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
06.11.2014, 16:15 Uhr
- Linlin Sun (MPI MIS, Leipzig):
- Dirac-harmonic maps and their heat flows
- A 01 (Sophus-Lie-SR)
- Abstract: The Dirac-harmonic map problem combines a second order harmonic map type system with a first order Dirac equation. With the aim of establishing a heat flow approach to the existence of Dirac-harmonic maps on spin manifolds with nonempty boundaries, we develop the existence, uniqueness and regularity for Dirac equations under a class of local elliptic boundary conditions. Finally, we get the local existence of Dirac-harmonic flow. Dirac-harmonic maps between Riemann surfaces will be considered. The existence, criterion of uncoupled Dirac-harmonic maps will be stated and a structure theorem of Dirac-harmonic map with suitable boundary condition will be given. It is a cooperation with CHEN Qun, Juergen Jost and ZHU Miaomiao.