Interactions between mathematics and physics: symmetries and randomness

08.11.2017, 14:00  –  Haus 9, Raum 2.22
Institutskolloquium

Simone Warzel (TU München), Frank Rupert (LMU München)

Das Kolloquium beginnt um 14:00 Uhr mit einem Vortrag

von Rupert Frank (LMU, München):

"Symmetry and Reflection Positivity"

Abstract:

There are many examples in mathematics, both pure and applied, in which problems with symmetric formulations have non-symmetric solutions. Sometimes this symmetry breaking is total, as in the example of turbulence, but often the symmetry breaking is only partial. One technique that can sometimes be used to constrain the symmetry breaking is reflection positivity. It is a simple and useful concept that will be explained in the talk, together with some examples. One of these concerns the minimum eigenvalues of the Laplace operator on a distorted hexagonal lattice. Another example that we will discuss is a functional inequality due to Onofri.

15:00 Uhr Teepause

Um 15:30 Uhr folgt ein Vortrag

von Simone Warzel (TU, München):

"The Phase Diagram of Random Matrix Models via Dyson's Brownian motion"

Abstract:

Real symmetric random matrices arise in many branches of mathematics. Historically, they first featured as effective descriptions of energy spectra in atomic physics. In this talk, I will illustrate the basic interesting questions in the field using the simple example of the Rosenzweig-Porter (RP) model. The latter interpolates between  the trivial random diagonal and the famous

Gaussian Orthogonal Ensemble and serves as a prototype for the universality class of localisation-delocalization transitions expected in more general disordered quantum systems. The RP model  can be described by a set of  stochastic differential equations known as Dyson's Brownian motion (DBM).

In the second part of the talk I will explain how the inherent energy- and time-scales in DBM manifest themselves in the phase diagram.

A method of characteristics well known from the Burger's equation is a basic ingredient.

(This talk is partially based on a series of joint works with Per von Soosten.)