Grant: Discovery Grants Program - Individual

I work on questions in analysis and geometry that have strong connections to the classical theory of automorphic forms (where the foundational work was done by Baily, Borel, Harish-Chandra, Piatetski-Shapiro, and others), and, on the other hand, to geometric quantization, which is an area of symplectic geometry whose development was pioneered by Kirillov, Kostant, Souriau and others. My earlier work provided information about asymptotic behaviour of sections of line bundles, about curvature of certain connections in vector bundles, about Toeplitz operators, and also explicit description of spaces of automorphic forms, often associated to some geometric data. I am also interested in spaces that parametrize various structures (e.g. complex structures on a symplectic manifold that are compatible with the symplectic form, or algebraic varieties that parametrize structure constants of n-dimensional algebras of a given kind). Techniques from the Kaehler geometry or algebraic geometry seem to be effective in these problems and allow to obtain topological information about varieties described above. I plan to obtain stronger and deeper results working further in all these directions.