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2001 Summer Program for Graduate Students on
Poisson and Quantum Structures
University of Cincinnati
June 18-July 13, 2001


Organizer: Timothy J. Hodges
Department of Mathematical Sciences
University of Cincinnati

timothy.hodges@uc.edu
http://math.uc.edu/~hodgestj/

Week One: June 18-22

Ken Meyer (University of Cincinnati)

Classical Hamiltonian Systems

I will start with classical examples of mechanical systems and reformulate them as a Hamiltonian systems of differential equations. Then I will discuss some of the features of the solutions, symmetries, integrals etc. of Hamiltonian equations.

Then I will go global and introduce symplectic manifolds. Hamiltonian flows live naturally on such manifolds. I will end with a discussion of the MMW reduction theorem for Hamiltonian systems with symmetries.

Everyone should know the basics of differential manifolds. The first few chapters of "Differential Topology" by M. W. Hirsch should be good enough.

Week Two: June 25-29

Jiang-Hua Lu (University of Arizona)   jhlu@math.arizona.edu

Introduction to Symplectic and Poisson Geometry

In this 10 hour mini-course, we will attempt to cover some basic materials in symplectic and Poisson geometry and point out some research problems in these fields. The course outline is as follows:

Day 1: Definition of Poisson manifolds, examples, and basic properties;

Day 2: Symplectic manifolds and examples;

Day 3: Moment maps and symplectic reduction;

Day 4: Poisson Lie groups and Lie bialgebras;

Day 5: Poisson homogeneous spaces

Prerequisites: First year course on differential manifolds; One semester of Lie theory (for the second half of the course).

Textbooks:

  1. McDuff, D., and D. Salamon, "Introduction to symplectic topology", Oxford Mathematical Monographs, Oxford, 1995.

  2. Korogodski, L. and Soibelman, Y., Algebras of functions on quantum groups, part I", AMS, Mathematical surveys and monographs, Vol. 56, 1998.

Week Three: July 2-6

Tetsuji Miwa (Kyoto University)   tetsuji@kusm.kyoto-u.ac.jp

Physical Combinatorics

I will explain the combinatorial aspects of integrable models. Corner transfer matrix method, crystal base and paths, fermionic and bosonic character formulas and coinvariants in conformal field theory.

Reading List:

Kac-Raina
Highest weight representations of Infinite
Dimensional Lie Algebras
World Scientific, 1987

Jimbo-Miwa
Algebraic Analysis of Solvable Lattice Models
CBMS 86, AMS, 1995

Week Four: July 9-13

Michael Gekhtman (University of Notre Dame)   mgekhtma@darwin.helios.nd.edu

Integrable Systems

In this mini-course, I plan to review basic techniques and constructions of the theory of integrable systems. We shall discuss Lax formalism, several versions of the inverse problem method, Lie algebraic approach to constructing exactly solvable Hamiltonian equations and an interplay between complete integrability and exact solvability. Examples to be used to illustrate these concepts include integrable equations of classical Hamiltonian mechanics as well as finite-dimensional models of the modern soliton theory such as the Toda lattice. If time permits, infinite dimensional examples (KP and KdV hierarchies) will also be considered.

Reading List (by no means comprehensive)

1. Arnold, V. I., Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.

2. Moser, J. Geometry of quadrics and spectral theory. The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), pp. 147--188, Springer, New York-Berlin, 1980.

3. Perelomov, A. M., Integrable Systems of Classical Mechanics and Lie Algebras, Birkhauser-Verlag, 1990.

4. Reyman, A.G. and Semenov-Tian-Shansky, M. A. Group-theoretical methods in the theory of finite-dimensional integrable systems Dynamical Systems VII, Encyclopedia of Math. Sci., Springer-Verlag, Berlin-Heidelberg-New York, 1994.

2001 Summer Program for Graduate Students on "Poisson and Quantum Structures"

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