CAREER: Stable sheaves, stable quotients, stable pairs

PI将致力于与代数几何中的模空间的几何和相交理论相关的广泛研究项目。首先，PI将研究广义theta函数空间的Le Potier奇异对偶猜想。沿着的方式，作者将调查有关的重言式环的模空间的K3表面，阿贝尔表面，和卡-丘流形，积极的theta线性级数，和Brill-Noether理论的层在高维基地。第二个方向涉及相交理论计算的模空间的稳定的复曲面和超曲面几何，和他们的比较Gromov-Witten理论。第三，作者计划通过与Quot格式的相交理论的比较来计算局部曲线和曲面几何的稳定对不变量，这是PI近年来研究的中心课题。PI工作于代数几何，即研究多项式方程的解。更具体地说，PI的研究重点是模理论和枚举几何。模空间将具有相似结构的对象放在一起，目的是获得所研究结构的定性和定量的全局统一视图。这将有利于代数几何，但也为一些相关领域。近年来，PI指导本科生和研究生在接近他的研究领域，并将继续这样做。他计划开发新的代数几何课程，并撰写本科代数几何教科书。他将共同组织一个研讨会在他的机构，以及南加州代数几何研讨会。此外，PI将组织研讨会，重点关注与层的模空间相关的主题。最后，PI将参与高中和大学的比赛，并在天才高中生的准备。

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.