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Advanced Graduate Courses 2006/07

Here is the schedule for all graduate-level courses for 2006-2007. See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2006 WINTER 2007 SPRING 2007
513 Introduction to analysis I 514 Introduction to analysis II 515 Introduction to analysis III
       M. Bownik (11.00)        M. Bownik (11.00)        M. Bownik (11.00)
520 Differential equations I 521 Differential equations II 522 Differential equations III
       J. Isenberg (10.00)        J. Isenberg (10.00)        J. Isenberg (10.00)
531 Introduction to topology I 532 Introduction to topology II 533 Introduction to diff geometry
       D. Dugger (9.00)        D. Dugger (9.00)        J. Isenberg (9.00)
544 Introduction to algebra I 545 Introduction to algebra II 546 Introduction to algebra III
       A. Vaintrob (13.00)        A. Vaintrob (13.00)        A. Vaintrob (13.00)
556 Networks and combinatorics 557 Discrete dynamical systems
       W. Kantor (13.00)        D. Sinha (10.00)
564 Mathematical statistics I 565 Mathematical statistics II 566 Mathematical statistics III
       D. Levin (12.00)        D. Levin (12.00)        D. Levin (12.00)
607 Introduction to PDEs I 607 Introduction to PDEs II 607 Markov chains
       P. Lu (12.00)        J. Isenberg (12.00)        D. Levin (14.00)
607 Graded commutative algebras 607 Noncommutative graded algebras 607 Differential graded algebras
       S. Yuzvinsky (14.00)        B. Shelton (14.00)        D. Sinha (14.00)
607 Infinite dim’l Lie algebras 607 Infinite dim’l Lie algebras
       A. Kleshchev (10.00)        A. Kleshchev (10.00)
616 Real analysis I 617 Real analysis II 618 Real analysis III
       N.C. Phillips (11.00)        N.C. Phillips (11.00)        N.C. Phillips (11.00)
634 Algebraic topology I 635 Algebraic topology II 636 Algebraic topology III
       H. Sadofsky (9.00)        H. Sadofsky (9.00)        H. Sadofsky (9.00)
637 Differential geometry I 638 Differential geometry II 639 Differential geometry III
       G. Landweber (10.00)        P. Gilkey (10.00)        P. Gilkey (10.00)
647 Abstract algebra I 648 Abstract algebra II 649 Abstract algebra III
       V. Ostrik (13.00)        V. Ostrik (13.00)        V. Ostrik (13.00)
681 Intro to algebraic geometry I 682 Intro to algebraic geometry II 683 Topics in algebraic geometry
       A. Polishchuk (13.00)        A. Polishchuk (13.00)        A. Vaintrob (12.00)
684 Topics in functional analysis 685 Topics in functional analysis 686 Topics in functional analysis
       H. Lin (11.00)        H. Lin (11.00)        H. Lin (11.00)
690 Spectral sequences and 691 Morse theory and surgery 692 WETSK
           characteristic classes        B. Botvinnik (9.00)        G. Landweber (9.00)
       D. Sinha (9.00)



681/2 Introduction to algebraic geometry, Polishchuk, Fall 2006/Winter 2007.

Algebraic geometry is one of the most highly developed and beautiful branches of mathematics with wide spectrum of connections and applications in other disciplines (both within and outside mathematics). Its ideas and methods play an important role in the development of various areas of mathematics, such as number theory, commutative and non-commutative ring theory, representation theory, complex analytic geometry, algebraic combinatorics, and mathematical physics. In essence, it is the study of solutions of polynomial equations in several variables. The algebraic part is in the polynomial nature of the equations, while the geometry lies in the curves, surfaces and higher dimensional objects the equations represent. This course will provide a basic introduction to algebraic geometry.

The topics covered will include affine and projective varieties, Hilbert’s Nullstellensatz, Zariski topology and regular functions, regular and rational maps of varieties, dimension, degree, blowing up, line bundles and divisors, the Riemann-Roch theorem for curves. We will spend a considerable amount of time looking at specific examples and applications.

Pre-requisites: 600 algebra.


683 Further topics in algebraic geometry, Vaintrob, Spring 2007.

We will develop some further topics in algebraic geometry based on the interests of the class (and on precisely what was covered in 681/682).

Pre-requisites: 681/682 Introduction to Algebraic Geometry.


607 Graded commutative algebras, Yuzvinsky, Fall 2006.

Every cohomology algebra of a topological space is graded commutative. Another source of such algebras is combinatorics. This course will start with some of the general theory of such algebras (e.g., definition of their singular loci) and then will focus on the class of those algebras called Orlik-Solomon algebras. This subject is a hot topic of current research. These algebras first appeared as the cohomology algebras of the complement in a complex linear space of a bunch of hyperplanes. Then it became clear that they are determined only by the combinatorics of hyperplanes – so called matroids. These algebras support cochain complexes whose cohomology is important for many problems. The course will tentatively contain the following topics:

  • A short excursion into graded commutative algebra: Hilbert series, Groebner bases, sigular locus, etc.
  • Definition of Orlik-Solomon algebras by generators and relations.
  • Their relation to geometric lattices.
  • Their relation to hyperplane arrangement complements.
  • The cochain complexes they define and the cohomology of these complexes.
  • Open problems.

Pre-requisites: 600 algebra.


