Conformal Field Theory, Solitons, and Elliptic Calogero-Sutherland Models.

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

The nonlinear Schrödinger equation on the half-line with homogeneous Robin boundary conditions.

We consider the nonlinear Schrödinger equation on the half-line with a Robin boundary condition at and with initial data in the weighted Sobolev space . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution.

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib-Borodin Ensembles.

We consider the limiting process that arises at the hard edge of Muttalib-Borodin ensembles. This point process depends on and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles.

A new approach to integrable evolution equations on the circle.

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

A Periodic Hexagon Tiling Model and Non-Hermitian Orthogonal Polynomials.

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there.