# Advanced Graduate Courses 2005/06

Here is the schedule for all graduate-level courses for 2005-2006. See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2005 |
WINTER 2006 |
SPRING 2006 |

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

H. Lin (10:00) | J. Isenberg (10:00) | J. Isenberg (10:00) |

541 Abstract Linear Algebra | ||

M. Vitulli (12:00) | ||

551 Numerical Analysis I | 552 Numerical Analysis II | 555 Mathematical Modelling |

Y. Xu (14:00) | Y. Xu (14:00) | J. Isenberg (12:00) |

556 Networks and Combinatorics | 557 Discrete Dynamical Systems | |

A. Polishchuk (13:00) | C. Phillips (10:00) | |

513 Introduction to Analysis I | 514 Introduction to Analysis II | 515 Introduction to Analysis III |

H. Lin (13:00) | H. Lin (13:00) | P. Gilkey (13:00) |

531 Introduction to Topology I | 532 Introduction to Topology II | 533 Introduction to Diff Geometry |

A. Polishchuk (11:00) | A. Polishchuk (11:00) | P. Lu (11:00) |

544 Introduction to Algebra I | 545 Introduction to Algebra II | 546 Introduction to Algebra III |

A. Kleshchev (9:00) | A. Kleshchev (9:00) | A. Kleshchev (9:00) |

564 Mathematical Statistics I | 565 Mathematical Statistics II | 566 Mathematical Statistics III |

D. Xu (12:00) | D. Xu (12:00) | D. Xu (12:00) |

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

M. Bownik (13:00) | M. Bownik (13:00) | M. Bownik (13:00) |

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

D. Dugger (11:00) | D. Dugger (11:00) | D. Dugger (11:00) |

647 Abstract Algebra I | 648 Abstract Algebra II | 549 Abstract Algebra III |

A. Vaintrob (9:00) | A. Vaintrob (9:00) | A. Vaintrob (9:00) |

671 Probability Theory I | 672 Probability Theory II | 673 Probability Theory III |

H. Wang (10:00) | H. Wang (10:00) | H. Wang (10:00) |

681 Lie Algebras | 682 Chevalley Groups | 683 Quantum Groups |

J. Brundan (9:00) | J. Brundan (9:00) | A. Berenstein (9:00) |

607 Curves and Jacobians | 607 Reflection Groups | 607 Homological Algebra |

A. Polishchuk (12:00) | S. Yuzvinsky (14:00) | V. Ostrik (10:00) |

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

C. Phillips (13:00) | C. Phillips (13:00) | C. Phillips (13:00) |

607 Radon Transform I | 607 Radon Transform II | 619 Complex Analysis |

Y. Xu (14:00) | Y. Xu (14:00) | H. Lin (14:00) |

690 Characteristic Classes | 691 Cobordism and K-theory | 692 WETSK |

P. Gilkey (11:00) | D. Sinha (11:00) | H. Sadofsky (11:00) |

607 Conformal Geometry | 607 Symplectic Geometry | |

B. Botvinnik (10:00) | G. Landweber (10:00) |

ADVANCED COURSE DESCRIPTIONS:

**619 Complex Analysis**, Lin, Spring 2006.

This is a second year course in analysis offered alternate years only, dealing with some aspects of complex analysis on manifolds.

Prerequisites: 600 analysis.

**690 Curvature, characteristic classes and Chern-Gauss-Bonnet**, Gilkey, Fall 2005.

We will discuss the characteristic classes from a differential geometry viewpoint. By using de Rham cohomology, one loses information concerning torsion but gains geometric insight.

In order to make things accessible to students who have taken the algebraic topology sequence as well as to students who have taken the differential geometry sequence, I will spend the beginning of the course discussing (briefly) the notion of a differentiable manifold, the tangent space, the cotangent space, differential forms, the generalized Stokes theorem, and the de Rham theorem which links topological cohomology with complex coefficients to the de Rham cohomology groups of closed modulo exact forms. Treatment will depend on the background of the students involved.

