I study the geometry and topology of differentiable manifolds, particularly of Lie groups, Riemannian symmetric spaces, and related homogeneous spaces. Important examples of such spaces are flag manifolds, i.e. adjoint orbits of compact Lie groups or, more generally, orbits of the isotropy representations of compact symmetric spaces. Their study has motivated important developments in mathematics. The following instances are relevant for my research:
The cohomology ring of flag manifolds has been investigated by A. Borel in the 1950's; at about the same time, R. Bott and H. Samelson approached this topic with different methods, most notably Morse theory, which they have also used for spaces of based loops in compact symmetric spaces. These results established the foundations of what is nowadays called generalized Schubert calculus, a mathematical theory whose origins go back to H. Schubert in the second half of the nineteenth century and which deals with (equivariant) cohomology, (equivariant) K-theory, (equivariant) quantum cohomology, Pontryagin ring etc.
Principal orbits of isotropy representations of symmetric spaces have the property that their image under the orthogonal projection to the normal space at any point is a convex polytope. This theorem of B. Kostant was extended by C.-L. Terng to a more general class of submanifolds in Hilbert spaces, the isoparametric submanifolds; at the same time, it is the precursor of the convexity theorems of M. Atiyah, V. Guillemin - S. Sternberg, F. Kirwan, and H. Duistermaat concerning Hamiltonian Lie group actions on symplectic manifolds.
The aforementioned Morse-theoretical approach to spaces of based loops in compact symmetric spaces has led R. Bott to the famous periodicity theorem concerning the homotopy groups of the unitary, orthogonal, and symplectic groups.
At the beginning of the 1990's physicists, especially string theorists, became aware of the associativity of a certain deformation of the cohomology ring of a smooth projective variety. Rigorous mathematical constructions of the deformed ring, called the (small) quantum cohomology ring, were done shortly after that. Flag manifolds are important examples of projective manifolds. Their quantum cohomology is by now fairly well understood, i.e. reduced to purely combinatorial models, due to A. Givental, B. Kim, D. Peterson, and other mathematicians. Theoretical physics keeps breaking through into this area, like for instance a mysterious connection with the Toda lattice (a mechanical system describing the motion of finitely many particles which move on a line and are connected by springs with certain prescribed potential).