# 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

**26.11.2015, 16:15 Uhr**

- Mircea Petrache
*(MPI MIS, Leipzig):* **Smirnov theorem and partial regularity with integer charges**- A 01 (Sophus-Lie-SR)
**Abstract:**I will present two applications of Smirnov's decomposition theorem for 1-currents to1) a minimization problem for 3D vector fields with integer charges

2) the regularity theory of Yang-Mills fields on bundles featuring topological defects.

In both cases the decomposition theorem in combination with

a) suitable approximation theorems

b) the integrality of the underlying degrees

allows to translate minimization problems at the continuum into discrete combinatorial problems.

Both settings 1), 2) are super-critical but the mixture of the above ingredients allows to surpass the difficulty that an epsilon-regularity result is not available.

**01.12.2015, 14:00 Uhr**

- Martin Kell
*(Universität Tübingen, Germany):* **On Cheeger and Sobolev differentials in metric measure spaces**- A 02 (Leon-Lichtenstein-SR)
**Abstract:**Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. He shows that the relaxed notion of gradient is sufficient to obtain 1-forms and make it possible to define Sobolev differentials which resemble the ones in the smooth setting. In this talk I will show that his theory fits nicely into the theory of Lipschitz differentiable spaces initiated by Cheeger, Keith and others. For this I present a new relaxation procedure for L^{p}-valued subadditive functionals and give a relationship between the module generated by a functional and the module generated by its relaxation. In the framework of Lipschitz differentiable spaces, which include so called PI- and RCD(K,N)-spaces, the Lipschitz module is pointwise finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of such spaces are reflexive.

**10.12.2015, 16:15 Uhr**

- Frank Bauer
*(MPI MIS, Leipzig):* **Gradient estimates on graphs**- A 01 (Sophus-Lie-SR)
**Abstract:**I will present gradient estimates on graphs satisfying a new curvature dimension inequality. Using these gradient estimates I will show how one can derive parabolic Harnack inequalities and Buser type eigenvalue estimates. If time permits I also will discuss a recently obtained Davies-Gaffney-Grigoryan Lemma on graphs that can be used to derive heat kernel estimates for negatively curved graphs. This is joined work with Paul Horn, Bobo Hua, Yong Lin, Gabor Lippner, Dan Mangoubi, and Shing-Tung Yau.