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.