Moduli of surfaces, vector bundles, and mirror symmetry

PI在代数曲面上稳定向量丛的分类和曲面变形模空间的紧致化之间建立了联系。PI将详细研究一般类型的曲面的这种对应关系，特别是它与光滑4-流形的瞬子不变量的Donaldson理论的关系。与Mark Gross和Sean Keel合作，PI描述了2维非紧Calabi-Yau流形的镜像伙伴的显式构造。PI将进行两个相关的项目：非紧Calabi-Yau流形的辛上同调环的代数描述，以及K3曲面上采样线丛的全局截面基的标准基的构造，类似于极化阿贝尔变种的theta函数。PI将研究某些4维几何空间的分类，以及这种空间的子集可以连续变形或经历向一点的“退化”的方法。在以前的工作中，PI将这种退化与给定几何空间上的线性空间丛的分类联系起来。他将详细研究这种通信，特别是它与唐纳森在理论物理推动下的工作的联系。镜像对称是弦理论中出现的一对几何空间之间的神秘对应，称为Calabi-Yau流形。PI将进行两个受镜面对称启发的项目。第一个是用显式术语描述由Calabi-Yau流形内的面积最小化曲面的计数建立的代数结构。第二个问题是Calabi-Yau流形上自然函数的构造。在最简单的圆环(甜甜圈的表面)的情况下，这些函数是经典已知的，并且在许多数学领域中都很重要。

The PI has established a connection between the classification of stable vector bundles on an algebraic surface and the compactification of the moduli space of deformations of the surface. The PI will study this correspondence in detail for surfaces of general type, in particular, its relation to the Donaldson theory of instanton invariants of smooth 4-manifolds. Jointly with Mark Gross and Sean Keel, the PI has described an explicit construction of the mirror partner to a non-compact Calabi--Yau manifold of complex dimension 2. The PI will pursue two related projects: an algebraic description of the symplectic cohomology ring of a non-compact Calabi-Yau manifold, and the construction of a canonical basis of global sections of an ample line bundle on a K3 surface, analogous to theta functions for polarized abelian varieties.The PI will study the classification of certain 4-dimensional geometric spaces and the ways in which such a space can be continuously deformed or undergo a "degeneration" given by a subset of the space collapsing to a point. In prior work the PI related such degenerations to the classification of bundles of linear spaces over the given geometric space. He will study this correspondence in detail, in particular its connection with work of Donaldson motivated by theoretical physics. Mirror symmetry is a mysterious correspondence between pairs of geometric spaces called Calabi-Yau manifolds arising in string theory. The PI will pursue two projects inspired by mirror symmetry. The first is a description in explicit terms of an algebraic structure built from counts of area-minimizing surfaces inside a Calabi-Yau manifold. The second is the construction of natural functions on Calabi-Yau manifolds. In the simplest case of a torus (the surface of a donut) these functions were known classically and are important in many areas of mathematics.