Here is the schedule for the graduate courses at the 600 level and above for 2004-2005.
| FALL 2004 | WINTER 2005 | SPRING 2005 |
| 607 Non-Commutative Rings | 607 Algebraic Number Theory | 607 Algebraic Groups |
| 607 Quantum Field Theory | 607 Commutative Algebra | 607 Algebraic Combinatorics |
| 607 Adv. Topics in Probability | ||
| 616 Real Analysis | 617 Real Analysis | 618 Real Analysis |
| 634 Algebraic Topology | 635 Algebraic Topology | 635 Algebraic Topology |
| 637 Differential Geometry | 638 Differential Geometry | 639 Differential Geometry |
| 647 Abstract Algebra | 648 Abstract Algebra | 649 Abstract Algebra |
| 681 Commutative Algebra | 682 Algebraic Geometry | 683 Adv. Topics in Algebra |
| 684 Fourier Analysis | 685 Fourier Analysis | 686 Ricci Flow |
| 690 Adv. Topics in Topology | 690 Adv. Topics in Topology | 692 WETSK |
607 Introduction to non-commutative ring theory, Shelton, Fall 2004.
We will cover the classical beginning results in non-commutative ring theory including rings of fractions, the Goldie Theorems, Morita equivalence, and perhaps one or more dimension theories (GK-dimension, projective dimension, Krull dimension). Time permitting I will cover the basic theory of filtered rings and almost commutative rings. I will try to keep everything motivated with lots of examples, but I will tend to stick to the Noetherian world.
Prerequisites: 600 algebra.
607 Quantum field theory and topology, Vaintrob, Fall 2004.
Quantum topology of knots and three-manifolds.
During the last two decades there has been a dramatic change in the relationship between geometry and physics. Ideas coming from quantum field theory have brought spectacular new results and even helped to create new areas of study in geometry and topology. The main focus of this course will be on topological quantum field theories (TQFTs) and their applications in low-dimensional topology. will start by introducing the mathematical formalism of TQFTs and related algebraic structures (tensor categories, Frobenius algebras, quantum groups, e.a.). After constructing first non-trivial examples of TQFTs and a crash course in knot theory, we will study related invariants of links and three-manifolds. These invariants include the famous Jones polynomial and the Witten-Reshetikhin-Turaev invariants of homology spheres. In the third part of the course we will study the perturbation theory approach to the Witten-Chern-Simons TQFT and its relationship with Vassiliev knot invariants and the Kontsevich integral. If time permits, we will include additional topics, such as connection between TQFTs and conformal field theories, moduli spaces and operads.
Prerequisites: 600 algebra, 600 topology. No prior knowledge of
physics or knot theory will be assumed.
607 Algebraic number theory, Polishchuk, Winter 2005.
This will be an introduction to modern algebraic number theory. We will concentrate on the study of local fields and their extensions with some applications to number fields. Then we'll review group cohomology and will proceed to the local class field theory.
Prerequisites: 600 algebra, 600 topology.
681 Introduction to commutative algebra, Vitulli, Fall 2004.
Localization of rings and modules. Primary decomposition integral dependence and valuations. ascending and descending chain conditions. Noetherian and Artinian Rings. Discrete rank one valuation rings and Dedekind domains graded rings. Filtrations and completions. Hilbert Functions and dimension theory.
Text: Introduction to commutative algebra by Atiyah and MacDonald.
Prerequisites: 600 algebra.
607 Topics in commutative algebra, Vitulli, Winter 2005.
Regular sequences and depth Cohen-Macaulay Rings. The canonical module and duality Gorenstein rings. Hilbert functions and multiplicities. Stanley-Reisner rings. Simplicial homology, cellular homology and local cohomology. Semigroup rings and invariant theory. Determinantal rings. Big Cohen-Macaulay Modules. (The precise topics to be covered might depend on the interests of the students in the class.)
Text: Cohen-Macaulay Rings by Bruns and Herzog supplemented by Notes by Bernd Ulrich.
Prerequisites: Introduction to commutative algebra.
682 Algebraic geometry I, Kleshchev, Winter 2005.
Affine and projective varieties (Zariski topology, irreducible components, product of varieties, flag varieties), dimension, morphisms (fiber of a morphism, constructive sets, open morphisms, birational morphisms), tangent spaces (simple points, local ring of a simple point, separability criterion), complete varieties (completeness of projective varieties), notion of algebraic group, identity component, action of algebraic groups on varieties.
Text: Linear algebraic groups by J. E. Humphreys.
Prerequisites: 600 algebra.
683 Algebraic Geometry II, Vaintrob, Spring 2005.
Varieties, sheaves and schemes.
Algebraic Geometry is one of the most highly developed and beautiful branches of mathematics. Its ideas and methods play an important role in the development of mathematics as the whole as well as of its various areas, such as number theory, ring theory, representation theory, complex analysis, combinatorics, and more. Recently it found exciting applications in computer science (coding theory) and theoretical physics (string theory, solutions).
