# Advanced Graduate Courses 2010/11

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

FALL 2010 |
WINTER 2011 |
SPRING 2011 |

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 |

H. Lin | H. Lin | P. Gilkey |

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

H. Lin | Y. Xu | Y. Xu |

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

A. Vaintrob | A. Vaintrob | W. He |

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

V. Vologodsky | V. Vologodsky | V. Vologodsky |

556 Networks and combinatorics | ||

A. Polishchuk | ||

557 Discrete dynamical systems | ||

V. Ostrik | ||

564 Mathematical statistics I | 565 Mathematical statistics II | |

D. Levin | C. Sinclair | |

607 Several complex variables | 607 Top. quantum field theories | |

W. He | D. Dugger | |

607 Random matrices, point processes | ||

C. Sinclair | ||

607 Toric varieties | 607 Etale cohomology | 607 Number theory |

N. Proudfoot | A. Polishchuk | V. Vologodsky |

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

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

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

H. Sadofsky | B. Botvinnik | B. Botvinnik |

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

P. Lu | P. Lu | P. Lu |

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

S. Yuzvinsky | S. Yuzvinsky | S. Yuzvinsky |

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

V. Ostrik | V. Ostrik | B. Webster |

684 Functional analysis | 685 Introduction to C*-algebras | 686 C*-algs: K-thy, Cuntz semigp |

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

690 Characteristic classes | 691 Cohomology ops, spectral seqs | 692 WETSK |

D. Sinha | B. Botvinnik | TBA |

ADVANCED COURSE DESCRIPTIONS:

The course serves as an introduction to modern algebraic and analytic number theory. It will cover the following topics:

- Unique Factorization Domains, Application: Fermat’s Theorem on sums of two squares.
- Quadratic reciprocity law.
- Dedekind domains. Ramification. Finiteness of the ideal class group. Dirichlet’s Theorem on units.
- Dirichlet’s Theorem on primes in arithmetic progressions.
- Class number formula for quadratic extensions.

Prerequisites: Math 544-546 (basic algebra including Galois Theory), Math 511-512 (elementary complex analysis).

Polishchuk, 607 **Etale cohomology**

Etale cohomology were invented by Grothendieck as a replacement of the usual cohomology of complex algebraic varieties for varieties defined over an arbitrary field. The main motivation (which does not exhaust all applications) for this was to prove the Weil conjectures about numbers of points of algebraic varieties over finite fields. The theory is developed in the context of theory of sheaves with respect to a new “topology” defined for algebraic varieties, called “etale topology”. Hopefully, we will have time to indicate how etale cohomology is applied to the proof of the Weil conjectures. I will assume knowledge of the relevant homological algebra including derived functors and sheaf theory. Another prerequisite is at least two terms of algebraic geometry.

Sinclair, 607 **Random matrix theory and point processes**

Point processes are probability spaces whose outcomes consist of finite subsets of points of some prescribed space (e.g. C or R). For instance, in statistical physics, one may be interested in the spacial distribution (in R^3) of charged particles when the temperature, pressure, etc are at prescribed values. Typical questions of interest for point processes are the probability that a given region contains a prescribed number of points, the probability density of points and the expected number of points in a given region.

A particularly important class of point processes are given by sets of eigenvalues of random matrices. For instance, choosing an N x N complex matrix with iid (complex) standard normal entries produces a point process on C—that is, induces a probability measure on cardinality N subsets of C as identified with sets of eigenvalues of matrices. Like identically charged particles, the eigenvalues of these (and other) random matrices repel each other—an important property in the application of random matrices to other domains. The statistics of random matrices appear experimentally in such diverse areas as energy levels of atomic spectra and the conjectural distribution of zeros of the Riemann zeta function.

Topics to be covered include: point processes, correlation functions (i.e. joint intensities), determinantal and Pfaffian point processes, Gaussian ensembles of random matrices, scaling limits as the size of matrices goes to infinity, Fredholm (and other operator) determinants, applications of orthogonal and skew-orthogonal polynomials, asymmetric ensembles and universality.

The graduate probability sequence is not a prerequisite. The first quarter of graduate analysis (i.e. measure theory) is.

Dugger, 607 **Topological quantum field theories**

TQFTs first arose in the work of Witten, who was considering “toy models” of quantum field theories in which one removed all of the physics: afterwards all that is left is the topology of spacetime. Witten and subsequent researchers have used TQFTs to produce subtle topological invariants of knots and manifolds. The subject is very much a developing one: people are still struggling to come up with the right definitions and tools for studying these phenomena.

In the first half of this course we will follow some of the early literature and consider several specific examples of TQFTs. We will spend some time talking about the motivation coming from physics, and about the connections with the theory of n-categories. In the second half of the course we will look at the Baez-Dolan cobordism hypothesis and try to understand some of Lurie’s work on this topic.

The course should be accessible to students who have successfully completed the 600-level algebra and topology sequences.

Proudfoot, 607 **Toric varieties**

Toric varieties are the most concrete algebraic varieties, as they can be described using combinatorial data such as polytopes in vector spaces. This allows us to understand their geometry and topology in much more explicit terms than we can with more general varieties. For example, we can give a complete description of the cohomology ring of any smooth, projective toric variety. Toric varieties are interesting for their own sake, and they provide an accessible collection of examples to such topics as geometric invariant theory, equivariant cohomology, and singularity theory. This class will be an introduction to toric varieties, with an emphasis on topological properties. Some background in algebraic geometry would be very helpful, but not absolutely necessary.

