

681 Representation theory | 682 Representation theory | 683 Representation theory |

V. Ostrik | V. Ostrik | A. Kleshchev |

684 Operator theory/C*-algebras | 685 Non-commutative C*-algebras | 686 Crossed product C*-algebras |

H. Lin | C. Phillips | C. Phillips |

690 Characteristic classes | 691 K-theory | 692 WETSK |

J. Tu | C. Phillips | N. Proudfoot |

ADVANCED COURSE DESCRIPTIONS:

Victor Ostrik and Sasha Kleshchev, 681/2/3: **Representation theory**

The Fall will cover the important classification/structure theory of complex semi simple Lie algebras and the related Weyl groups. The connection to simple Lie groups will also be explained. Then the finite dimensional representation theory of these objects will be developed in the Winter quarter, especially the Weyl character formula and its geometric significance. The Spring quarter will be concerned with the at first sight unrelated representation theory of quivers. These turn out to be intimately related to semi simple Lie algebras and the more general Kac-Moody Lie algebras.

Prerequisites: 600 algebra.

Arkady Berenstein, 607: **Simple rings**

In modern algebra, ring theory is of great importance. Historically, rings emerged from various branches of mathematics: number theory (rings of integers, orders, Hecke rings), representation theory (group rings, enveloping algebras, rings of matrices), differential Geometry and functional analysis (Algebras of functions and operators, e.g. differential operators), homological algebra (cohomology rings, abelian categories, Grothendieck’s K-groups). The aim of this course is to give a systematic introduction to non-commutative ring theory with emphasis on semisimple and simple rings (semisimple rings can be thought of as associative analogues of semisimple Lie algebras).

I will begin by reviewing the definitions of semisimple modules and rings, the Jacobson radical of a ring, and the Wedderburn structure theorem (all of which should be familiar from 600 algebra). Then I will introduce central simple algebras and the Brauer group. If time permits I will explain the cohomological interpretation of the Brauer group and connection with K-theory, emphasizing connections with representations of finite groups.

Text: Non-commutative algebra, by Farb and Dennis

Prerequisites: 600 algebra (some knowledge of homological algebra would also be helpful).

Chris Sinclair, 607: ** 607 Algebraic and analytic number theory**

Algebraic number theory is the study of the multiplicative structure of algebraic integers in number fields. A number field K is a finite degree extension of Q. The ring of integers o_K consists of the elements in K which are roots of monic integer polynomials. A first disappointing realization is the failure of unique factorization in o_K for most number fields K. The class number h_K is an invariant of K which gives, in some sense, how far o_K is from having unique factorization. This, and the other classical invariants: the number of real and complex embeddings, the discriminant, and the regulator will be discussed. Unique factorization can be recovered if, instead of integers, we consider ideals in o_K (which can be uniquely factored into prime ideals). The factorization, norm and trace of ideals (and elements) will be discussed. We will also discuss the structure of units and roots of unity in number fields. Associated to each number field is an analytic function zeta_K(s) called the Dedekind zeta function (the Riemann zeta function being that for K = Q. We will discuss this object, explain how it encodes information about primes and the invariants of the number field (via the analytic class number formula). We will then introduce zeta functions ‘twisted’ by characters; the simplest examples being L-functions of Dirichlet characters. We will see how analytic properties of these functions can be used to demonstrate the infinitude of primes in arithmetic progressions (for instance to claim that there are infinitely many primes which have remainder 1 when divided by 17).

