Geometry and Topology

In the course of the 20th century there have been remarkable developments in the study of geometry and the topology of manifolds. However, difficult and significant problems still remain, especially in 3 and 4 dimensions. Various approaches to such problems exist but the main trend is to investigate certain extra geometric structures on manifolds.

Troughout their development, geometry and physics have exerted a strong influence on each other. For example in 4 dimensions important achievements came about through interaction with gauge theory and in particular, the study of gauge theory as infinite dimensional geometry has been very fruitful in topology of 4 dimensional manifolds. On the other hand, 4 dimensional manifolds are also studied from the viewpoint of Lefschetz fibrations and symplectic geometry.

Recently, there has been important progress in Thurston's geometrization program in 3 dimensions. Geometry and topology in 3-dimensions are related to 2 dimensional quantum field theory as well. We have studied invariants of 3 dimensional manifolds by means of conformal field theory. This method has also been applied to representations of braid groups. Our research covers methods of contact geometry, foliations and dynamical systems as well.

The moduli spaces of Riemann surfaces and mapping class groups are important subjects in our research. We investigate characteristic classes of surface bundles - Mumford-Morita-Miller classes - from various points of view in relation with cohomology of mapping class groups. We also study moduli spaces of graphs concerning automorphisms of free groups.

The local to global study of geometries was a major trend of geometry that led to remarkable developments, particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, the familiar geometry of space-time in relativity theory, as well as in various other kinds of geometry (symplectic, complex geometry, ...) surprisingly little is known about global properties of the geometry. In this context, we study how the local geometric structure affects the global nature of non-Riemannian manifolds by investigating discrete subgroups of Lie groups.

April 2022