# Mathematical challenges of materials science and condensed matter physics:

From quantum mechanics through statistical mechanics to nonlinear pde

Hausdorff Trimester Program

May 2 - August 31, 2012

**Organizers:** Sergio Conti, Richard James, Stephan Luckhaus, Stefan Müller, Manfred Salmhofer, Benjamin Schlein

The last ten to fifteen years have seen a fascinating growth of possibilities to probe matter on the scale of single atoms, both experimentally and by numerical simulation. This has raised hopes for the systematic bottom-up design of materials with largely improved or completely new properties. Material behavior is, however, by no means determined by the properties at the atomic scale alone. Indeed, it is often the collective behavior of a huge amount of atoms, e.g. in the motion and pinning of dislocations, which crucially determines key material features like brittleness, formabillity and strength. Between the atomic scale and the macroscopic scale there are a large number of scales on which important features including lattice imperfections, precipitates, grains and phase domains arise. An at least equally challenging separation of scales arises in the time domain, from the femtosecond scale of elementary processes to life time analysis of engineering devices on the scale of decades. Thus so called multiscale modelling has become a very common term in materials science and engineering. To go beyond an ad-hoc coupling of phenomenological models is, however, a major unsolved problem and poses fundamental questions for mathematics and theoretical physics.

The aim of this Hausdorff Trimester Program was to bring together researchers with a broad mathematical and scientific background to identify prototype problems where mathematics can provide new insight into the passage between different scales of description.

The focus was on the following three subthemes (ordered roughly by increasing length scale):

- Quantum many-body systems and effective models
- Statistical mechanics of solids and metastability
- Multiscales at the continuum level and material properties