I study the geometry and topology of differentiable manifolds, with a particular focus on Lie groups , Riemannian symmetric spaces, and related homogeneous spaces. Important examples are flag manifolds, which arise as adjoint orbits of compact Lie groups and, more generally, as orbits of isotropy representations of compact symmetric spaces. These spaces have played a central role in several major developments in geometry, topology, and representation theory. Some examples that are especially relevant to my research are described below.

The cohomology of flag manifolds was investigated by A. Borel in the 1950s. Around the same time, R. Bott and H. Samelson introduced new methods, most notably Morse theory, to study these spaces and the based loop spaces of compact symmetric spaces. Their work laid the foundations of what is now called generalized Schubert calculus, a subject whose roots go back to the work of H. Schubert in the nineteenth century and which encompasses equivariant cohomology, equivariant K-theory, quantum cohomology, and related topics.

Principal orbits of isotropy representations of symmetric spaces enjoy remarkable convexity properties. A theorem of B. Kostant states that the image of such an orbit under orthogonal projection onto any normal space is a convex polytope. This result was later generalized by C.-L. Terng to isoparametric submanifolds in Hilbert spaces and may be viewed as a precursor of the convexity theorems of M. Atiyah, V. Guillemin, S. Sternberg, F. Kirwan, and H. Duistermaat concerning Hamiltonian group actions on symplectic manifolds.

The Morse-theoretic study of based loop spaces on compact symmetric spaces also led R. Bott to his celebrated periodicity theorem, one of the fundamental results in algebraic topology, describing the stable homotopy groups of the unitary, orthogonal, and symplectic groups.

In the early 1990s, developments in theoretical physics, particularly in string theory, drew attention to a remarkable deformation of the cohomology ring of a smooth projective variety. This led to the rigorous mathematical construction of the small quantum cohomology ring. Since flag manifolds are important examples of projective varieties, they became a natural testing ground for these ideas. Through the work of A. Givental, B. Kim, D. Peterson, and others, the quantum cohomology of flag manifolds has become largely understood in terms of combinatorial and representation-theoretic models. Connections with integrable systems, notably the Toda lattice, continue to reveal unexpected interactions between geometry, algebra, and mathematical physics.

Flag manifolds may also be viewed as spaces of isospectral Hermitian matrices. From this perspective, the classical Schur-Horn theorem appears as an early manifestation of the convexity phenomena described above. More recently, these ideas have found applications in frame theory, where connections between certain spaces of frames and level sets of the diagonal projection map have led to new proofs of the frame homotopy conjecture and related results.