In this lecture, analytical tools for quantum many-body physics will be theoretically discussed and practically applied. We will cover many-body perturbation theory both in thermal equilibrium and out of equilibrium, as well as methods for the exact solution of correlated low-dimensional systems, such as bosonization. The participants will be guided by accompanying exercises towards independently solving interesting quantum many-body problems using the aforementioned methods.

N-body problem; Hamiltonian systems, invariant objects (fixed points, periodic trajectories, invariant tori, and stable and unstable manifolds), nonlinear resonances, bifurcations, Arnold diffusion. Examples: coupled maps, Fermi-Pasta-Ulam-Tsingou problem, many-body systems with classical limit

1. Introduction 2. Quantum states, geometry, entanglement, correlations, non-locality 3. Quantum communication 4. Quantum computation 5. Quantum noise 6. Information, entropy and capacity

This class is divided in three parts and will cover hydrodynamics of passive fluids, hydrodynamics of active fluids, and biological pattern formation processes. We will review basics of continuum mechanics and equilibrium and non-equilibrium thermodynamics, linear response theory, Onsager Relations, and the derivation of hydrodynamic equations both from microscopic rules and from a phenomenological approach using broken symmetries and conservation laws. Topics for passive fluids include thin film fluids and the statics and dynamics of nematic liquid crystals. Topics for active fluids include active nematic or polar fluids and biological fluids. Moving beyond linear response we will discuss spatial chemical systems, bifurcation theory, and biological pattern formation. The class will end on coupled chemical and hydrodynamic pattern formation processes relevant for morphogenesis.

• Introduction to the path integral formalism • Bose-Hubbard model and the superfluid-Mott insulator transition • Quantum O(N) non-linear sigma model (NLSM) • Hubbard model • Sachdev-Ye-Kitaev (SYK) model

random variables, some distributions, statistical inference; law of large numbers, central limit theorem; Gaussian stochastic processes in discrete and continuous time; Brownian motion, Ornstein-Uhlenbeck-process; Markov processes, Langevin equation, Fokker Planck equations; Ito versus Stratonowich stochastic integrals

The aim of this lecture is to give an insight into the exciting research in the field of magnetism and magnetic materials on the nanoscale. We will start with an introduction in the basics of (ferro)magnetism and magnetic materials with particular focus on magnetic anisotropy, domains and exchange bias and we will give an introduction to magnetic microscopy. Using this knowledge, superparamagnetism and molecular magnets will be discussed. In addition, we will cover aspects of spin transport phenomena such as giant and tunneling magneto resistance, including e.g.a discussion on spin transfer torque.