Analysis Seminar

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.

Global existence and convergence of solutions to gradient systems and applications to Yang-Mills flow

Paul Feehan
Rutgers University; Member, School of Mathematics
February 24, 2016
We discuss our results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional over a closed, four-dimensional, Riemannian manifolds: If the initial connection is close enough to a minimum of the Yang-Mills energy functional, in a norm or energy sense, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection.

Stochastic quantization equations

Hao Shen
Columbia University
February 23, 2016
Stochastic quantization equations are evolutionary PDEs driven by space-time white noises. They are proposed by physicists in the 80s as the natural dynamics associated to the (Euclidean) quantum field theories. We will discuss the recent progress on well-posedness problems of such equations. The examples include the Phi4 model, the sine-Gordon model, and perhaps models with gauge symmetries if time allowed.

Calibrations of Degree Two and Regularity Issues

Constante Bellettini
Princeton University; Member, School of Mathematics
April 9, 2013

Calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of calibrated currents. We will review some of these issues, then focusing on the two-dimensional case where we will show a surprising connection with pseudo-holomorphic curves and an infinitesimal regularity result, namely the uniqueness of tangent cones

Resonances for Normally Hyperbolic Trapped Sets

Semyon Dyatlov
University of California
April 2, 2013

Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.