Moduli Problems in Algebraic Geometry, their Structures, and their Applications

该奖项支持代数几何的研究，代数几何是数学的中心领域。研究者将研究以代数性质为特征的数学对象参数化的空间的某些几何性质。其中一个空间参数化代数空间中的一维复维子对象；另一个空间参数化代数空间上的域。这些是数学中重要的研究结构，部分原因是它们内在美丽的结构以及这些结构结果在几何，拓扑和表示理论中的应用，部分原因是这些空间是理论物理中对象（场，弦）的数学类似物。这个项目的完成将提高我们对数学和理论物理的理解。本研究项目的长期目标是通过理解理论物理的新思想来拓宽数学研究，并通过提供对理论物理发展至关重要的数学基础来促进理论物理的发展。本课题为研究五次Calabi-Yau三倍型的所有格Gromov-Witten不变量提供了一个新的理论。这个理论是超弦理论中设想的几何实现。这将为代数几何中相似模空间的研究提供新的途径。研究者将进一步发展一种理论，通过Kirwan去量子化的角度来理解Calabi-Yau三倍的广义Donaldson-Thomas不变量，并在理解最近构造的关于Calabi-Yau三倍上的轴的模的反常轴方面取得进展。

This award supports research in algebraic geometry, a central area of mathematics. The investigator will study certain geometric properties of spaces that parameterize mathematical objects characterized by algebraic properties. One such space parameterizes one-complex-dimensional sub-objects in an algebraic space; the other space parameterizes fields on an algebraic space. These are important constructions to study in mathematics, in part due to their intrinsically beautiful structures and the applications of these structural results to geometry, topology, and representation theories, and in part due to the fact that these spaces are mathematical analogues of objects (fields, strings) in theoretical physics. The completion of this project will improve our understanding of mathematics and of theoretical physics in general. This research project addresses the long-term goal of broadening mathematical research by understanding new ideas from theoretical physics, and contributing to the development of theoretical physics by providing the mathematical foundations vital to its advancement. This research project will develop a new theory to study all genus Gromov-Witten invariants of the quintic Calabi-Yau threefolds. This theory is a geometric realization of that envisioned in superstring theories. It will provide new avenues to study similar moduli spaces in algebraic geometry. The investigator will further develop a theory to understand the generalized Donaldson-Thomas invariants of Calabi-Yau threefolds through the angle of Kirwan desingularizations, and to make an inroad to understand the recently constructed perverse sheaves on the moduli of sheaves on Calabi-Yau threefolds.