Working seminar on
		integrable systems and random matrices
	Spring 2007
	Thursdays 3:30 - 4:30pm, Courant Institute WWH Room 513
	Organizers: Jinho Baik and Percy Deift (baik@cims.nyu.edu, deift@cims.nyu.edu)
	Date	Speaker	Title and Abstract
	18-Jan	Percy Deift	On the Pfaffian
		(CIMS)	The solution of many problems in mathematics and mathematical
			physics can be expressed in terms of the Pfaffian of a skew symmetric matrix.
			This talk is an introduction to some basic properties of the Pfaffian.
			Various applications will also be discussed.
	8-Feb	Rowan Killip	A road less travelled
		(UCLA)	I will outline an approach to random matrix theory first
			pursued by H. Trotter.  The route appears to lead in interesting
			directions, but also has plenty of potholes.
	15-Feb	Peter Miller	The Semiclassical Modified Nonlinear Schrodinger
		(U. Michigan)	Equation:  Facts and Artifacts
			I will discuss some recent work (joint with J. DiFranco) on semiclassical Cauchy
			problems for an integrable perturbation of the focusing nonlinear Schr\"odinger
			equation.  The perturbation is singular in the sense of inverse-scattering and also in the
			more practical sense that the modified equation admits solutions with surprisingly
			different properties than the unmodified equation.  "Facts and Artifacts" is a reference to
			a similarly-titled paper by E. V. Doktorov, whose lecture on the subject in Edinburgh in
			2004 originally piqued our interest.
	22-Feb	Rowan Killip	Eigenvalue statistics for CMV mstrices
	starts at	(UCLA)	Per request, I will discuss some of the technicalities
	4:00pm		of my recent joint work on eigenvalue statistics for CMV matrices
	1-Mar	Alexei Onatski	How Large Random Matrices Help Macroeconomists
		(Columbia U.	The central bank processes a huge amount of information before it decides on the level of
		Economics dept.)	the short-term interest rate. However, formal economic and statistical models
			employed to inform this decision rely only on few economic indicators. I will describe
			undesirable consequences of this contrast, discuss modern econometric techniques
			which reduce the contrast, relate these techniques to recent research on large random
			matrices, describe some new results, and point out directions for future research.
	8-Mar	Robert Buckingham	Asymptotics of the Tracy-Widom distribution functions
		(U. Michigan)	We find the constant term of the asymptotic expansions of the Tracy-Widom distribution
			functions as the distribution parameter approaches minus infinity.  The Tracy-Widom distribution
			functions are written in terms of an integrals of a Painlev\'e II function up to positive infinity. The
			constants are computed by deriving alternate formulas for the distribution functions starting from
			minus infinity.  The results for the GOE and GSE distribution functions are new, while the result
			for the GUE distribution function provides an alternate proof of the recent work of Deift, Its, and
			Krasovsky. The new integral formula also allows the evaluation of the total integral of the
			Hastings-McLeod solution of the Painlev\'e II equation.  This is joint work with Jinho Baik and
			Jeff DiFranco.
	15-Mar	No seminar	Spring Break
	22-Mar	No seminar
	29-Mar	Dmitry Shepelsky	Asymptotics for the Camassa-Holm equation
		(Institutte for Low	We discuss the long-time asymptotis of solutions of the Camassa-Holm equation
		Temperature Physics	$$u_t-u_{txx}+2u_x+3uu_x=2u_xu_{xx}+uu_{xxx},$$ which is an integrable
		and Engineering)	model equation describing the shallow-water approximation in inviscid hydrodynamics.
			The method is the adaptation of the nonlinear steepest descent method for Riemann-Hilbert
			problems. This is joint work with Anne Boutet de Monvel.
	5-Apr	No seminar
	12-Apr	Lauren Williams	Tableaux combinatorics for the asymmetric exclusion process
		(Harvard)	The partially asymmetric exclusion process (PASEP) is an important model from statistical
			mechanics which describes a system of interacting particles hoppingleft and right on a one
			dimensional lattice of N sites.  It is partially asymmetric in the sense that the probability of hopping
			left is q times the probability of hopping right. Additionally, particles may enter from the left with
			probability alpha and exit to the right with probability beta. It has been observed that the (unique)
			stationary distribution of the PASEP has remarkable connections to combinatorics. We will describe
			how in fact the (normalized) probability of being in a particular state of the PASEP can be viewed as
			a certain weight generating function for the permutation tableaux (certain tableaux that come
			indirectly from the totally non-negative part of the Grassmannian) of a fixed shape.  Our first proof is
			algebraic and uses the matrix ansatz of Derrida et al. Our second proof involves defining a Markov
			chain -- which we call the PT chain -- on the set of permutation tableaux which projects to the PASEP
			in a very strong sense, thus revealing a hidden structure behind the PASEP. Via the bijection from
			permutation tableaux to permutations, the PT chain can also be viewed as a Markov chain on the
			symmetric group. Another nice feature of the PT chain is that it possesses a certain symmetry which
			extends the particle-hole symmetry of the PASEP.  This is joint work with Sylvie Corteel.
	19-Apr	Sandrine Peche	The spectral radius of large real symmetric random matrices
		(UC Davis)	with non symmetrically distributed entries
			We consider random symmetric matrices H_N, of size NXN, with independent
			entries above the diagonal. Consider $H_N=\frac{1}{\sqrt N}(Hij)$ where the H_{ij}
			are bounded random variables which are centered with variance $\sigma^2$
			but non symmetrically distributed. It is known that the spectral measure of such
			random matrix ensembles converges as $N \to \infty$ to the semi-circle law. Using
			the moment methods developped by A. Soshnikov, we prove that the spectral
			radius is bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon})$,
			where $\epsilon $ is an arbitrary small positive number.
	26-Apr