# Advanced Graduate Courses 2008/09

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

FALL 2008 |
WINTER 2009 |
SPRING 2009 |

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 |

Y. Xu | Y. Xu | P. Gilkey |

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

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

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

S. Yuzvinsky | S. Yuzvinsky | P. Gilkey |

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

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

556 Networks and combinatorics | 557 Discrete dynamical systems | |

V. Ostrik | H. Lin | |

607 Arithmetic combinatorics | 607 Combinat’l commut’ve algebra | 607 Cluster algebras |

M. Bownik | N. Proudfoot | A. Berenstein |

607 Symplectic geometry | 607 Advanced topics in probability | 607 Calculus of functors |

N. Proudfoot | D. Levin | H. Sadofsky |

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

H. Lin | H. Lin | H. Lin |

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 |

P. Lu | J. Isenberg | J. Isenberg |

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

B. Shelton | B. Shelton | B. Shelton |

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

V. Ostrik | V. Ostrik | V. Ostrik |

684 Operator theory/C*-algebras I | 685 Operator theory/C*-algebras II | 686 Operator theory/C*-algebras III |

N. C. Phillips | N. C. Phillips | N. C. Phillips |

690 Characteristic classes | 691 Morse theory | 692 WETSK |

D. Spivak | D. Dugger | A. Vaintrob |

ADVANCED COURSE DESCRIPTIONS:

** 607 – Combinatorial commutative algebra**, N. Proudfoot

Combinatorial commutative algebra is the study of commutative rings that are defined using combinatorial data. The flow of information between combinatorics and commutative algebra (and, by extension, algebraic geometry) occurs in both directions, with each field contributing significantly to the other. We will focus primarily on the study of rings that are obtained as quotients of polynomial rings by collections of monomials, which are intimately related to such objects as simplicial complexes and planar graphs. Other possible topics include toric varieties, the Hilbert scheme of points in the plane, and tropical algebraic geometry.

**607 – Cluster algebras**, A. Berenstein

Cluster algebras were introduced in 2001 by Fomin and Zelevinsky in order to study the total positivity in algebraic groups and canonical bases in their representations A cluster algebra is a commutative algebra given by combinatorial data as a subalgebra of the field of fractions of finitely many indeteminants. A spectacular “Laurent phenomenon” asserts that each cluster algebra, is, in fact, a subalgebra of the algebra of Laurent polynomials in those indeterminates. Cluster algebras have links with a variety of other fields, including Stasheff polytopes (associahedra), the Bethe ansatz, toric varieties, and representation theory. In particular, all cluster algebras of finite type have been classified by the Dynkin diagrams and are strongly related to finite root systems.

** 607 Arithmetic combinatorics**, M. Bownik.

Additive combinatorics is a theory of counting additive structures in sets. A remarkable feature of this subject is the use of tools from many diverse areas of mathematics such as elementary combinatorics, number theory, harmonic analysis, graph theory, discrete geometry, probability, and ergodic theory. While the methods used in additive combinatorics are quite sophisticated, most results have very simple formulation. For example, the Szemeredi theorem states that any subset of positive integers with positive density has arbitrary long arithmetic progressions. A stunning result of Green and Tao says that primes have arbitrary long arithmetic progressions. The aim of this course is to give a flavor of this area. We will cover selected topics from the book of Tao and Vu “Additive Combinatorics”, Cambridge Univ. Press, 2006.

** 607 Symplectic geometry**, N. Proudfoot

Symplectic geometry is the mathematical formalism of hamiltonian mechanics, in which the world (a symplectic manifold) evolves over time (the action of the Lie group of real numbers). It turns out that this generalizes nicely to actions of other Lie groups, with applications to many areas of mathematics, including topology, combinatorics, and representation theory.

** 607 Calculus of functors**, H. Sadofsky

Goodwillie gives a method for taking a functor which preserves homotopy equivalances (say from spaces to spaces) and approximating it by homotopy theoretically simpler functors. In fact he assembles these functors in a tower of functors which he calls the “Taylor tower” where the fibers between stages of the tower are the “derivatives” of the functor.

In this course we will describe the construction and properties of the functors in these towers. In particular, we will study the Goodwillie tower of map(K,X) where K is a finite complex, and interpret the usual stable splittings in this context when K is a sphere, we will study Kuhn’s result that polynomial functors from spectra to spectra split after localization, and Arone-Mahowald’s result that the Goodwillie tower of the identity functor evaluated on spheres realizes the chromatic filtration.