I will then introduce connections and curvature. The characteristic classes are built using Chern-Weyl theory from the curvature and are invariantly defined closed differential forms whose de Rham cohomology classes are independent of the particular connection chosen. We will study real and complex vector bundles — these give rise to the Pontrjagin and Chern forms. We will also study oriented even dimensional real vector bundles — this gives rise to the Euler form. A more general treatment involving principle bundles with more general structure groups will be given. I have some notes I have written on the subject for several expository articles that will be distributed.

A discussion of the Chern-Gauss-Bonnet (which relates the Euler characteristic to geometry information) will be given which follows Chern’s original treatment. I also hope to discuss the Hirzebruch signature formula and the Riemann-Roch formula. There will be occasional homework assignments — created and graded by the TA in the course (Ecaterina Puffini).

Prerequisites: 600 topology.

**691 Cobordism and K-theory**, Sinha, Winter 2006.

We will develop cobordism theory and K-theory, leading up to the Signature Theorem. Basic topics will include: characteristic numbers, the Pontryagin-Thom construction, Bott periodicity, Thom isomorphisms, K-theory characteristic classes. We will also connect both with the more general framework of stable homotopy theory and to geometry, as developed in the fall 690.

Prerequisites: Math 690.

**692 What Every Topologist Should Know**, Sadofsky, Spring 2006.

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 homotopy of spheres, spectral sequences, cobordism theory, generalized cohomologies, topology of Lie groups and homogeneous spaces, power operations in cohomology and K-theory, index theory and many more.

Prerequisites: Math 691.

**607 Conformal Geometry**, Botvinnik, Fall 2005.

No description available.

Prerequisites: 600 topology and differential geometry.

**607 Introduction to Symplectic Geometry**, Landweber, Winter 2006.

In Riemannian geometry, one considers smooth manifolds with a metric, a symmetric bilinear form which allows you to measure distances and curvature. In contrast, symplectic geometry considers smooth manifolds with an ANTI-symmetric bilinear form. Such a symplectic structure shares some properties with metrics, in that it determines a volume form and identifies the tangent and cotangent spaces. However, there is no symplectic analogue of the curvature or other local properties of differential geometry, as the Darboux theorem states that all symplectic manifolds look the same locally. Instead, symplectic geometry is used as a tool to study the global properties of manifolds, such as their cohomology, or their number of pseudo-holomorphic curves. Examples of symplectic manifolds include projective spaces, flag manifolds, Riemann surfaces, Kahler manifolds, moduli spaces of connections, and the infinite dimensional analogues of these. In addition, the cotangent bundle of a smooth manifold admits a canonical symplectic structure which is fundamental to both classical Hamiltonian mechanics and quantization. Symplectic geometry has also played a significant role the study of smooth 4-manifolds. If time permits, this course will introduce equivariant symplectic geometry, covering topics such as moment maps, symplectic reduction, equivariant cohomology, and Kirwan surjectivity.

Prerequisites: 600 topology, 690, manifolds and differential forms (does NOT require 600 differential geometry).

**684/5/6 Operator theory and C*-algebras**, Phillips, Fall/Winter/Spring 2004/05.

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).)

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.

Prerequisites: 600 analysis.

**607 Radon Transform and Computerized Tomography**, Xu, Fall/Winter 2004/05.

Have you ever wondered if mathematics has any real applications? Are you tired of being asked this question? If so, come to this course.

In its simplest form, a Radon transform of a function is just a line integral of the function. It turns out that an X-ray through a body is related to such a transform. The central problem of Computerized Tomography (CT) is to recover an image (say the image of a heart, a lung) from Radon projections (X-rays). In the past 30 years this wonderful application of mathematics in the medical field has grown into an industry [sic] by itself.

This course will cover the mathematical theory of CT. It will study the Radon transform on the Euclidean space (often just R^2 or R^3) and go over the basics (its properties, inverse transform, relation to Fourier transform, etc.). The main goal is to understand how CT works; that is, how to recover a function from its Radon projections. Along the way, mathematical ideas from several areas (Fourier analysis, approximation theory, orthogonal polynomials, numerical analysis) will be introduced.

Prerequisites: 600 analysis.

