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Analysis/PDE Reading Seminar

Fall 2017

Date:September 19, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Eviatar Procaccia, TAMU
Title:Introduction to SLE (Schramm-Lowner Evolution)
Abstract:SLE is a family of random curves, which is proved to be the scaling limit of many discrete random interfaces, common in statistical physics. The study of SLE gave birth to beautiful techniques such as Smirnov's discrete holomorphic functions, and study of fractal dimension of random curves. In this first talk I will introduce the subject of SLE, starting from a background of models in statistical physics (percolation, loop erased radnom walk, etc), and then continuing with the fantastic discoveries of Oded Schramm, Greg Lawler, Wendelin Werner, Steffen Rhode and others.

Date:September 26, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Eviatar Procaccia, TAMU
Title:Introduction to SLE (Schramm-Lowner Evolution) Part 2

Date:October 10, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Isaac Harris, TAMU
Title:Some analytical tools for non-linear eigenvalues applied to inverse scattering.
Abstract:We will discuss some results pertaining to non-linear eigenvalue problems and how they can be used to study the transmission eigenvalue problem that comes from inverse scattering. Transmission eigenvalue problems are a new class of eigenvalue problems that are non-selfadjoint and non-linear (in the proper function spaces) which can not be handled by standard theory (such as the Hilbert–Schmidt theorem).

Date:October 17, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Robert Rahm, TAMU
Title:Asymptotic density of eigenvalues for 1 dimensional Schroedinger operators
Abstract:Consider the Schr\odinger Equation -y''(x) + q(x)y(x) = \lambda y(x) on $[0,\infty)$. We will initially assume only that q is non-negative and increasing and \lim_{x\to\infty}q(x) = \infty but will later put some restrictive assumptions on it. We will discuss the asymptotic density of eigenvalues of the equation. In particular, if N(T) is the number of eigenvalues less than T$ we will show: N(T) = \int_{0}^{q^{-1}(T)}\{T-q(x)\}^{1/2}dx + o(1).

Date:October 31, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Dean Baskin, TAMU
Title:A brief introduction to resonances
Abstract:Spectral theory for the Laplacian on compact manifolds gives you a discrete set of eigenvalues and an orthonormal basis of eigenfunctions. For non-compact problems, there is typically continuous spectrum and only finitely many eigenvalues. Is there anything resembling a discrete family of eigenvalues in this context? In some sense, the answer is yes: these are the resonances. In this talk I will provide a brief introduction to the theory of resonances. I will provide some motivation for their study by working through the case of the one-dimensional wave equation. I will then talk about resonances on Euclidean and hyperbolic spaces, where they can be calculated explicitly. Finally I will provide some discussion of how to define them in the case of potential scattering (and maybe some geometric scattering) via the analytic Fredholm theorem. In a future talk I will use the explicit calculation of the resonances on hyperbolic space to calculate the asymptotic behavior of the wave equation on Minkowski space.

Date:November 7, 2017
Time:4:00pm
Location:BLOC 624
Speaker:Boris Hanin, TAMU
Title:Introduction to the Quantum Hall Effect
Abstract:The Quantum Hall effect concerns the low temperature behavior of many electrons confined to a two dimensional sample, such as the surface of a semi-conductor, and subjected to a strong perpendicular magnetic field. In their seminal experiments in the early 1980’s, Von Klitzing-Dorda-Pepper and Tsui-Stormer-Gossard discovered that the so-called Hall conductance is robustly and precisely quantized. This lead to much to much work so-called topological phases of matter and topological insulators. In this talk, I will give a gentle introduction to the QHE, with a focus on the basic examples. I will then briefly describe what is known about the partition function for the simplest and most important QHE wavefunction introduced by Laughlin. I will end with an open question about partition functions for important generalizations of the Laughlin wavefunctions, called incompressible states boundaryless states. The analysis of these wavefunctions is thought to involve conformal field theory and topological recursion in more than one dimension.