Abstracts/Reading List:
2001
Summer Program for Graduate Students on
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/
Daily
Schedule General
Information
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:
-
McDuff, D., and D. Salamon, "Introduction to symplectic topology",
Oxford Mathematical Monographs, Oxford, 1995.
-
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"
Daily
Schedule General
Information
