Kaehler manifolds, automorphic forms, and quantization

我工作的问题在分析和几何有很强的联系，以古典理论的自守形式（其中的基础工作是由Beveli，博雷尔，哈里什-钱德拉，皮亚特斯基-夏皮罗，和其他人），并在另一方面，几何量化，这是一个领域的辛几何的发展是开创性的基里洛夫，科斯坦特，苏里奥和其他人。我早期的工作提供了有关渐近行为的部分线丛，曲率的某些连接在向量丛，关于Toeplitz运营商，并明确描述空间的自守形式，往往与一些几何数据。 我也对参数化各种结构的空间感兴趣（例如，与辛形式兼容的辛流形上的复杂结构，或参数化给定类型的n维代数的结构常数的代数簇）。从Kaehler几何或代数几何的技术似乎是有效的，在这些问题，并允许获得拓扑信息的品种上述。 我计划在所有这些方向上进一步努力，取得更有力、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.