22.7 - 26.7.2024

**Organized by:** C. Hainzl, B. Schlein, R. Seiringer

• Speaker: Serena Cenatiempo (GSSI, L'Aquila)

Title: The interacting Bose gas: a mathematical challenge

Abstract: Interacting bosons are unique quantum systems whose low-temperature phases exhibit fascinating quantum mechanical effects at a macroscopic scale. Over the past two decades, the mathematical understanding of these systems has improved tremendously. In these lectures, we will review some of these advances, with a perspective on the largely open challenge of understanding their general behavior in the thermodynamic limit, the appropriate large-scale framework for investigating the occurrence of phase transitions.

• Speaker: Christophe Garban (University of Lyon 1)

Title: Phase transitions in continuous spin systems

Abstract: The goal of this mini-course will be to give an introduction to spin systems defined on a lattice Z^d with an underlying continuous symmetry. This includes the following examples:

- The Gaussian Free Field whose 'spins' are R valued

- The Villain and XY models whose spins are S^1 valued

- The classical Heisenberg model with values in S^2

- Lattice gauge theory with continuous gauge group G

Some of the classical techniques which are very powerful when dealing with a discrete symmetry spin system (for example the so-called Peierls argument for the Ising model) do not apply for continuous symmetry spin systems. This mini-course will explain how such continuous symmetries affect the fluctuations in the system and will introduce some of the main relevant techniques, among which:

- Mermin-Wagner theorem (on the absence of long-range order in 2d)

- Reflection positivity

- Group synchronization on graphs and Nishimori line

• Speaker: Phan Thanh Nam (LMU Munich)

Title: Bosonization and correlation of the electron gas

Abstract: In a series of seminal papers from the 1950s, Bohm and Pines proposed the random phase approximation as an effective description for the correlation of electrons. In this theory, the leading-order behavior of the correlation structure of fermions arises from a quasi-bosonic Bogoliubov theory, where each virtual boson corresponds to a suitable pair of fermions. I will discuss recent joint work with Martin Ravn Christiansen and Christian Hainzl concerning a rigorous derivation of the Bohm-Pines approach for the electron gas in the mean-field regime.

• Speaker: Klaus Widmayer (University of Vienna and University of Zurich)

Title: An introduction to the mathematics of Landau damping

Abstract: While "Landau damping" is regarded as an important effect in the dynamics of hot, collisionless plasmas, its mathematical understanding is still in its infancy. In particular, the terminology has evolved to include several types of stabilizing effects, the mathematical description of which can differ markedly between various settings of relevance.

The goal of this minicourse is to present an introduction to this realm of ideas, without being too technical or requiring deep prior knowledge in the analysis of PDE. In particular, we will focus first on linearized dynamics for some of the basic, illustrative examples, and describe nonlinear results only later and in as much detail as time permits.