**607 Advanced topics in probability**, D. Levin

This course would follow 672-673, and pick up where that left off. This would cover “what every young probabilist should know” from the theory of continuous stochastic processes. In particular:

- Brownian motion and its sample-path properties
- continuous martingales
- Ito integral and stochastic calculus
- diffusion processes and stochastic differential equations
- Gaussian processes

** 681-3 Representation theory**, V. Ostrik

This course will cover basic theory of finite dimensional Lie algebras and their representations over the field of characteristic zero. I will emphasize the theory of semisimple Lie algebras and their classification; finite dimensional representations of semisimple Lie algebras, their characters, tensor products and branching rules; the theory of Chevalley groups.

** 684-6 Operator theory and C*-algebras**, N. C. Phillips

Topics to include:

- Some basic results of operator theory, especially Fredholm operators and index theory, spectrum, and the spectral theorem for selfadjoint operators on Hilbert space.
- Some basic results on Banach algebras, especially functional calculus and the Gelfand transform for commutative Banach algebras. (The Gelfand transform simultaneously generalizes Fourier series, the Fourier transform, the Poisson integral formula, and the fact that if X is compact then X can be recovered from the algebra C (X) of continuous complex valued functions on X.)
- The basic theory of C*-algebras, including the basics of their representation theory.
- K-theory for Banach algebras and C*-algebras. (This is a generalization of the K-theory that algebraic topologists make from vector bundles. It is where “noncommutative” index theory lives. No previous knowledge of algebraic topology is required.)
- Topics from group C*-algebras and discrete crossed products, including connections with dynamical systems.

There are many related topics that I will not have time to treat, but I will at least try to mention the existence of some of them.

Prerequisite: Math 616-618 or equivalent. Here is some more information.

** 690 Characteristic classes**, D. Spivak

In this course, we will discuss characteristic classes from a homotopy-theoretic viewpoint. After some basic category theory, we’ll discuss classifying spaces.

It turns out that there is a natural one-to-one correspondence between vector bundles on a space X and homotopy-classes of maps from X to a manifold called the Grassmannian. In other words, the Grassmannian “represents” vector-bundles in a natural and formal way.

We will discuss classifying spaces for various types of fiber bundles; one views these classifying spaces as “fully embodying fiber-bundleness”! By finding a classifying space C for a type of bundle, we give ourselves two perspectives on the same concept: one is “fiber bundles on X,” and the other is “maps from X to C.”

Now let’s suppose you know the cohomology of your classifying space C very well. This is the case for many types of fiber bundles; for example the cohomology of the Grassmannian is well-understood. Then we are in good shape to know some cohomology of any random space X. The reason is that every fiber bundle on X gives a map f: X—>C, and we can use the contravariance of cohomology to get a pullback map f*, from cohomology on C (which we understand well) to cohomology on X. The pulled-back cohomology classes are called characteristic classes.

This course will be very categorical in nature, but I won’t assume you have much category-theoretic background. Instead of feeling technical, I hope that the category-theoretic perspective will enlighten students as to the naturality of the concepts involved.

** 691 Morse Theory**, D. Dugger

Morse theory is a set of techniques for studying the topology of a smooth manifold M by looking at the critical points of a smooth, real-valued function on M. Some of the classical applications are to prove the Lefschetz theorem on hyperplane sections of algebraic varieties, and to the original proof of Bott periodicity. This course will develop the basics of Morse theory, and then we will apply it to prove the so-called h-cobordism theorem. Time permitting, we will be able to deduce the Poincare conjecture in dimensions greater than 5.

** 692 What Every Topologist Should Know (WETSK),** A. Vaintrob

This is a seminar-style course where the students read various classic topology papers and give lectures on them. This is intended to enhance your knowledge of topology and fill gaps in your background. In the past, each student has read two papers and given a total of 2-4 lectures. The precise material covered depends on your own interests and the papers you choose. Possible topics include spectral sequences, cobordism theory, generalized cohomologies, topology of Lie groups and homogeneous spaces, power operations in cohomology and K-theory, surgery theory, index theory and many more. This year we might also choose papers which are fundamental in low-dimensional topology, both “classical” and “quantum”.

Pre-requisites: Math 691.