607 Non-commutative graded algebras, Shelton, Winter 2007.

This will be a general introductory course to non-commutative ring theory with emphasis on all things graded.


607 Differential graded algebras, Sinha, Spring 2007.

We focus on chain complexes with a compatible graded-commutative multiplication, also known as differential graded algebras (DGA’s). These first arose in studying differential forms on a manifold but now appear in many areas throughout mathematics. A guiding problem is classification up to quasi-isomorphism, an equivalence relation generated by maps which induce an isomorphism in homology rings. But DGA’s with isomorphic homology rings need not be quasi-isomorphic (unless the DGA’s are formal), and we will focus on this distinction, developing Sullivan’s minimal models as a standard tool. The other main conceptual starting point is that one can do homotopy theory with DGA’s, mimicing in the algebraic world constructions well-known for spaces. Indeed by work of Quillen the homotopy theory of rational, 1-connected DGA’s perfectly reflects the homotopy theory of rational, 1-connected spaces.

The theory is greatly enriched because of a wealth of close ties to other interesting objects. We will highlight Quillen’s original connections to DG Lie algebras in particular through Lie algebra cohomology (sometimes referred to as Koszul duality between commutative and Lie algebras), my closely-related recent connection with DG Lie coalgebras, and Pirashvili’s connection with A-infinity algebras.

Pre-requisites: 600 algebra and topology sequences.


607 Infinite dimensional Lie algebras, Kleshchev, Fall/Winter 2006/7.

Kac-Moody algebras, Lie algebras of infinite matrices, Heisenberg algebras, and Virasoro algebras as main examples; construction of affine Kac-Moody algebras; real and imaginary roots; affine Weyl group; highest weight modules; Weyl-Kac character formula.

Boson-fermion correspondence, applications to soliton equations, the Kac determinant formula for Virasoro algebras, Sugawara construction, vertex construction for basic representations, vertex algebras, W-algebras.

Books: Kac, Infinite dimensional Lie algebras; Kac-Raina, Highest weight representations of infinite dimensional Lie algebras; Kac, Vertex algebras for beginners.

Pre-requisites: Some knowledge of finite dimensional complex semisimple Lie algebras.


684/5/6 Topics in functional analysis, Lin, Fall/Winter/Spring 2006/07.

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.


607 Introduction to PDEs I: Basic elliptic and parabolic equations, Lu, Fall 2006.

We will cover the following topics.

  • Basic Laplace equations (section 2.2 in [E]).
  • Basic Heat equation (section 2.3 in [E]).
  • Sobolev spaces (embedding theorems and some basic inequalities setion 5.1-5.3, 5.7, part of 5.6 and 5.8 in [E]).
  • Weak solution method to ellptic equation (section 6.1- 6.4 in [E]).
  • Weak solution method to parabolic equation (section 7.1 in [E]).
  • Schauder theory for elliptic equation (section chapter 3 to 6 in[K]).
  • A brief introduction to Schauder theory of parabolic equation (section 10.1-10.4 in [K]).


  • [E] Lawrence C. Evans, partial differential equations;
  • [K] N.V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces


607 Introduction to PDEs II: Nonlinear elliptic equations, Isenberg, Winter 2007.

Relying on some of the machinery built up in the first course above, we develop some basic tools and apply them to the study of some of the nonlinear elliptic PDEs that arise in geometry. Here are some of the topics we will cover:


  • Maximum Principle
  • Sub and Super Solution Method
  • Contraction Mapping
  • Gluing
  • Einstein Constraint Equations
  • Minimal and Constant Mean Curvature Submanifolds
  • Harmonic Maps

690 Spectral sequences and characteristic classes, Sinha, Fall 2006.

As is customary for 690, we cover characteristic classes for vector bundles. The point of view in this course is through first developing the Leray-Serre spectral sequence for fiber bundles more generally and then seeing charactertistic classes through LSSS computations. If time permits at the end we may also use the LSSS to investigate the topology of Lie groups, homogeneous spaces, and other spaces obtained by quotienting group actions.

Pre-requisites: 600 topology.


691 Morse theory, surgery theory, and applications, Botvinnik, Winter 2007.

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), Landweber, Spring 2007.

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 Markov chains and random walks on graphs, Levin, Spring 2007.

There has been a resurgence of interest in Markov chains and other discrete probability models because of their connection to diverse areas such as physics (statistical mechanics) and theoretical computer science (combinatorics, randomized algorithms.) The proposed course would cover background material on Markov chains, along with these newer developments for random walks on graphs and the rigorous analysis of “mixing time” – the time required for a chain to achieve equilibrium.

Likely topics to include:

  • Classical ergodic theory for Markov chains.
  • Random walks on graphs: recurrence, transience, heat kernel bounds, electrical network analysis, cover times.
  • Probabilistic techniques for bounding mixing times: coupling and its variants, strong stationary times.
  • Bounding spectral gap, tools from geometry: Poincare inequality, Cheeger constant, comparison of Dirichlet forms.
  • Applications to statistical mechanics: Ising model. Connection between rapid mixing and spatial phase transitions.

References for the course:

  • R. Lyons, Y. Peres. Probability on Trees and Networks (book in preparation)
  • D. Aldous, J. Fill. Random walks on graphs and reversible Markov chains (book in preparation)
  • W. Woess. Random Walks on Infinite Graphs and Groups (Cambridge U.P.)