# Lecture Series

## Course on the statistics and dynamics of disordered systems

Date: every Tuesday, 18:00 - 20:00 (if not stated otherwise)

Venue: HIM lecture hall, Poppelsdorfer Allee 45

Tuesday, October 19, 18:00: Nicola Kistler, The random energy model (REM) - statics

Abstract: The random energy model (REM) - statics. The "REM" is a simple model which has played an important role in the understanding of the Parisi Theory for (mean field) Spin Glasses. It is constructed outgoing from a centered Gaussian random field of independent random variables. A crucial question in order to understand the behavior of the REM is the asymptotics of the largest values of the field. I will introduce some rather robust methods in order to answer such a question, which allows us to prove that the Gibbs measure converges (in low temperature) to the Poisson-Dirichlet distribution.

Tuesday, October 26, 18:00: Nicola Kistler, The generalized random energy model (GREM) - statics

Abstract: The generalized random energy model (GREM) - statics. The "GREM" is an extension of the REM where the random field is "hierarchically organized". Still, it is completely solvable. I will present a proof that shows that the associated Gibbs measure still converges in a certain region to the Poisson-Dirichlet distribution. Yet the Gibbs measure of the GREM has a richer (somewhat hidden) structure, which I will also describe. Keywords in this case are the Derrida-Ruelle probability cascades.

Tuesday, November 2, 18:00: Onur Gün, Bouchaud s REM-like trap model

Abstract: Bouchaud s REM-like trap model. I will start with some basic tools needed for the study of the dynamics of spin glasses such as heavy-tailed random variables and their convergence to stable laws, Levy processes, stable subordinators, point processes and Continuous Mapping Theorem. Then I will describe Bouchaud s REM-like trap model which is a simple model introduced by Bouchaud as an ansatz for the dynamics of spin glasses. I will prove the convergence of the clock process and aging for this model on all time scales using a point process approach established by Veronique Gayrard.

Wednesday, November 10, 18:00: Onur Gün, Dynamics of the REM and GREM-like trap model

Abstract: Dynamics of the REM and GREM-like trap model. To give an insight, I will first state some results about the exponential sums of Gaussian random variables. I will prove the convergence of the clock process and aging for all time scales for the dynamics of the REM in the light of the results from the Bouchaud's REM-like trap model, using an extended point process approach. Then I will describe the GREM-like trap model and prove the results we have reached so far for various time scales.

## Hatem Zaag: Blow-up for the semilinear wave equation

A series of three lectures.

Date: November 9 - 11, 2010, 10:00 - 11:30

Venue: Lipschitz-Saal, Endenicher Allee 60

## Thomas Duyckaerts: Dynamics of the energy-critical focusing wave equation

A series of two lectures.

Date: November 18 + 19, 2010, 10:00 - 11:30

Venue: HIM lecture hall, Poppelsdorfer Allee 45

## Course on duality for interacting particle systems

Date: Tuesday, November 23, and Wednesday, November 24, 18:15 - 20:00

Venue: HIM lecture hall, Poppelsdorfer Allee 45

Tuesday, November 23, 18:15: Frank Redig, Old examples of duality from a new perspective

Abstract: Old examples of duality from a new perspective. 1. Context: Interacting particle systems/ semigroup/ generator 2. Duality/self-duality 3. Examples: Wright-Fisher diffusion, independent random walkers, exclusion process, k-exclusion process 4. Duality functions and symmetries of the generator 4.1. Duality functions and different representations 4.2. Self-duality functions and symmetries

Wednesday, November 24, 18:15: Frank Redig, The brownian momentum process, inclusion process and related models

Abstract: The brownian momentum process, inclusion process and related models. 1. Brownian momentum process, brownian energy process, relation with the KMP-model 2. Duality between inclusion process and brownian energy process 3. Self-duality of inclusion process 4. Adding boundary generators: duality for non-equilibrium models 5. (Introduction to) the asymmetric inclusion process