He, 607 **Several Complex variables**

1. We shall cover basic properties of functions of several complex variable and basic material of Hormander’s L^2 estimate, which, for example, has numerous applications in complex analysis and algebraic geometry.

2. We shall also cover basic material of complex manifolds, in particular Kähler manifolds and complex algebraic varieties, in particular some material in Griffths & Harris, Chapter 0 and I.

3. If time permits, we shall cover the proof of beautiful Kodaira embedding theorem. This is a perfect example which combines complex analysis and algebraic geometry together.

The prerequisite of the course is grasp of basic ideas of differential manifolds, topology and elementary complex analysis (for example, a undergraduate complex analysis course).

Ostrik, 681 **Introduction to representation theory, I**

This class is devoted to study of finite dimensional Lie algebras and their representations. We start with definition of a Lie algebra and finish with a classification of semisimple Lie algebras over the field of complex numbers.

Ostrik, 682 **Introduction to representation theory, II**

This class is devoted to study of finite dimensional representations of semisimple Lie algebras over the field of complex numbers. Our main goal is a classification of such representations and the Weyl character formula.

Webster, 683 **Representation theory III**

This class will focus on geometric methods in the representation theory of Lie algebras and groups. Our central goal will be the theorem of Beilinson and Bernstein relating D-modules on the flag variety to representations of Lie algebras. The majority of the class will be spent building up the basics of the geometry of the flag manifold, and of the theory of D-modules in order to understand and prove this theorem. Other topics like perverse sheaves and Kazhdan-Lusztig theory may also be discussed if time allows.

The most important prerequisite is 681/2; some knowledge of algebraic geometry (for example, what a projective variety is) would also be quite valuable, but not absolutely essential.

Phillips, 684 **Functional analysis**

This course will cover some standard basic topics in the theory of Banach spaces and linear operators. Rough outline:

- A little review of Banach spaces and Hilbert spaces. Quotient spaces.
- Topological vector spaces.
- The weak and weak* topologies, Alaoglu’s Theorem, Krein-Milman Theorem.
- Basic definitions of linear operators; examples.
- Compact operators. Fredholm operators; Fredholm index.
- Basic definitions of Banach algebras; many examples.
- Spectrum, spectral radius, holomorphic functional calculus.

Prerequisite: Math 616–618.

Lin, 685 **Introduction to C*-algebras**

This is the course following 684. It will introduce the general concept of Banach algebras and C*-algebras as subalgebras of operators on Hilbert spaces. The course will cover commutative C*-algebras and finite dimensional C*-algebras, hereditary C*-subalgebras and ideals, representations and GNS construction, von Neumann algebras and inductive limits of C*-algebras, examples of C*-algebras. In the later part of the course, we will discuss the Stinespring representation theorem and a brief introduction of K-theory.

There will be no exams. But students are expected to participate in the in-class presentations.

Phillips, 686 **C*-algebras: K-theory and the Cuntz semigroup**

This is the course following Math 685. We will pick up where Math 686 finishes. The particular topic in operator algebras can be changed to suit the interests of the students. My current plan is to discuss the K-theory of operator algebras in reasonable depth, including Bott periodicity, the classification of AF algebras, and some form of KK-theory (most likely asymptotic morphisms and the E-theory of Connes and Higson). Then I plan to present the theory of the Cuntz semigroup. It has recently become important in the classification theory of C*-algebras. The Cuntz semigroup is similar to the K_0 group, but is made from positive elements instead of projections, and one does not adjoin inverses.

Sinha, 690 **Characteristic classes**

Math 690 is meant to follow the 600 topology sequence. This year as is typical the topic will be characteristic classes, an application of cohomology to the study of vector bundles. Sections of vector bundles are like “twisted functions” of real or complex variables so their study, including that of characteristic classes, is a central tool in algebraic and geometric topology, differential geometry, and algebraic geometry.

We will first emphasize the intersection theoretic approach to the theory, for example seeing the Euler class in terms of the zeroes of sections. Here it will be clear that cohomology has some distinct advantages over homology. We will then develop the axiomatic and homotopy theoretic viewpoints and do plenty of computations. Applications will be to questions such as vector fields on spheres and cobordism theory, and we can also practice with Riemann-Roch formulas (though we won’t be able to prove them). As a bridge to 691, we will be sure to discuss the Wu formula, which allows us to view Steenrod operations through characteristic classes, and I’ll share recent calculations in the cohomology of symmetric groups.

Botvinnik, 691 **Spectral sequences and cohomology operations**

First we will study the Steenrod Algebra of cohomology operations, then we prove several applications including the results on vector fields on spheres. Next we will work first with the Serre spectral sequence and then with the Adams spectral sequence to prove several classical results in homotopy theory and compute particular cobordism groups.

To be announced, 692 **WETSK (What every topologist (and geometer) should know)**

In this course students read significant papers in geometry and topology and present them to the rest of the class. Students typically do two papers. In this year, papers which are more geometric will be encouraged.