# Advanced Graduate Courses 2007/08

Here is the schedule for all graduate-level courses for 2007-2008. See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2007 |
WINTER 2008 |
SPRING 2008 |

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

P. Gilkey (9.00) | P. Gilkey (9.00) | P. Gilkey (9.00) |

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

Y. Xu (10.00) | Y. Xu (10.00) | Y. Xu (10.00) |

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

P. Lu (1.00) | P. Lu (1.00) | J. Isenberg (1.00) |

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

A. Berenstein (11.00) | A. Berenstein (11.00) | A. Berenstein (11.00) |

556 Networks and combinatorics | 557 Discrete dynamical systems | |

N. Proudfoot (1.00) | D. Dugger (12.00) | |

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

Q.-M. Shao (12.00) | Q.-M. Shao (12.00) | Q.-M. Shao (12.00) |

607 General relativity | 607 Topics in functional analysis | 607 Group actions on C*-algebras |

J. Isenberg (12.00) | H. Lin (2.00) | N. C. Phillips (2.00) |

607 Numerical analysis | 607 Formal group laws and cobordism | |

Y. Xu (2.00) | H. Sadofsky (12.00) | |

607 Homological algebra | 607 Commutative algebra, I | 607 Commutative algebra, II |

N. Proudfoot (10.00) | M. Vitulli (10.00) | M. Vitulli (10.00) |

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

P. Lu (9.00) | P. Lu (9.00) | P. Lu (9.00) |

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

D. Sinha (1.00) | D. Sinha (1.00) | D. Sinha (1.00) |

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

A. Polischuk (11.00) | A. Polischuk (11.00) | A. Polischuk (11.00) |

672 Probability theory I | 673 Probability theory II | |

D. Levin (2.00) | D. Levin (2.00) | |

681 Algebraic groups I | 682 Algebraic groups II | 683 Algebraic group actions |

J. Brundan (11.00) | J. Brundan (11.00) | V. Ostrik (11.00) |

684 Topics in Fourier analysis | 685 Topics in Fourier analysis | 686 Topics in wavelet analysis |

M. Bownik (10.00) | M. Bownik (10.00) | M. Bownik (10.00) |

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

P. Gilkey (1.00) | D. Dugger (1.00) | D. Dugger (1.00) |

ADVANCED COURSE DESCRIPTIONS:

** 607 – Topics in functional analysis**, Lin

This is a seminar course. Topics include essential normal operators on the Hilbert spaces, K-theory and classical dynamical systems.

There will be no exams. However, students may be required to present some materials related to the topics of the course.

** 607 – Group actions on C*-algebras and their crossed products **, Phillips

This is a continuation of Huaxin Lin’s Math 607 in Winter 2008. The course will be about group actions on compact spaces and on simple C*-algebras, and the structure of the resulting crossed products, with emphasis on the cases (minimal actions on spaces, outer actions on simple C*-algebras) in which the crossed product is expected to be simple. The groups will mostly be finite groups or the group of integers.

The formal prerequisite is a version of Math 684-686 which dealt with C*-algebras. However, I will try to make the course accessible to students who have had some other version of Math 684-686 and who are willing to take on faith some standard theorems about C*-algebras.

**607 – General relativity: The mathematics of black holes, cosmology, and gravitational waves **Isenberg

Classical (ie, nonquantum) gravitational physics is very accurately modeled by spacetime solutions of Einstein’s equations of general relativity. The physics of black holes, the large scale behavior of the universe, and the nature of gravitational radiation can be analyzed by studying various families of spacetime solutions.

Assuming that the students are familiar with the basics of Riemannian geometry, this course first introduces the mathematics of Lorentzian geometries and spacetimes, and discusses the Einstein partial differential equations on spacetimes. We then discusses models for black holes, for cosmology, and for gravitational radiation in turn. In each case, we start by considering archetypal explicit solutions (Schwarzschild for black holes, Friedmann for cosmology), and then we address some of the main mathematical questions of current interest, such as Cosmic Censorship.

Pre-requisite: a course in Riemannian geometry

** 607 Formal group laws and complex cobordism**, Sadofsky.

A formal group law of dimension n over a (commutative) ring R is a power series in 2n variables satisfying associativity and identity axioms with respect to composition. Such objects occur in nature when considering analytic Lie groups or (affine) group schemes. They occur specifically in algebraic topology when considering E^{*}(CP^{{infty}}) for certain nice (complex oriented) generalized cohomology theories E. CP^{{infty}} is an abelian topological group, and E^{*} of the group structure map leads to a formal group law in a natural way when E^{*} is complex oriented.