**681 Lie Algebras**, Brundan, Fall 2005.

This will be a first course on the classification and structure of finite dimensional semisimple Lie Algebras. We will prove the classification theorem of finite dimensional simple Lie algebras (by Dynkin diagrams), then go on to classify the finite dimensional irreducible representations. Note this course is a pre-requisite for the courses on Chevalley groups and Quantum groups in winter and spring.

Text: Introduction to Lie algebras and representation theory by J. E. Humphreys.

Prerequisites: 600 algebra.

**682 Chevalley Groups**, Brundan, Winter 2006.

This is a continuation of the course in Lie algebras in the fall. The goal is to define and study the (untwisted) finite simple groups of Lie type. These are the most important family of finite simple groups, and are constructed starting from finite dimensional simple Lie algebras over the complex numbers by a beautiful modular reduction procedure developed by Chevalley in the 1950s. Indeed, aside from alternating groups and the twisted Chevalley groups (which are constructed in a very similar but slightly more complicated process) there are only 26 more sporadic examples of finite simple groups… Along the way, we will develop a little more of the representation theory of finite dimensional semisimple Lie algebras, such as the Weyl character formula.

Text: Lectures on Chevalley groups by R. Steinberg.

Prerequisites: Math 681.

**683 Quantum Groups**, Berenstein, Spring 2006.

The course will be about algebraic aspects of Quantum Groups. Quantum groups (or more precisely quantized enveloping algebras) were introduced independently by Drinfeld and Jimbo around 1985, as an algebraic framework for quantum Yang-Baxter equations. Since then numerous applications of Quantum Groups have been found in areas ranging from theoretical physics via symplectic geometry and knot theory to ordinary and modular representations of reductive algebraic groups. The course provides a systematic introduction to the structure theory and representation theory of quantum groups.

Here is a tentative content of the course.

- 1. Hopf algebras and their representations
- 2. Quantum linear algebra (after Manin)
- 3. Quantum algebraic groups, quantized enveloping algebras, and their representations

Text(s): Manin, Quantum groups and noncommutative geometry. Brown and Goodearl, Lectures on algebraic quantum groups.

Prerequisites: Math 682.

**607 Curves and Jacobians**, Polishchuk, Fall 2005.

We will study the following topics:

- 1) 2 constructions of the Jacobian: through symmetric powers of a curve and through periods of holomorphic 1-forms;
- 2) Classical geometry of Jacobians, Abel’s theorem, principal polarization on the Jacobian, and theta-functions;
- 3) Explicit coordinates on Jacobians of hyperelliptic curves;
- 4) Varieties of special divisors and singularities of theta-divisor;
- 5) Introduction to Shottky problem (if time allows).

Prerequisites: basic algebraic geometry and a little analysis on complex manifolds.

**607 Invariants of Reflection Groups**, Yuzvinsky, Winter 2006.

This course may possibly be extended to Spring as well if there seems to be enough demand and the resources are available.

This course would tentatively start from the classification of finite groups generated by reflections via Shephard and Todd, including Coxeter groups and Shephard groups. Then we would discuss reflection arrangements. These are sets of the hyperplanes fixed by reflections of a group generated by reflections. They have special properties in the world of arbitrary arrangements of hyperplanes. This part would include the following.

- a) Differential operators annihilating a polynomial and coinvariants of groups.
- b) Modules of derivations and differential forms.
- c) Free arrangements.
- d) Solomon -Terao theorem that all reflection arrangements are free.

Text: Invariants of reflection groups, R. Kane.

Prerequisites: 600 algebra.

**607 Homological Algebra**, Ostrik, Spring 2006.

This is a foundational course in homological algebra. The aim of the course will be to develop a core background knowledge of homological techniques, which the student can then take away and apply to his or her own specialized field. The course will start by introducing abelian categories, then projective and injective resolutions, Yoneda extensions, derived functors, Ext and Tor. In addition to the general theory, we’ll spend a lot of time learning to work explicit examples, depending on the interests of the students in the class.

Textbook: Weibel, An introduction to homological algebra.

Prerequisites: 600 algebra.