# Advanced Graduate Courses 2009/10

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

FALL 2009 |
WINTER 2010 |
SPRING 2010 |

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

To be announced | To be announced | |

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

P. Gilkey | P. Gilkey | P. Gilkey |

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

Y. Xu | New hire? | New hire? |

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

D. Sinha | D. Sinha | P. Lu |

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

M. Vitulli | M. Vitulli | M. Vitulli |

556 Networks and combinatorics | ||

N. Proudfoot | ||

607 Homological algebra | 607 Invariant theory | 607 Quantum groups? |

S. Yuzvinsky | J. Brundan | A. Berenstein |

607 Loop spaces | 607 Fourier for probabilists | 607 Model categories |

D. Sinha | Y. Xu | D. Spivak |

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 |

D. Dugger | D. Dugger | D. Dugger |

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

A. Kleshchev | A. Kleshchev | A. Kleshchev |

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

New hire? | New hire? | |

681 Non-commutative ring theory | 682 Algebraic geometry | 683 Algebraic geometry |

B. Shelton | N. Proudfoot | N. Proudfoot |

684 Functional analysis | 685 Functional analysis | 686 Functional analysis |

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

690 K-theory | 691 Morse theory | 692 WETSK |

C. Phillips | B. Botvinnik | H. Sadofsky |

ADVANCED COURSE DESCRIPTIONS:

** 607 – Fourier analysis for probabilists**, Y. Xu

The aim of the course is an introduction to Fourier series and Fourier transform. We will cover the basic results in the Fourier series and Fourier transforms in sevreral variables, and will also talk about discrete Fourier transforms. The course will end up at Fourier analysis on the fundamental domain of the translation tiling. The prerequisite of the course is 600 analysis.

**607 – Quantum groups**, A. Berenstein

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 an introduction to the structure theory and representation theory of quantum groups.

Here is a tentative content.

- 1. Introduction to Hopf algebras
- 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.

** 607 Invariant theory**, J. Brundan.

I’ll cover a selection of topics in classical invariant theory. The basic theme is to let a group (say a finite one to start with) act on some space (for example it might be acting linearly on a finite dimensional vector space). There is then an induced action of the group on the algebra of functions (e.g. polynomial ones) on that space. What can you say about the algebra of invariants, i.e. the subalgebra consisting of all functions fixed by the group action What does this subalgebra tell you about the action of the group on the original space?

I’ll spend a substantial part of the course talking about the situation when the group is a finite group acting on a finite dimensional vector space by reflections. Here there are rich and beautiful results tied up with symmetric functions, semisimple Lie algebras, Coxeter groups, hyperplane arrangements,…. But I’ll also talk a little about some more general situations, including some where the group is a Lie group like the general linear group — when issues surrounding geometric reductivity and Hilbert’s 14th problem crop up. This is the jumping off point for geometric invariant theory (something I’m not going to seriously get into).

The pre-requisites for the course are 600 algebra. But later in the term I’ll make use of a few basic notions from (mostly affine) algebraic geometry — so it might be helpful though not essential if you are taking 682 Algebraic Geometry in parallel with this (or have done a little algebraic geometry before).

** 682/3 Algebraic geometry**, N. Proudfoot

The first term of this class will be an introduction to classical algebraic geometry, which means studying subvarieties of affine and projective spaces. The second term could go in any one of many different directions. Here are some possibilities:

1. Continue the study of classical algebraic geometry.

2. Re-approach the subject from a very formal, modern perspective. For example, a space X can be identified with the functor from spaces to sets that takes any space Y and outputs the set Hom(Y,X).

3. Focus on toric varieties, which are a very special class of algebraic varieties. Their combinatorial nature makes it possible to understand them in a very concrete way, which would allow us to do a lot of cool and detailed computations, and thereby shed light on some pretty abstract definitions.

** 681 Non-commutative ring theory**, B. Shelton

This will be a general introductory course to non-commutative ring theory, starting with basic properties of projective indecomposable modules.

**607 Homological algebra**, S. Yuzvinsky

This will be a general introductory course to homological algebra.

