UCF/USF special workshop on Complex Analytic Methods with Applications in Orthogonal Polynomials, Integrable Systems, and Random Matrix Theory

Flyer:

Program:

Saturday, 12:40 pm. Recent developments in spectral theory of soliton gases for integrable equations (Alex Tovbis, UCF).

TBA

Saturday, 1:30 pm. Non-generic focusing of the semiclassical nonlinear Schrödinger equation (Robert Jenkins, UCF).

TBA

Saturday, 2:20 pm. Nonlocal integrable equations and their Riemann-Hilbert Problems (Wen-Xiu Ma, USF).

We will talk about how to construct nonlocal integrable equations through group reductions of matrix spectral problems. The reduced matrix spectral problems are used to formulate a kind of Riemann-Hilbert problems, whose reflectionless cases generate soliton solutions.

Saturday, 3:40 pm. SLE, Or Sturm-Liouville Entropy (Razvan Teodorescu, USF).

Which integral equations of Fredholm type are compatible with the classical Bethe Ansatz? Does two-dimensional integrability giving rise to the “Hofstadter butterfly” imply the existence of a one-parameter group of 1D integrable models? And will we ever run out of new meanings for SLE? These questions and others besides will be discussed in the context of the concept of averaging of solutions for Sturm-Liouville problems on the real line.

Saturday, 4:30 pm. Criticalities in Random Normal Matrices (Seung-Yeop Lee, USF).

While the universalities in criticality are well-studied for General Unitary Ensemble, the analogous study for Normal Matrix almost does not exist. I will show a specific model with a merging singularity where the critical behavior of the kernel could be observed. It showcases the Painleve II as in the merging singularity in One matrix model. This is a joint work with Meng Yang.

Sunday, 9:00 am. Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves (Constance Schober, UCF).

TBA

Sunday, 9:50 am. Non-standard Green energy problems in the complex plane (Abey Lopez-Garcia, UCF).

We consider several non-standard discrete and continuous Green energy problems in the complex plane and study the asymptotic relations between their solutions. In the discrete setting, we consider two problems; one with variable particle positions (within a given compact set) and variable particle masses, the other one with variable masses but prescribed positions. The mass of a particle is allowed to take any value in the range [0,R], where R is a fixed parameter in the problem. The corresponding continuous energy problems are defined on the space of positive measures with mass at most R and supported on the given compact set, with an additional upper constraint that appears as a consequence of the prescribed positions condition. We prove that the equilibrium constant and equilibrium measure vary continuously as functions of the parameter (the latter in the weak-star topology). In the unconstrained energy problem we present a greedy algorithm that converges to the equilibrium constant and equilibrium measure. This is a joint work with Alex Tovbis.

Sunday, 10:40 am. “It is useful to solve extremal problems” –Almost I. Newton (Dmitry Khavinson, USF).

Everyone one knows how useful the Schwarz Lemma is. ( Not really due to H.A. Schwarz.) But what if we replace the origin by interpolating at several points? What if we switch from bounded functions to Hardy or Bergman spaces functions? What if we replace the disk by a general domain, or a Riemann surface? What if instead of analytic functions we investigate the Schwarz lemma versions for harmonic functions in R^n ? Or, for analytic functions, in C^n? What if we move from linear to nonlinear problems, e.g. , Schwarz’ lemma for nonvanishishing functions? Some of these questions have been at least partially solved, having moved analysis in substantial ways forward. And some are waiting for a new trick, new high ground, new minds.

Sunday, 1:00 pm. Arc length null quadrature domains (Erik Lundberg, FAU).

A planar domain is referred to as an arclength null quadrature domain if integrals of analytic functions with respect to arclength along the boundary always vanish. We discuss recent progress on the classification problem for arclength null quadrature domains, and we explain how conformal mappings can be used to construct explicit examples. Time permitting, we will also review some exactly solvable problems in fluid dynamics where various types of quadrature domains appear as solutions.

Sunday, 1:50 pm. Hermite-Pade approximation, equilibrium problems and Riemann surfaces (Evguenii Rakhmanov, USF).

Classical Stahl’s theorem describes convergence of diagonal Pade approximants for functions with branch points. Possible generalizations of this theorem for the case of Hermite-Pade approximations attract common attention but the progress is slow. A few particular results in this direction are obtained by di erent authors but a general case is essentially wide open. It is not even clear how a general conjecture on convergence may be formulated.

We plan to discuss this questions in some details. There are two known approaches to the problem. A conjecture on convergence may be stated in terms of an equilibrium problem for a logarithmic potential. An alternative way is to use so-called G-function on an algebraic Riemann surface associated with the problem at hand. Either way meet certain challenges.

Sunday, 2:40 pm. The Ising Model Coupled to 2D Gravity (Nathan Hayford, USF).

One of the most celebrated exactly solvable models in statistical mechanics is the two-dimensional Ising model. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been described in the literature since the original paper of V. Kazakov in 1986. In this talk, we present a rigorous proof of Kazakov’s results, using steepest descent analysis for biorthogonal polynomials. We are able to calculate the genus 0 partition function, and we also find that the phase transition is described by the string equation of a 3rd order reduction of the KP hierarchy, in agreement with the predictions of G. Moore, M. Douglas, and their collaborators. This is joint work with Maurice Duits and Seung-Yeop Lee.