Analysis Seminar

Nematic liquid crystal phase in a system of interacting dimers

Ian Jauslin
Member, School of Mathematics
October 25, 2017
In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.

On the number of nodal domains of toral eigenfunctions

Igor Wigman
King's College, London
April 19, 2016
We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions. This work is joint with Jerry Buckley.

Spectral gaps via additive combinatorics

Semyon Dyatlov
Massachusetts Institute of Technology
April 19, 2016
A spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a finite dimensional space; in a related case of Pollicott--Ruelle resonances, a spectral gap gives an exponential remainder in the prime geodesic theorem. We study spectralgaps in the classical setting of convex co-compact hyperbolic surfaces, where the trapped trajectories form a fractal set of dimension $2\delta + 1$.

Quantum Yang-Mills theory in two dimensions: exact versus perturbative

Timothy Nguyen
Michigan State University
April 6, 2016
The conventional perturbative approach and the nonperturbative lattice approach are the two standard yet very distinct formulations of quantum gauge theories. Since in dimension two Yang-Mills theory has a rigorous continuum limit of the lattice formulation, it makes sense to ask whether the two approaches are consistent (i.e., do perturbative computations yield asymptotic expansions for the nonperturbative ones?).

Local eigenvalue statistics for random regular graphs

Roland Bauerschmidt
Harvard University
March 16, 2016
I will discuss results on local eigenvalue statistics for uniform random regular graphs. For graphs whose degrees grow slowly with the number of vertices, we prove that the local semicircle law holds at the optimal scale, and that the bulk eigenvalue statistics (gap statistics and averaged energy correlation functions) are given by those of the GOE. This is joint work with J. Huang, A. Knowles, and H.-T. Yau.

Topology of the set of singularities of viscosity solutions of the Hamilton-Jacobi equation

Albert Fathi
École normale supérieure de Lyon
March 15, 2016
We will mainly report on the progress done recently the connectedness properties of the set of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation. To make the lecture accessible to people with no previous knowledge in the subject, after stating the main result in its full generality, we will restrict to distance functions to closed subsets of Euclidean space, which contain all the relevant aspects of the problem.

The hidden landscape of localization of eigenfunctions

Svitlana Mayboroda
University of Minnesota
March 8, 2016
Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

Supersymmetric approach to random band matrices

Tatyana Shcherbyna
Member, School of Mathematics
March 2, 2016
Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition even in 1d. In this talk we will discuss an application of the supersymmetric method (SUSY) to the analysis of the bulk local regime of some specific types of RBM.