ASPECTS OF INTEGRABLE SYSTEMS

IAS - Spring 2003


Organizers: Percy Deift (IAS and NYU) 
Emma Previato (IAS and Boston University) 
Day: Wednesdays
Time: 3:30 p.m - 5:30 p.m
Place: S-101 for all meetings, except the first which will be in S-114


Each meeting will feature two talks, both concerning the same aspect of integrable systems. Aspects to be considered will include: random matrix theory, algebraic geometry (theta-function theory and intersection theory on moduli spaces), tau-function on the infinite-dimensional Grassmann manifold. 

The series should serve as an introduction to these topics, speakers will be asked to provide some background for non-experts and the audience will be encouraged to pose informal questions.

 

Meeting Date

January 29
Emma Previato, IAS and Boston University
Integrable Hierarchies of Nonlinear PDEs
ACI (Algebraically Completely Integrable) Hamiltonian Systems



3:30 p.m
4:30 p.m

S-114

February 12
Eugene Strahov, Brunel University
Random Characteristic Polynomials: Riemann-Hilbert Approach
3:30 p.m. S-101
February 26
David Ben-Zvi, University of Chicago
Solitons and Many-Body Systems in Algebraic Geometry
3:30 p.m. S-101
March 12
No Meeting - Postponed to April 30
3:30 p.m. S-101
March 26
Luen-Chau Li, Pennsylvania State University
Hidden Symmetries in Integrable Systems and Dynamical Poisson Groupoids

3:30 p.m.

S-101
Igor Krichever, Columbia University
Integrable Chains on Algebraic Curves
4:30 p.m. S-101
April 9
Nick Ercolani, University of Arizona at Tucson
The Large N Expansion for a Random Matrix Partition Function

1:30 p.m.

S-101
Jinho Baik, Princeton University
Universality of Discrete Orthogonal Polynomial Ensembles
2:30 p.m. S-101
April 30
Andrei Okounkov, Princeton University

3:30 p.m.

S-101

ABSTRACTS

Emma Previato, IAS and Boston University
Integrable Hierarchies of Nonlinear PDEs

Two geometric structures for certain (hierarchies of) integrable PDEs emerged in the late 1960s and through the '70s: a (finite-genus) spectral curve and its theta functions; and an infinite Grassmannian. This talk will provide the setting and two main questions that remain open, namely the higher-rank and higher-dimensional cases, together with some of their applications.

Emma Previato, IAS and Boston University
ACI (Algebraically Completely Integrable) Hamiltonian Systems

Many examples of completely integrable Hamiltonian systems have been linearized on abelian varieties (ACI case), chiefly by means of Lax pairs with parameters and their spectral curves. This survey will present: the symplectic-geometric interpretation of reduction for such systems; their relation with hierarchies of integrable PDEs; and the generalization due to N.J. Hitchin.

 


David Ben-Zvi, University of Chicago
Solitons and Many-Body Systems in Algebraic Geometry

A puzzling discovery of the theory of integrable systems is that the motion of the poles of meromorphic solutions to soliton equations (PDE such as the Korteweg--deVries equation) is often governed by integrable many--body systems (ODE such as the Calogero-Moser system). I will present an explanation of this phenomenon (joint work with T. Nevins) using (noncommutative) algebraic geometry. We study the space of "configurations of points on the quantum plane" and other spaces of noncommutative vector bundles as a natural bridge between solitons and particles. Namely, the soliton equations are realized as flows on these configurations, and a geometric Fourier transform converts the flows into the linear flows along tori (Jacobians of spectral curves) which give the "integration" of the many--body system.


Luen-Chau Li, Pennsylvania State University
Hidden Symmetries in Integrable Systems and Dynamical Poisson Groupoids

Many well-known examples of integrable systems are related to Lie groups and Lie algebras. In this talk, I will present examples which are related to the coboundary dynamical Poisson groupoids of Etingof and Varchenko. In particular, this will include several classes of models known under the common name "spin Calogero-Moser systems". As is well-known, the coboundary dynamical Poisson groupoids provide a natural geometric interpretation of the classical dynamical r-matrices which first appeared in the context of Wess-Zumino-Witten conformal field theory. An important feature of our general scheme is the exact solvability of the equations via factorization problems on Lie groupoids. I will discuss the structures underlying this solution method and explain why they are also of interest in other problems.

 


Nick Ercolani, University of Arizona at Tucson
The Large N Expansion for a Random Matrix Partition Function

This talk will present some recent work, joint with Ken McLaughlin, which uses Riemann-Hilbert techniques to describe the large N asymptotics of the partition function for N X N Hermitian random matrices wrt a general class of exponential weights. Applications to some graphical enumeration problems will be discussed.


Jinho Baik, Princeton University
Universality of Discrete Orthogonal Polynomial Ensembles

In the random matrix theory, it is known that in the bulk scaling limit, the correlation functions of the scaled eigenvalues are universal (sine kernel) for a general class of unitary invariant measure on Hermitian matrices. The density function of the eigenvalues of unitary invariant measure is given by the Coulomb gas of beta=2 with certain external (continuous) potential. In this talk, we replace the potential by pure point measure. We prove the universality for a general class of pure point measures when we take continuum limit and bulk scaling limit simultaneously. An application of this result is the computation of the local correlation functions of random hexagon tiling. This is a joint work with Thomas Kriecherbauer, Ken McLaughlin and Peter Miller.

Updated - Monday, March 31, 2003