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
11.02.2016, 16:15 Uhr
- Felix Finster (Universität Regensburg):
- Causal variational principles and conservation laws
- A3 01 (Sophus-Lie-SR)
- Abstract: The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for desribing and analyzing non-smooth geometries. The dynamics is described by so-called causal variational principles.
The talk focuses on causal variational principles from the perspective of analysis and the calculus of variations. After a simple introduction to causal variational principles, I shall explain how Noether-like theorems can be formulated in this setting. The basic method is to work with so-called surface layer integrals, which I will introduce. The simplest version of a Noether-like theorem is proven. At the end of the talk, I will give an outlook on causal fermion systems and explain how our Noether-like theorems are related to the conservation laws for charge and energy-momentum in relativistic quantum theory.
This is joint work with Johannes Kleiner (Regensburg).
18.02.2016, 16:15 Uhr
- Ruijun Wu (MPI MIS, Leipzig):
- Regularity aspects of super nonlinear sigma models
- A3 01 (Sophus-Lie-SR)
- Abstract: We will talk about the full nonlinear sigma model but without supersymmetries involved. Weak solutions of the Euler-Lagrangian equations will be shown to be smooth.