# An Outline of the Institute's Activities

Our institute is one of two mathematical oriented institutes in the Max Planck Society. We have approx. 130 scientists and PhD students which are working on a wide area of problems.

It is the institute's mission to do research work in the field of pure and applied mathematics and promote the interlinking of ideas between mathematics and the sciences in both directions. Experience in history shows that the fundamental problems of physics, chemistry, biology and other sciences have led to important new developments in mathematics while mathematics has had a profound impact on these fields of knowledge. For instance, Fourier's studies of the thermal conduction equations led to the development of the theory of Fourier series and in general to the creation of harmonic analysis. Beyond this, his practical work as a surveyor inspired Gauss, one of the greatest mathematicians of all times, to develop his theory of surfaces and differential geometry. That, in turn, forms the basis for Einstein's general relativity theory and the standard model in elementary particle physics today. Heisenberg's formulation of quantum mechanics also accelerated the development of functional analysis, especially the spectral theory for operators. Finally, the standard model of elementary particles is formulated in the setting of gauge field theories that are based upon a profound synthesis of physics, geometry (topology) and analysis.

The main fields of mathematical research at the Max Planck Institute for Mathematics in the Sciences are analysis, geometry, mathematical physics and scientific computing. A key research topic is the theory of non-linear partial differential equations. Its special focus encompasses

- Riemannian, Kählerian and algebraic geometry including their interrelation with modern theoretical physics
- mathematical models in material sciences (microstructures, micromagnetism, homogenisation, phase transitions, refraction phenomena, interfaces and thin films)
- continuum mechanics (the theory of elasticity and hydro- and gas dynamics)
- many-particle systems in statistical physics and neural networks
- general relativity theory and quantum field theory
- problems of mathematical biology
- scientific computing

Most of these mathematical models typically lead to partial differential equations with strong non-linearities whose solutions have singularities or describe complicated oscillation and concentration effects. In practice, these mathematical effects correspond to shock waves, turbulence, material defects or microstructures that are seen under the microscope. In order to understand these phenomena, it is necessary to develop analytical tools which allow one to identify the significant mathematical objects. Co-operation between mathematics and modern sciences includes a wide range of topics incorporating both areas with a strong interaction with mathematics such as statistical physics, elementary particle physics, cosmology, celestial mechanics or continuum mechanics, and fields of research that are only at the beginning of their mathematisation such as a myriad of questions of material sciences or biology. The institute has only a few long-term positions for academics. As in other international mathematical research institutes, it has a predominant programme of guests that is open to mathematicians from all countries of the world. Guests may come to the institute for a maximum of two years to work on a variety of research areas. Leading scientists are invited at regular intervals to occupy an outstanding position at the institute as the Sophus-Lie guest research professor.