# Mathematical physics

The mathematical physics deals with mathematical problems that their motivation or their application in the (theoretical) physics have. On the one hand, the rigorous mathematical formulation of physical theories and the analysis of the underlying mathematical structures, and on the other hand the application of mathematical solution methods and strategies to physical problems are of particular importance. Furthermore, ideas from (mostly theoretical) physics are taken up in the context of mathematical physics, which then serve as motivation for creating new mathematical concepts. Because of this nature, mathematical physics can be viewed as both a branch of mathematics and physics.

## Questions in mathematical physics

Mathematical physics deals with the mathematically rigorous treatment of models of physical phenomena. The transitions to theoretical physics are fluid. Important sub-areas of mathematical physics are:

### Classic mechanics

In classical mechanics, methods of differential geometry and the theory of Lie groups are mainly used. Specifically, the phase space of a physical system is modeled by a symplectic or a Poisson manifold , on which a Lie group may act. For example, the effects of symmetries and constraints can be studied in detail. Another field of research is the stability theory of dynamic systems , such as our solar system.

### Classical field theories

In order to understand the various classical field theories such as electro- and hydrodynamics or classical Yang-Mills theories , a broad spectrum of mathematical fundamentals is required , especially from the theory of partial differential equations , the calculus of variations , distribution theory , Fourier analysis and the main fiber bundle . Some of the most important unsolved questions in mathematics come from this area: The question of the existence and regularity of solutions to the Navier-Stokes equations, for example, is one of the seven Millennium Problems of the Clay Mathematics Institute .

### general theory of relativity

The general theory of relativity is based on pseudo-Riemannian geometry . In addition to the solution theory of Einstein's field equations , differential topological methods and singularity theorems from mathematics are used to obtain statements about the global topology of the universe or black holes.

### Quantum physics

Quantum physics allows the description of nature on atomic scales. Their mathematical formulation uses, among other things, the spectral theory of unrestricted operators on Hilbert spaces , in particular of Schrödinger operators . Furthermore, C * algebras and the representation theory of Lie groups and Lie algebras are of central importance. Mathematical physics continues to deal with the mathematically rigorous formulation of axiomatic quantum field theories , such as algebraic quantum field theory or constructive quantum field theory , as well as the analysis of various quantization methods , for example in relation to classical limits or well-definedness.

One of the Millennium Problems comes from this area too , namely the proof of the existence of a mass gap in the quantized versions of the Yang-Mills equations.

### Statistical Physics

Systems of many interacting particles are described by statistical physics. Central questions are existence and properties of phase transitions , symmetry breaking and, for systems with finite particle numbers, a thermodynamic limit . Some of the areas of mathematics that are relevant here are the theory of stochastic processes or random matrices .

### Approaches to new physical theories

Mathematical physics is not only concerned with the mathematical investigation of existing physical models. Rather, the search for new theories - for example a quantum physical description of gravitation - is an important area of ​​work, since both physical knowledge and mathematical methods are required here. Some prominent approaches here are string theory , loop quantum gravity , non-commutative geometry or topological quantum field theory .

## Some research areas and associations

The international organization for mathematical physics is the International Association of Mathematical Physics (IAMP), which organizes international congresses every three years.

In Germany, the Max Planck Institute for Mathematics in the Natural Sciences in Leipzig is dedicated to some aspects of mathematical physics. In Vienna there is the Erwin Schrödinger Institute for Mathematical Physics on the initiative of Walter Thirring , who set up a strong school of mathematical physics in Vienna. In Paris, the Henri Poincaré Institute traditionally focuses on mathematical physics.

The German DMV specialist group "Mathematical Physics" states that it aims to be open to all mathematicians who are interested in the mathematical treatment of physically motivated questions. It promotes contact between mathematical physicists in Germany (conferences, specialist literature, mailing list).

There are similar mathematical specialist groups in other countries or in the context of physics. The DMV - German Physical Society cooperation is to be deepened in order not to come into competition between the two subject areas.

The Dannie Heineman Prize and the Henri Poincaré Prize of the IAMP are awarded especially for achievements in mathematical physics .

A list of open problems was published by Barry Simon in 1984 and 2000 ( Simon Problems ).

## Education

A degree in mathematical physics is currently possible in Germany (as of 2014) at the following universities:

These courses of study include courses from theoretical physics as well as mathematics and are therefore usually offered in cooperation with the faculties of physics and mathematics. The focus here is on the interplay between the two disciplines, although the thematic focus, especially in the master’s degree programs, can vary from university to university. Graduates of these degree programs should ideally have acquired the ability to effectively apply modern mathematical concepts and structures to problems in theoretical physics, which in practice enables an introduction to current research topics in both areas.

A distinction must be made between the actual mathematical physics and the lectures and courses on mathematical methods of physics offered at many universities , which are intended to teach physicists the necessary basic mathematical knowledge. The focus is on the broadest possible and application-related presentation of mathematical methods, specially tailored to the needs of physics, and less on proof techniques or proof of mathematical theorems as is the case in the pure mathematics lectures of the physics course. Important focal points are the subjects of vector spaces and vector algebra , simple tensor calculus ( linear algebra ), vector analysis and potential theory , function theory ( residual theorem ), special functions ( spherical functions , Legendre polynomials , etc.), ordinary and partial differential equations , Fourier analysis and probability theory including stochastic processes .