**Work of the Institute**The Max-Planck-Institute for Mathematics in the Sciences, founded in 1995, is concerned with mathematical challenges which arise through theoretical questions in the natural sciences. The common denominator of all teams in the institute is the field of partial differential equations and the calculus of variations. The purpose of the institute is to perform research in areas of pure and applied mathematics and to promote the flow of ideas between mathematics and the natural sciences in both directions. It is a lesson of history that fundamental problems of physics, chemistry, biology and other sciences have led to new important advances in mathematics, while mathematics has in turn had a deep influence in these sciences. For example, Fourier’s research into the diffusion equation led to the development of Fourier series and more generally to the founding of harmonic analysis. Gauss, one of the greatest mathematicians ever, was inspired by his practical work as a surveyor to develop his theory of surfaces, which led to the creation of differential geometry, and the basis for both Einstein’s general theory of relativity and the standard model in elementary particle physics. Heisenberg’s formulation of quantum mechanics accelerated the development of functional analysis, especially the spectral theory of operators. Important progress in the general theory of non-linear dynamical systems in the context of Kolmogorov-Arnold-Moser theory (KAM theory) was initiated by questions in celestial mechanics. The modern theory of concentrated compactness has its origin in problems of atomic bonding. Finally, the standard model in elementary particle physics is formulated as a gauge theory, which is based on a deep synthesis of physics, analysis, geometry, and topology. The key areas of research of the Max Planck Institute for Mathematics in the Sciences are analysis, geometry, mathematical physics, and scientific computing. A central theme in all these areas is the theory of non-linear partial differential equations. The main aspects of work are:

- Riemannian, Kählerian and algebraic geometry and its interaction with modern theo-retical physics (harmonic maps, minimal surfaces, Ricci flow, mean curvature flow, Yang-Mills and Seiberg-Witten equations)
- Mathematical models in materials science (microstructures, micromagnetism, homogenization, phase transitions, fault phenomena, boundary surfaces and thin films)
- Mathematical models in biology
- Continuum mechanics (elasticity, hydrodynamics, dynamics of gases)
- many-particle systems in statistical physics and neural networks
- The theory of general relativity and quantum field theory
- The numerical solution of partial differential equations and integral equations (multi-grid methods and boundary element methods).

Most of these mathematical models lead quite naturally to partial differential equa-tions with strong non-linearities, whose solutions have singularities, complicated oscillations or concentration effects. In the real world, these effects can be seen in shock waves, turbulence, material defects or microstructures, which can be observed under the microscope. Very ingenious model building is necessary for a better understanding of these phenomena, as well as their analytical investigation, in order to find the essential mathematical objects. The concurrent development of mathematics and the natural sciences leads to a wide spectrum of topics, which include areas with a traditionally strong interaction with mathematics such as statistical mechanics, elementary particle physics, cosmology, celestial mechanics or continuum mechanics, as well as areas for which a mathematical foundation has yet to be established. There are few long-range visitors at the Institute. As in several other international research institutions, the main emphasis is on visiting mathematicians from all over the world, who may visit the institute for a maximum of two years to work in fluctuating teams on special topics. For more information on applying for a visiting position, see the institute’s website. The Institute also offers a special Sophus-Lie Visiting Professorship, which will be awarded to leading scientists on a regular basis. At the end of 1998 there were 37 members of staff working at the institute, including 26 scientists. In 1998, the Institute also hosted 150 short- and medium-term visitors.

**Max-Planck-Institut für Mathematik in den Naturwissenschaften**

Inselstr. 22-26

04103 Leipzig

Tel: +49-341-9959-50

Fax: +49-341-9959-658

E-Mail: rluebke@mis.mpg.de

Further information: http://www.mis.mpg.de