# 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

**21.08.2014, 16:15 Uhr**

- Felix Finster
*(Universität Regensburg):* **Topological fermion systems: Geometry and analysis on quantum spaces**- A 02 (Leon-Lichtenstein-SR)
**Abstract:**Topological fermion systems provide a general framework for desribing and analyzing non-smooth geometries. They can also be used to describe "quantum spaces" or "quantum space-times" as considered in quantum gravity. Moreover, they set the stage for the so-called fermionic projector formulation of relativistic quantum field theory. The aim of the talk is to give a simple introduction, with an emphasis on conceptual issues. Starting from a collection of functions on $\R^3$ (which can be thought of as Schrödinger wave functions), we ask the question whether the geometry of the Euclidean space is encoded in these functions. Bringing this question into a precise mathematical form leads us to the abstract definition of topological fermion systems. This definition will be illustrated by the examples of vector fields on the sphere, a vector bundle over a manifold, and a lattice system. As an example motivated from physics, we briefly consider Dirac spinors on a globally hyperbolic Lorentzian manifold and introduce the setting of causal fermion systems. The inherent geometric and analytic structures on a topological fermion system are introduced and explained. A brief outlook on the applications to quantum field theory is given.

**16.10.2014, 16:15 Uhr**

- José M. Gracia Bondía
*(Universidad de Zaragoza, Spain, & Universidad de Costa Rica):* **The Klyachko paradigm in theoretical chemistry**- A 02 (Leon-Lichtenstein-SR)
**Abstract:**It is well known that the problem of determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a twelve-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is: the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory), and remains without satisfactory solution. The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them, and historically the first, is density functional theory, based on a three-variable object for which N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Ensemble representability for 1-RDMs was solved 50 years ago. However, only recently, thanks to outstanding work by Klyachko on generalized Pauli constraints, real progress has been made on pure representability for 1-RDM. These constraints determine small polytopes of admissible pure N-representable sets of 1-RDMs. Somewhat mysteriously, physical states seem to cling to the boundary of the polytopes. We speak of pinning when there are saturated constraints, implying strong selection rules which drastically simplify the configurations. Quasi-pinning appears to be ubiquitous. We review recent numerical evidence and theoretical justification for this phenomenon.