## Sept. 1st - Sept. 7, 2013: Summerschool at CIRM

Related to the trimester at IHP, there will be a summerschool at CIRM in Marseille, on

*"Current Topics in Mathematical Physics"*

Organizers: Maria J. Esteban (CNRS), Mathieu Lewin (CNRS), Robert Seiringer (Vienna)

Lectures:

Lectures:

Lectures:

#### Eric Carlen (Rutgers, New Brunswick, USA)

*"Mathematical Problems in Kinetic Theory*"

**Abstract:** We give an introduction, starting from the beginning, to propagation of chaos, spectral gaps and entropy production for N-particle master equations and the Boltzmann equation, discussing both recent progress and open problems.

#### Rupert Frank (Caltech, Pasadena, USA)

*"Ground states of non-linear scalar field equations"*

**Abstract:** Ground states of semi-linear PDE appear in a number of physical and mathematical applications including, for example, the blow-up analysis for non-linear Schroedinger equations. In this lecture we present tools from non-linear functional analysis to study the existence, regularity, symmetry and uniqueness of these solutions.

#### Christian Hainzl (Tuebingen, Germany)

*"The BCS-functional of superconductivity (superfluidity)"*

**Abstract:** We start by analyzing the translation invariant BCS functional, and derive the non-linear BCS-gap equation. We show that the existence of a non-trivial solution is equivalent to a linear criterion. Applying methods of spectral theory, such as Birman-Schwinger type techniques, to the corresponding linear operator it is possible to evaluate the critical temperature. Finally, we outline how the Ginzburg-Landau theory can be derived from the BCS functional close to the critical temperature via a semiclassical limit.

#### Jens Marklof (Bristol, UK)

*"From Newton's laws to macroscopic transport equations: The Boltzmann-Grad limit of the Lorentz gas"*

**Abstract:** Since the revolutionary ideas of Boltzmann well over a century ago, the rigorous derivation of kinetic transport equations for a macroscopic particle cloud remains one of the key challenges in mathematical physics. In these lectures I will describe recent progress in the case of the Lorentz gas, a dilute gas of point particles moving in an infinite array of scatterers, and describe the transport equations that emerge in the limit of small scatterer density. The lectures will introduce a non-expert audience to the relevant techniques from dynamical systems, ergodic theory, number theory and mathematical physics.

#### Daniel Ueltschi (Warwick, UK)

"Quantum spin systems and phase transitions"

"Quantum spin systems and phase transitions"

**Abstract:** These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences

The lectures are aimed at PhD students and post-docs and they will cover different subjects in Mathematical Physics. This is the 4th event of this type. There has been two previous summerschools at the ESI in Vienna in 2008 and 2011, and one in Aarhus in 2010.

There is a limited number of participants and a selection will be made. The local expenses of all of them will be covered.

**Application**

The deadline for applying to the summerschool is passed.