These L-functions and zeta functions are initially defined only in the half plane Re(s) > 1. Riemann’s famous (and sole) publication in number theory showed how the Riemann zeta funtion can be analytically continued to a meromorphic function on the complex plane with a single simple pole at 1 (he also conjectured that all the non-trivial zeros are on the line Re(s) = 1/2 — the proof of this conjecture will earn you a cool 1 million dollars and tenure at the university of your choice). Fifty years after this, Hecke introduced a wide class of L-functions (which include those of Dirichlet, and the Dedekind zeta function) and showed how to extend Riemann’s idea to derive a functional equation, and therefore a meromorphic continuation for these L-functions. Fifty years after this Tate gave an elegant argument which derived the functional equation using the adeles and idles — restricted direct products of completions of K with respect to all the various equivalence classes of absolute values on K. We will discuss adeles and ideles (and a bit about local fields) before proceeding to Tate’s proof, which uses the notion of Fourier analysis on local fields, to derive Hecke’s functional equation.

Time allowing, implications of the functional equation will be explored.

Pre-requisites: 600 algebra and basic complex analysis.

Dev Sinha, 607: **Group cohomology**

This will be an introductory course in group cohomology, emphasizing its connections to invariant theory, and also discussing Hopf rings in these contexts as well as representation theory.

We will start by following the book of Adem and Milgram: basic definitions, the use of classifying spaces, restriction and induction, transfer, and the role of invariant theory. But I will use my own approach to the cohomology of symmetric groups and symmetric invariants of polynomial rings (in multiple sets of variables) rather than theirs.

Pre-requisites: 600 algebra, 600 topology.

Weiyong He, 607: **Hodge theory and its applications on Kahler manifolds**

We shall mainly present Hodge’s theorem on the existence/uniqueness of harmonic form in a de Rham cohomology class on compact Riemannian manifolds, with emphasis on its applications to Kahler manifolds. Students are supposed to be familiar with smooth manifolds and de Rham cohomology. Other notions are optional: Kahler manifolds, an introduction to sheaf cohomology, very basic idea of elliptic PDEs. Here are the topics:

1. Give a detailed proof of Hodge theorem.

2. The applications of Hodge theorem on compact Kahler manifolds, including Lefschetz theorem and others.

3. If time permits we shall also present a gentle cohomology theory on Kahler manifolds and its applications, including line bundles, first Chern classes, Kodaira’s vanishing theorem and embedding theorem.

Barlomiej Siudeja, 607: **Potential theory and Brownian motion**

The goal is to look at Brownian motion from a very functional analytic perspective, and explore its relation to classical potential theory. We will study heat kernels, Green functions and other PDE related concepts using formulas involving Brownian motion. This approach will give us a perfect excuse to generalize most of the concepts to more abstract settings.

The class requires good background in probability with basic knowledge of stochastic processes and a hint of PDEs.

Outline of topics:

1. Vector bundles.

2. Vector bundles over compact spaces.

3. Bott periodicity.

4. K-theory as a cohomology theory.

5. Products in K-theory.

6. The Thom isomorphism and computations of K-theory in other examples.

7. Applications to the Hopf invariant problem.

This is a course on the K-theory of (locally) compact Hausdorff spaces, not on the K-theory of C*-algebras. C*-algebras will be mentioned only occasionally, and no knowledge of them (or interest in them) is assumed. Nevertheless, this course is important background for anyone interested in the K-theory of C*-algebras.

Text: K-theory by M. F. Atiyah.

Prerequisites: 600 topology.

Huaxin Lin, 684: **Introduction to operator theory and C*-algebras**

We will discuss the following topics:

1. Hilbert spaces.

2. Elements of operator theory, compact operators.

3. A brief discussion of weak topologies in operator theory.

4. Banach algebras and spectral theory.

5. Some elements of C*-algebras.

7. Normal operators and Fredholm theory (if time permits).

Prerequisites: 600 analysis.

Chris Phillips, 685/686: **Introduction to noncommutative C*-algebras**/ **Crossed product C*-algebras and their structure.**

These courses will be a continuation of Huaxin Lin’s Math 684 in Fall 2012. Therefore its starting point will depend on how far Lin gets. This description is based on the assumption that Lin has done the functional analysis background related to C*-algebras, an introduction to Banach algebras, the Gelfand transform, and the characterization of commutative C*-algebras.