Geometry and Topology

In differential geometry we deal with metric, topological and integral geometry questions. In particular, we are interested in universal inequalities on closed Riemannian or Finsler manifolds involving metric and topological invariants, in the interaction with geometric group theory, in valuation theory, or in geometric structures on low dimensional manifolds. In symplectic geometry, we work with symplectic manifolds admitting singularities, including normal forms and quantization. A highlight of the group is the construction of an abstract fluid computer using techniques from contact geometry, symbolic dynamics and Turing completeness. Using topological quantum field theory we aim to abstractly construct a hybrid machine combining the fluid computer with a quantum computer. In algebraic topology the techniques of homotopy theory and infinity categories have infiltrated other areas, with impact in classification problems in algebraic geometry and representation theory. We also consider the interactions of algebraic topology with algebraic and complex geometry: topology of algebraic varieties and complex and almost manifolds, mixed Hodge theory, formality, and multiplicative structures in homotopical algebra and cohomological operations.