We’ll study one dimensional formal group laws and applications to algebraic topology. Topics covered will include at least some of the following:

- Lazard’s theorem identifying the universal one dimensionalformal group law.
- Quillen’s work identifying Lazard’s formal group law with the natural formal group law over the complex cobordism ring MU
^{*}, leading to Quillen’s construction of the BP-spectrum. - p-typical formal group laws and classification of p-typical formal group laws by height over an algebraic closed field (of characteristic p).
- The Lubin-Tate theory of lifts and Morava stabilizer groups.
- How the previous two topics lead to the chromatic filtration of stable homotopy theory.

Pre-requisites: We will assume the 600-level algebra course and the 600-level algebraic topology course. A basic familiarity with stable homotopy theory will also be helpful, but I’ll try to review the relevant ideas as we need them. Students taking this course will be expected to turn in solutions to some fraction (probably about 1/2) of perhaps 20-30 exercises I’ll assign over the term.

** 607 Homological algebra**, Yuzvinsky

This will be a general introductory course to homological algebra.

** 607 Commutative algebra I**, Vitulli

In addition to being a beautiful and powerful subject on its own right, commutative algebra serves as the foundation for modern day algebraic geometry and algebraic number theory. This seminar will introduce notions from both areas but the bias will be towards algebraic geometry. We will use the text by H. Matsumura to gain a basic understanding of the subject and the newly released text by I. Swanson and C. Huneke to delve into some topics that are important in much current research by the commutative algebra community.

Pre-requisite: 600 Algebra

Probable Texts

Commutative Ring Theory by Hideyuki Matsumura [M], Translated by Miles Reid, Cambridge Studies in Advanced Mathematics (No. 8) Cambridge University Press, 1989.

Integral Closure of Ideals, Rings, and Modules by Irena Swanson and Craig Huneke, London Mathematical Lecture Note Series 336, Cambridge University Press, 2006 [optional first term].

Topics to be covered include the following.

- Basic Results: (Chapter 1 of [M] ) prime avoidance, reduced rings, radical ideals, Jacobson radical, local rings, Chinese Remainder Theorem (CRT), Nakayama’s lemma (NAK), chain conditions (Noetherian and Artinian rings)
- Prime Ideals and Localization: (Chapter 2 of [M]) the prime spectrum of a ring and the Zariski topology, consequences of Nakayama’s lemma, localization of rings and modules, multiplicative subsets, saturation, support of a module, functorial properties of localization, locally free modules, the Hilbert Nullstellensatz, associated primes and primary decomposition
- Extensions of Rings: (Chapter 3. [M] ) Flatness, Completion, and the Artin-Rees Lemma
- Integral Extensions of Rings: (Chapter 2 of [SH]) lying-over, incomparability, the going up and going down theorems, integral closure of graded rings.
- Valuation Theory (Chapter 4 of [M] or Chapter 6 of [SH]) Krull valuations, discrete (rank one) valuation rings, Dedekind domains, the Krull-Akizuki Theorem, Krull rings.
- Krull Dimension Theory: (Chapter 5 of [M]) graded rings and modules, homogeneous ideals, the Hilbert function and the Hilbert-Samuel function, systems of parameters, multiplicity, dimension of extension rings.
- Regular Sequences and Depth: (Chapter 6 of [M]) regular sequences and the Koszul complex, Cohen-Macaulay rings, the Auslander-Buchsbaum dimension formula, and Gorenstein rings.

** 607 Commutative algebra II**, Vitulli

Probable Texts:

Commutative Ring Theory by Hideyuki Matsumura [M], Translated by Miles Reid, Cambridge Studies in Advanced Mathematics (No. 8), Cambridge University Press, 1989. [optional second term]

Integral Closure of Ideals, Rings, and Modules by Irena Swanson and Craig Huneke, London Mathematical Lecture Note Series 336, Cambridge University Press, 2006.

Topics to be covered include the following.

- Integral closure of ideals: (Chapter 1 of [HS]) basic properties, reductions, case of monomial ideals.
- More on Noetherian Rings: Normalization theorems, complete rings, Jacobian ideals, Serre’s condtions, absolute integral closure.
- Regular rings: (Chapter 7 of [M]) basic properties, UFDs, complete intersection rings
- Rees algebras: (Chapter 5 of [HS]) constructions, integral closure of Rees algebras, integral closure of powers of an ideal, formal equidimensionality, blowing up (connections with algebraic geometry)
- Derivations: (Chapter 7 of [SH]) analytic approach, derivations and differentials
- Reductions: (Chapter 8 of [SH]) basic properties, connections with Rees algebras, minimal reductions, reducing to the infinite field case, superficial elements and reductions
- Rees Valuations: uniqueness of Rees valuations, construction, examples, properties, rational powers of ideals
- Multiplicity and Integral Closure: multiplicity, Ree’s theorem, equimulltiple families of ideals

**607 Numerical Analysis**, Y. Xu

Numerical analysis is a brach of mathematics devoted to methods and theorems that can be used to solve the “real” world problems. The course will be introductory and offer a quick survey on several topics. The emphasis is on mathematics; no programming skill is required. All graduate students are welcome.

