Analysis, Geometric Structures and Mathematical Physics

In this key research area, we further develop the interrelated methods of algebraic and enumerative geometry, geometric representation theory, differential geometry, functional analysis, the geometric calculus of variations, complex analysis, low dimensional topology, spectral theory, and the theories of non-linear partial differential equations and dynamical systems. We apply these methods to problems ranging from fluid mechanics and geophysics to gravitational physics, quantum physics, and string theory.

On the algebraic side, our work focusses on higher-dimensional algebraic geometry and its relationships with combinatorics, singularity theory, and representation theory. For example, we find new ways of constructing fibered Calabi-Yau manifolds, which play a key role in string theory, or give algebraic descriptions of moduli spaces, which are fundamental objects in field theories. We use methods of enumerative combinatorics to establish connections between algebraic invariants that arise in seemingly unrelated ways in combinatorics, low-dimensional topology, algebraic geometry, or quantum field theory. 

The focus of our work in complex analysis are the properties of spaces of holomorphic functions as well as CR geometries. We relate properties of solutions of partial differential equations such as the CR equations to invariant geometric properties in order to study regularity and uniqueness of their solutions, using tools from several complex variables, functional analysis, and algebraic geometry. 

We investigate geometric structures with special focus on parabolic geometries, which include conformal structures, hypersurface-type CR structures, and several types of generic distributions. Tools from differential geometry and the representation theory of semi-simple Lie algebras are applied to examine the properties of these structures and to construct differential operators and differential complexes that are intrinsic to them. Applications range from general relativity, complex analysis, and the geometric theory of differential equations to numerical analysis. In mathematical physics, we investigate the asymptotic behavior of solutions to certain non-linear partial differential equations that arise, for example, in Yang-Mills theory, continuum mechanics, kinetic theory, and quantum mechanics. A special focus lies on the study of dispersive effects, both in the context of singularity formation and in the question of stability of stationary configurations. This leads to natural connections with spectral theory, complex analysis, non-linear functional analysis, and harmonic analysis. We also apply non-linear partial differential equations and dynamical systems to model wave phenomena in the oceans or the atmosphere, and study these models using new analytical insights.

We study the geometry of space-times using both analytic and synthetic approaches. Invariants in general relativity are investigated on the level of initial data for the Einstein equations using methods of the geometric calculus of variations, making connections with classical results and problems in differential geometry.

In the area of low-dimensional topology, our research focusses on the interplay between contact topology, quantum field theory, and algebraic invariants arising in Heegard-Floer homology.