Schedule September-December 2021

Classical Multiplier Theorems and Recent Improvements

A multiplier operator alters the frequency of input functions/signals via multiplication by a fixed function called a multiplier. Multiplier theorems provide sufficient conditions for multiplier operators to preserve integrability. The classical multiplier theorems of Marcinkiewicz and of Hormander on Euclidean spaces will be reviewed and comparisons between different versions of these results (and examples) will be given. The main focus of the talk is to discuss recent optimal improvements of these theorems in terms of membership of the multipliers in appropriate scales of Sobolev classes.

Gibbs ensemble for the discrete nonlinear Schrodinger equation, circular beta-ensemble and double confluent Heun equation

The discrete defocusing nonlinear Schrodinger equation in its integrable version is called Ablowitz Ladik lattice. We consider the Gibbs ensemble for the Ablowitz Ladik lattice. In this setting the Lax matrix of the Ablowitz Ladik lattice turns into a random matrix that is related to the circular beta-ensemble at high temperature. We obtain the free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. The density of states is obtained via a particular solution of the double-confluent Heun equation.

Joint work with Guido Mazzuca.

Normal Stein spaces with Bergman-Einstein metric and finite ball quotients

In this talk, we will start with a conjecture posed by Cheng, which states that the Bergman metric of a bounded, strongly pseudoconvex domain in Cn with smooth boundary is Khaler-Einstein if and only if the domain is biholomorphic to the unit ball Bn. Then we will discuss the recent developments on solving and generalizing Cheng’s conjecture.

The talk is based on a joint paper with Huang, and a recent preprint with Ebenfelt and Xu.

Weil-Peterson curves, traveling salesman theorems and minimal surfaces

I will describe several new characterizations of Weil-Petersson curves. These curves are the closure of the smooth planar closed curves for the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Their work was motivated by problems in string theory, but the same class arises naturally in geometric function theory, Mumford's work on computer vision, and the theory of Schramm-Loewner evolutions (SLE). The new characterizations include quantities such as Sobolev smoothness, Mobius energy, fixed curves of biLipschitz involutions, Peter Jones's beta-numbers, the thickness of hyperbolic convex hulls, the total curvature of minimal surfaces in hyperbolic space, and the renormalized area of these surfaces. Moreover, these characterizations extend to higher dimensions and remain equivalent there.

On the Lack of Fredholm Solvability for the Lp Dirichlet Problem for Weakly Elliptic Systems in the Upper Half-Space

The $L^p$ Dirichlet Problem for constant coefficient second-order systems satisfying the Legendre-Hadamard strong ellipticity condition is well posed in the upper half-space. Surprisingly, this result may fail if only weak ellipticity is assumed, and the failure manifests itself at a fundamental level through lack of Fredholm solvability. In this talk I will discuss a couple of pathological cases, sought in the class of weakly elliptic systems that fail to possess a distinguished coefficient tensor. This is joint work with Dorina Mitrea and Marius Mitrea.