CAREER: Stable sheaves, stable quotients, stable pairs

PI将致力于与代数几何中模空间的几何和交集理论相关的广泛的研究项目。首先，PI将研究Le Potier关于广义theta函数空间的奇怪的对偶猜想。在此过程中，作者将研究与K3曲面、交换曲面和Calabi-Yau流形的模空间的重言环、theta线性级数的正性以及高维基上的层的Brill-Noether理论有关的问题。第二个方向是关于局部环面几何和超曲面几何稳定商的模空间的交集理论计算，以及它们与Gromov-Witten理论的比较。第三，作者计划通过与Quot格式的交集理论的比较来计算局部曲线曲面几何的稳定对不变量。Quot格式是近年来PI研究的中心课题。PI工作在代数几何中，它研究多项式方程的解。更具体地说，PI的研究集中在模理论和计数几何上。模空间将具有相似结构的物体组合在一起，目的是在定性和定量上获得所研究结构的全局统一视图。这不仅对代数几何有益，而且对许多相关领域也是有益的。近年来，PI在与他的研究相近的领域指导本科生和研究生，并将继续这样做。他计划开发新的代数几何课程，并撰写一本本科生的代数几何教材。他将在他所在的机构联合组织一个研讨会，以及南加州代数几何研讨会。此外，国际和平研究所将组织讲习班，重点讨论与滑轮模数空间有关的主题。最后，PI将参与高中和大学的比赛，以及为有天赋的高中生做准备。

The PI will work on a wide range of research projects related to the geometry and intersection theory of moduli spaces in algebraic geometry.Three main directions are proposed. First, the PI will study Le Potier's strange duality conjecture for spaces of generalized theta functions. Along the way, the author will investigate questions related to the tautological rings of the moduli space of K3 surfaces, abelian surfaces, and Calabi-Yau manifolds, the positivity of the theta linear series, and the Brill-Noether theory of sheaves over higher dimensional bases. A second direction concerns intersection theoretic calculations on the moduli space of stable quotients for local toric and hypersurface geometries, and their comparison with Gromov-Witten theory. Thirdly, the author plans to compute stable pair invariants of local curve and surface geometries by comparison with intersection theory of Quot scheme, a topic which has been at the center of the PI's research in recent years.The PI works in algebraic geometry, which is the study of solutions of polynomial equations. More specifically, the PI's research focuses on moduli theory and enumerative geometry. Moduli spaces put together objects with similar structure, with the goal of obtaining a global unified view of the structure under study, both qualitatively and quantitatively. This will be beneficial for algebraic geometry, but also for a number of related fields. In recent years, the PI mentored undergraduate and graduate students in areas close to his research, and will continue to do so. He plans to develop new algebraic geometry courses, and to write an undergraduate algebraic geometry text. He will co-organize a seminar at his institution, as well as the Southern California Algebraic Geometry Seminar. Furthermore, the PI will organize workshops focusing on topics related to moduli spaces of sheaves. Finally, the PI will be involved in high school and college competitions, and in the preparation of gifted high school students.