** 607 Loop, de-loop: mapping spaces and classifying spaces**, D. Sinha

Topologists study spaces and maps between them. The collection of all maps between two spaces is itself a space, and for example (with some weak hypotheses) the study of the components of that space corresponds to studying the homotopy classes of maps between those spaces. The most basic mapping spaces are spaces of maps from a circle, which are known as loop spaces. Spaces of maps from a sphere are known as iterated loop spaces. While mapping spaces, and in particular loop spaces, are “infinite-dimensional” and difficult to visualize at first, we can compute their homology fairly effectively in some cases.

It also turns out that any topological group is homotopy equivalent to the loop space of some other space. That other space is called the classifying space for the group, sometimes called a de-looping of the group. Classifying spaces arise in a wide variety of settings – for example cohomology theories are represented by spaces which have arbitrary deloopings. The study of classifying spaces naturally complements the study of loop spaces. We will interweave the study of loop spaces and classifying spaces, with the following outline. Let n denote the number of “loopings”, which for example is -1 for the classifying space functor.

List of topics:

- Map (S^1, S^d).
- Map (S^1, Sigma X) – James’ Theorem.
- Map (S^1, X) – the Eilenberg-Moore spectral sequence, take one.
- Basics of BG – the classifying space of G.
- The simplicial model for BG and arbitrary deloopings when G is abelian.
- Digression on spectra and cohomology theories.
- Iterated loop spaces and operations on their homology.
- Recognition principles for iterated loop spaces.
- The the cosimplicial model for Map (X, Y) in general and the Anderson spectral sequence.
- McClure-Smith theory, encompassing homology operations in the Anderson spectral sequence [n > 0] and beyond.

** 607 Model categories**, D. Spivak

Homotopical Algebra was invented by Daniel Quillen in 1967. Noticing that the projective resolution of an R-module (for a commutative ring R) is similar in nature to a CW approximation of a topological space, Quillen set out to determine what additional structure one needs on a category in order to “do homotopy theory.”

The resulting theory was that of model categories. The most important feature of a model category is that it comes with a subcategory of “weak equivalences,” morphisms which are not necessarily invertible, but which are somehow homotopically invertible. Making this precise requires additional data (including a subcategory of cofibrations and a subcategory of fibrations) and five axioms. However, once this structure is in place, one can discuss for example mapping cones, spectra, and homotopy colimits in any model category. For example, the homotopy colimit of a pushout diagram of commutative rings is the “derived tensor product,” (tensor^L) in homological algebra.

Model categories abound. Aside from the “obvious” model structures on the categories of topological spaces, simplicial sets, and chain complexes of R-modules, one also has interesting model structures for categories like sheaves of modules on a scheme, the category of categories, and the category of algebras of any given type.

In this course, we will use Hirschhorn’s excellent book “Model Categories and their Localizations.” We will define model categories, and give many examples. We will also spend a good amount of time reviewing simplicial sets, as they become very important in the theory. Finally, we will discuss localizations of model categories, invented by Bousfield, and end with Dan Dugger’s result that any model category C is canonically equivalent to a localization of a certain universal model category UC, built from C.

** 684/5/6 Topics in functional analysis**, H. Lin

Math 684 will give a self-contained introduction to C*-algebra theory. The main topics include: the basics of C*-algebras; commutative C*-algebras; positive linear functionals; von Neumann algebras; the spectral theory; inductive limits of C*-algebras; amenable C*-algebras; K-theory. Math 685 and 686 will continue from this with more advanced topics. There will be no tests. Students are expected to do home work and present some of their work in class.

Pre-requisites: 600 analysis.

** 690 K-theory**, C. Phillips

** 691 Morse theory, surgery theory, and applications**, B. Botvinnik

The course will start with the basic smooth topology: manifolds, smooth maps, transversality. Then we’ll study the basic Morse Theory on smooth manifolds and surgery theory. The next topic is to study the space of Morse functions and generalized Morse functions in order to describe the solution of so called concordance problem and the space of smooth structure on a given manifold. Special care will be given to the n-sphere. As applications, we will prove results on the homotopy type of moduli spaces of Riemannian metrics and Riemannian metrics of positive scalar curvature. It is required that student are familiar with basic 600-algebraic topology and are comfortable with basic 600-analysis. There will be given some homework assignments and projects for in-class presentations.

** 692 What Every Topologist Should Know (WETSK),** H. Sadofsky

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.

Pre-requisites: Math 691.