Mathematical Physics Seminar

We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogs of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system.

Date	Speaker	Title
Feb. 14	Percy Deift, Courant Institute and IAS	The Riemann-Hilbert Problem: applications The speaker will describe the application of Riemann-Hilbert techniques to a variety of problems in mathematics and mathematical physics. The nonlinear steepest descent method plays a key role. Areas of applications include: random matrix theory and orthogonal polynomials, integrable systems, problems in random permutations, and random particle systems.
Feb. 21	W. M. Wang, CNRS and UMass at Amherst	Stability of the quantum harmonic oscillator under time quasi-periodic perturbations We prove stability of quantum harmonic oscillator for a large set (of positive measure, asymptotically of measure 1) of perturbation frequencies. From the KAM point of view, this is a border line case. Generally speaking, stability under time quasi-periodic perturbation is a precursor toward stability under nonlinear perturbation.
Feb. 28	Irina Nenciu, Courant Institute and IAS	CMV: the unitary analog of Jacobi matrices We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogs of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system.
March 7	Jean Bourgain, IAS	On localization in lattice Schroedinger operators
March 14	Andrei Okounkov, Princeton University	Counting curves in 3-folds: a case of gauge/string duality in algebraic geometry This will be a nontechnical discussion of a conjecture about algebraic curves in 3-folds proposed by Maulik, Nekrasov, Pandharipande, and myself a few years ago and of the mathematical progress toward it so far.
March 21	Spring Break	TBA
March 28
April 4	Ari Laptev, KTH	Hardy inequalities for many particles We prove some inequalities of Hardy type for many particles. In particular, we show how introducing Aharonov-Bohm magnetic fields could give such inequalities for two-dimensional particles. It turned out that 2D Hardy inequalities hold also for fermions.
April 11	Detlev Buchholz, University of Goettingen	Integrable models and operator algebras Recently, it has been possible to establish rigorously the existence of an abundance of 1+1-dimensional relativistic quantum field theories with factorizing scattering matrices by operator-algebraic means. This novel approach, which is complementary to the advanced methods of constructive quantum field theory, settles some long-standing questions in the context of integrable models (form-factor program) and sheds new light on the problem of constructing quantum field theories. In this talk, a survey is given of the basic ideas, results and perspectives of this approach.
April 18	Y. Peres. University of California, Berkeley	Zeros of the IID gaussian power series: an exactly solvable Coulomb gas in the hyperbolic plane Analogies between zeros of random polynomials and the 2D Coulomb gas were pointed out by Lebouef in the 1990s. We show that in the case of i.i.d. Gaussian coefficients, this analogy can be taken much further, but the process of random zeros only becomes purely fermionic in the limit as the degree tends to infinity. (This contrasts with the work of Jancovici relating the planar Coulomb gas to the Ginibre ensemble.) More precisely, the zero set of a random power series with i.i.d. complex Gaussian coefficients form a determinantal process in the unit disk, governed by the Bergman kernel. We deduce the exact distribution of the number of zeros in a disk of radius r about the origin. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; unlike Dyson's Brownian motions, no drift is involved. (Talk based on joint work with Balint Virag).
April 25	Peter Hislop, University of Kentucky	Spectral Averaging, the Wegner Estimate and the Density of States for Random Schrodinger Operators I will discuss recent work with J. M. Combes and F. Klopp on the Wegner estimate for random Schrodinger operators with general probability measures. The new technical result is a spectral averaging method allowing us to deal with more general classes of probability measures. These results are applied to obtain better continuity estimates on the integrated density of states and the expectation of the spectral shift function.