# VARIOGEO

The project "**VARIOGEO**" is concerned with "**The geometric calculus of variations and its applications**" in a wide range of fields. It will start with fundamental examples of variational problems from geometry and physics, the Bernstein problem for minimal submanifolds of Euclidean spaces, nonabelian Hodge theory as a harmonic map approach to representations of Kähler groups, and Dirac harmonic maps as a mathematical version of the nonlinear supersymmetric sigma model of quantum field theory. These examples will motivate a general regularity and rigidity theory in geometric analysis that will be based in a fundamental way on convexity properties. Convexity will then be linked to concepts of non-positive curvature in geometry, and it should lead to a general theory of duality relations and convexity. That theory will encompass the formal structures of the new calculus of variations and statistical mechanics, information theory and statistics, and mathematical population genetics in biology. Also, the connection with symmetry principles as arising in high energy theoretical physics will be systematically explored.

The mathematical theories can also be applied to material science (nonlinear elasticity), the theory of cognition (invariant pattern recognition) and implementation in neural networks, efficient representation of networks and other structured data, and bioinformatics (population based concepts for DNA sequence comparison).

VARIOGEO is supported by the ERC Advanced Investigator Grant ERC-2010-AdG_20100224, Grant Agreement Number 267087.

## Next Working Seminars

**16.04.2015, 16:15 Uhr**

- Miaomiao Zhu
*(MPI MIS, Leipzig):* - The heat flow for Dirac-harmonic maps
- A 01 (Sophus-Lie-SR)

**23.04.2015, 16:15 Uhr**

- Tat Dat Tran
*(MPI MIS, Leipzig):* **The free energy method in population genetics**- A 01 (Sophus-Lie-SR)
**Abstract:**In this talk, I shall systematically construct free energy functionals for the Kolmogorov forward equations (known by physicists as Fokker-Planck equations) which derived from the Wright-Fisher models with general mutations and selection. I shall then use them to construct a necessary and sufficient condition for the Wright-Fisher diffusion processes to have a unique stationary reversible probability measure. When this condition is satisfied, I show that the flow of probability measures (densities) exponentially converges to the stationary reversible one under various notions of distance (total variation, entropy, L1, etc.) by using techniques in curvature-dimension method of Bakry.