Many of these developments became possible due to the overhaul of the foundations of Algebraic Geometry in the 1960s which replaced the classical algebraic approach with much more flexible and powerful language of sheaves and schemes. In this course we will study algebraic varieties using the now standard sheaf-theoretic methods. We will start with a general introduction to sheaves and their cohomology and then move to their application in Algebraic Geometry.
The topics we will discuss include: divisors, line bundles and rational maps; coherent sheaves on projective varieties and graded modules; Hilbert polynomial; Serre duality; Riemann-Roch theorem for curves and its applications. We will also introduce and discuss schemes (objects generalizing both commutative rings and algebraic varieties), but we will not study them in detail.
Pre-requisites: Algebraic geometry I
607 Algebraic groups, Kleshchev, Spring 2005.
Lie algebra of an algebraic group, derivations, homogeneous spaces, factors, semisimple and unipotent elements, solvable groups, Borel subgroups, centralizers of tori, structure of reductive groups, representations and classification of semisimple groups.
Text: Linear algebraic groups by J. E. Humphreys.
Prerequisites: Algebraic geometry I.
607 Advanced topics in probability, Wang, Spring 2005.
Probability theory is of major importance to a wide variety of disciplines, including Statistics, Economics, Physics, Chemistry, Biology, Epidemics, Finance, and Insurance. In this course we will give an introduction to the modern probability theory and its applications in different areas through examples. The following topics will be covered.
- Introduction to probability theory.
- Random walks and its applications in option pricing.
- Discrete Markov chains and its applications.
- The theory of continuous Markov chains.
- Birth and death processes and Branching processes and their applications.
- Brownian motion and its properties.
- Martingale theory and its applications.
- Stochastic calculus and its applications.
Prerequisites: 500 level real analysis and 400 level probability and statistics.
607 Algebraic combinatorics, Yuzvinsky, Spring 2005.
This course is suggested as a part of the second year graduate program in algebra. The general idea of the course consists of two parts: many algebraic (and not only algebraic) problems reduce to combinatorial ones the most interesting results in combinatorics have been obtained by applying theorems from commutative and skew-commutative algebra. The course would tentatively include the following topics:
- Short excursion into graded commutative algebra: Hilbert series, dimension, depth, cohomology, etc.
- Regular and Cohen-Macaulay rings and algebras.
- The face ring of a simplicial complex; Stanley's proof of the upper-bound conjecture.
- Lattices and matroids; Mobius function.
- The Orlik-Solomon algebra of a lattice.
- Latin squares and realization of Abelian groups by 3-nets in projective plane Alebraization of nets.
Prerequisites: 600 algebra.
684/5 Introduction to Fourier analysis and wavelets, Bownik, Fall 2004/Winter 2005.
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 applications in various branches of science including signal analysis, tomography, partial differential equations, potential theory, mathematical physics and number theory. A recent noteworthy area of focus in Fourier analysis is orthogonal expansions in wavelet bases. The theory of wavelets is a very active area of research with many real-world applications. This course is an introduction to the theory of Fourier series, Fourier integrals, wavelets, and related topics. More specifically, we are planning to cover the following:
- General properties of orthogonal systems, Riesz bases, and frames.
- Convergence and summability of Fourier series. Intro to lacunary series.
- Fourier transforms of $L^2$ functions, inversion formula, Plancherel's theorem.
- Multivariable Fourier series, the Poisson summation formula.
- Theory of distributions, Fourier transforms of tempered distributions, the Paley-Wiener theorem.
- General theory of wavelets, scaling functions, multi-resolution analysis.
- Construction of Stromberg wavelets, Meyer wavelets, and compactly supported Daubechies wavelets.
- Multivariable wavelets, tensor products.
- Wavelets and Calderon-Zygmund operators in various function spaces.
- Applications to signal processing, discrete Fourier and wavelet transforms.
There will be a couple of homework assignments. Since there will be no
final exam each student will give an oral presentation on a subject of
his/her choice related to this course.
Text: Y. Katznelson, An Introduction to Harmonic Analysis Dover, 1976;
P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Press, 1997
Pre-requisites: 600 analysis.
686 Ricci Flow, Lu, Spring 2005.
In this course the emphasis will be on the basic geometric pictures and the basic tools used in Ricci flow. The aim is to get greater understanding of sectons 1-10 of Perelman's preprint [2002]. The following topics will be covered.
- What is Ricci flow? Some special solutions for Ricci flow. Intuitive understanding of what Ricci flow does to Riemannian metrics.
- Survey of major results before Perelman. Short time existence, Three manifolds with positive Ricci curvature converges to quotient of standard sphere, Shi's derivative estimates.
- Maximum principle in Ricci flow and its application (Hamilton-Ivey pinching Theorem).
- Injectivity radius and Bishop-Gromov volume comparison theorem in Riemannian geometry.
- Ricci flow is a variational flow and non-collapsing theorem (from section 1-4 in Perelman).
- Dilation limit of singularities in Ricci flow.
- Remaining sections from Perelman's paper, spending time on any necessary preparation needed to understand this.
Prerequisites: Elementary Riemannian geometry, some knowledge of PDEs.
