The geometry of the moduli spaces of morphisms and sheaves

PI将探讨模理论中的几个问题。第一个方向涉及稳定商的模空间，特别是超曲面和局部几何的不变量的计算。还将研究稳定商结构的变化。其次，PI将逼近曲面上滑层的模空间的Verlinde公式和奇怪的对偶猜想。奇异对偶关系到层模空间上广义theta函数的空间，并应用于研究相关的theta线性级数。虽然曲线的情况早在几年前就已被理解，但曲面的情况在很大程度上仍然是开放的。其他项目涉及亏格0的稳定映射和稳定商的模空间的拓扑不变量，以及环面几何的高阶DT不变量。PI工作在代数几何中，粗略地说，它是研究多项式方程的解。目前的建议概述了几个研究方向，所有这些都涉及模空间的各个方面，例如，将具有相似性质的代数几何对象分类的空间。模理论传统上与组合学、辛几何、数论、表示论、理论物理等领域有着密切的联系。PI旨在更好地理解近几十年来出现的三种模空间。特别是，该提议将结合定量的方面，例如自然构造的不变量的计算，以及定性的方面，例如模空间的几何研究。

The PI will approach several questions in moduli theory. A first direction concerns the the moduli space of stable quotients, in particular the calculation of invariants for hypersurface and local geometries. Variations of stable quotient constructions will also be studied. Secondly, the PI will approach the Verlinde formula and the strange duality conjecture for moduli spaces of sheaves over surfaces. Strange duality concerns the spaces of generalized theta functions over moduli spaces of sheaves, and has applications to the study of the associated theta linear series. While the case of curves has been understood a few years ago, the case of surfaces is still largely open. Other projects concern topological invariants of the moduli spaces of genus 0 stable maps and stable quotients, and higher rank DT invariants for toric geometries.The PI works in algebraic geometry, which, roughly speaking, is the study of solutions of polynomial equations. The current proposal outlines several research directions all of which touching on aspects of moduli spaces e.g. spaces which classify algebro-geometric objects with similar properties. Moduli theory has traditionally been closely related to other fields such as combinatorics, symplectic geometry, number theory, representation theory, theoretical physics. The PI aims to better understand three kinds of moduli spaces all of which emerged in recent decades. In particular, the proposal will combine a quantitative side e.g. calculation of naturally constructed invariants, and qualitative aspects e.g. the geometric study of the moduli spaces.