Kaehler manifolds, automorphic forms, and quantization

我致力于分析和几何中的问题，这些问题与经典的自同构形式理论(其中基础工作是由Baly、Borel、Harish-Chandra、Piatetski-Shapiro和其他人完成的)有很强的联系，另一方面，我研究的是几何量子化，这是辛几何领域的一个领域，其发展是由Kirillov、Kostant、Souriau和其他人开创的。我以前的工作提供了关于线丛截面的渐近行为的信息，关于向量丛中某些连通曲率的信息，关于Toeplitz算子的信息，以及关于通常与某些几何数据相关的自同构形式空间的显式描述。 我还对将各种结构参数化的空间感兴趣(例如，辛流形上与辛形式相容的复杂结构，或将给定类型的n维代数的结构常数参数化的代数变体)。来自Kaehler几何或代数几何的技术似乎在这些问题上是有效的，并允许获得关于上述变化的拓扑信息。 我计划在所有这些方向上进一步努力，取得更强有力和更深入的成果。

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.