Kaehler manifolds, automorphic forms, and quantization

我研究的分析和几何问题与经典自同构形式理论（其中的基础工作是由Baily， Borel, Harish-Chandra， Piatetski-Shapiro等人完成的）有很强的联系，另一方面，与几何量子化，这是辛几何的一个领域，其发展是由Kirillov， Kostant， Souriau等人开创的。我早期的工作提供了关于线束截面的渐近行为的信息，关于向量束中某些连接的曲率，关于Toeplitz算子，以及对自同构形式空间的明确描述，通常与一些几何数据相关。

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.