The first quarter will likely cover standard topics in an introductry course, including Interpolation, Numerical Integration and Orthogonal Polynomials. I plan to give a concise account of what I regard as the most useful results in one variable, concentrating on ideas, and then give a tour of what is known, or possible, in several variables. The second quarter, if there is enough interests, will follow the same format on other topics.

** 681-2 Algebraic groups**, Brundan

This course will provide a basic introduction to the theory of (affine) algebraic groups. An algebraic group is an affine algebraic variety equipped with an additional group structure such that the group operations are algebraic functions. Everyone knows at least one example: the general linear group GL_{n}(k) over an algebraically closed field k. In fact, all algebraic groups can be viewed as matrix groups, since every algebraic group can be embedded as a closed subgroup of some GL_{n}(k), by the algebraic groups analogue of Cayley’s theorem.

The course will start from the basic language of algebraic geometry, at a level suitable for students who have just taken quals. In fact, studying algebraic groups is a good way to learn a little algebraic geometry too. For example, the homogeneous spaces G / H arising as coset spaces of an algebraic group G with respect to a closed subgroup H include very many fundamental examples of smooth algebraic varieties like Grassmannians and flag varieties.

We will develop the general theory of algebraic groups and their Lie algebras, homogeneous spaces and algebraic group actions, before specializing our study to reductive algebraic groups. The latter have a beautiful classification by combinatorial data in terms of root systems and Dynkin diagrams. We will stop short of actually proving this classification, but we will start to develop the structure theory for reductive algebraic groups and see many examples.

Pre-requisites: 600 algebra

** 683 Algebraic group actions and quivers**, Ostrik

In this course we will describe some interesting examples of actions of algebraic groups on algebraic varieties. Most of the examples we’ll consider arise from the representation theory of quivers. So at the same time as describing some beautiful applications of the general theory of algebraic groups, this course will provide an introduction to the representation theory of finite dimensional algebras and especially path algebras of quivers. In particular, we will discuss quivers of finite type, which are classified by Gabriel’s theorem in terms of ADE Dynkin diagrams. Then we will discuss representations of quivers of affine type and Ringel’s construction of quantum groups using representations of quivers.

Pre-requisites: algebraic groups

** 684-6 Fourier analysis and wavelets**, Bownik

Fourier analysis is a subject of mathematics that originated with the study of Fourier series and integrals. Nowadays, Fourier analysis is a vast area of research with connections and applications in various branches of science including partial differential equations, potential theory, mathematical physics, number theory, signal analysis, and tomography. A recent noteworthy area of focus in Fourier analysis are wavelet bases. The theory of wavelets is a very active area of research with many real-world applications to signal processing, e.g. JPEG 2000 image compression algorithm.

This course is an introduction to Fourier analysis and wavelets. More specifically, we are planning to cover the following topics:

- General properties of orthogonal systems, Riesz bases, and frames.
- Convergence and summability of Fourier series, lacunary series.
- Fourier transforms, inversion formula, Plancherel’s theorem.
- Multiple Fourier series and the Poisson summation formula.
- Theory of distributions, Schwartz class, and tempered distributions.
- Hardy-Littewood maximal function, Hilbert transform, and Calder’on-Zygmund singular integral operators.
- $L^2$ theory of wavelets, shift-invariant spaces, scaling functions, and multiresolution analysis.
- Construction of MSF wavelets, Str”omberg wavelets, Meyer wavelets, and compactly supported Daubechies wavelets.
- Wavelets in higher dimensions, expansive dilations.
- Wavelet bases in function spaces such as: Lebesgue spaces, Hardy spaces, Lipschitz spaces, the space BMO, Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces.
- Applications to signal processing, discrete Fourier and wavelet transforms.

**690 Characteristic classes**, Gilkey

The Chern, Pontrjagin, and Euler classes are invariants of complex, real, and oriented vector bundles (respectively) that can be defined either topologically using integer cohomology or differential geometrically using DeRham cohomology. This course will p resent a brief introduction to the subject — relevant results of vector bundle theory will be developed as needed. Open to students who have had had either the 600 Algebraic topology sequence (and in exceptional circumstances to students who have only had the 600 Differential geometry sequence).

** 691 K-Theory**, Dugger

K-theory is a generalized cohomology theory which is constructed out of vector bundles. Next to singular cohomology, it is perhaps the simplest cohomology theory to work with—and it comes up often in geometry and topology. In this course we’ll develop the basics of working with this theory, and also cover some applications. I hope to discuss Thom isomorphism, some versions of the Grothendieck-Riemann-Roch theorem, as well as a tiny bit of index theory. Time permitting, we’ll look at how K-theory solves the vector fields on spheres problem.

Course requirements are the 600 topology sequence, and also some familiarity with vector bundles and characteristic classes (like one can get from 690).

** 692 What Every Topologist Should Know (WETSK),** Dugger

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.

Pre-requisites: Math 691.