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
12.12.2013, 16:15 Uhr
- Jürgen Tolksdorf (MPI MIS, Leipzig):
- Dirac operators with torsion - Part 1: The geometrical set up and the action functional
- Abstract: In this talk I'll introduce a specific class of Dirac operators and discuss the corresponding geometrical background. This class of Dirac operators canonically generalizes twisted spin-Dirac operators over spin manifolds. This class allows to describe various functionals, like the Yang-Mills action and the functional of Dirac harmonic maps, from a unified geometrical perspective. In the talk, however, I'll put emphasize on the case of Dirac operators with torsion. It will be shown that torsion can be dynamically described rather similar to Yang-Mills gauge fields.
19.12.2013, 16:15 Uhr
- Jürgen Tolksdorf:
- Dirac operators with torsion - Part 2: Torsion, gravitino and the N =1 super-symmetric action seen from a classical geometrical perspective.
- Abstract: In this talk, I'll focus on Dirac operators with torsion in the case of Riemann surfaces. The relation between torsion and what is called ``gravitino'' will be discussed in some detail. I'll briefly discuss how the action functional of N = 1 super symmetry over Riemann surfaces is related to Dirac operators with torsion.