Projects on Modular Algebraic Geometry

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。继续他以前在椭圆稳定映射上的工作，PI的近期目标是完成他在亏格2稳定映射的模空间结构上的联合工作;然后加强他们在高亏格情况下已经获得的结果;对于任何亏格，他们最近使用稳定映射的模空间的主分量上的导出分辨率获得了枚举不变量;这些新的不变量然后被用来制定一个精确的递归关系的高亏格GW不变量的光滑五次;这些应该是有用的验证物理学家的高亏格镜像对称性预测。此外，PI计划将他们开发的技术推广和应用到Gromov-Witten理论，镜像对称，可能还有双有理几何。除此之外，PI还引入了射影平面上一般线性位置的n点空间的模紧化;这可能对奇点理论产生重要影响。最后，他还致力于通过GIT方法研究最坏情况下具有有限商奇点的投影簇的加权强因子分解。物理学的镜像对称性领域在20世纪90年代爆发到数学舞台上。这是数学和物理之间，或者更具体地说是几何/分析和量子场论之间极其丰富的相互作用的一部分。物理学家基于他们的物理直觉和实验，提出两个不同的物理模型之间存在着深刻的对偶关系，即镜像对称。基于这种二重性，他们发现了一些令人惊讶的数学公式和语句。数学家面临的挑战是理解物理理论背后的数学，并证明物理学家所做的一些预言。稳定映射的模空间是连接物理学家预言和数学证明的主要数学工具。这些空间加在一起可以是相当任意的。PI的主要目标之一是提供这些空间的局部结构，以便可以应用一些重要的几何工具。该项目中研究的所有数学空间都与数学的不同分支有关，也与理论物理学中超弦理论的积极研究直接相关。因此，这个项目的潜在影响肯定会超越代数几何和数学研究。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." Continuing his previous work on elliptic stable maps, the PI's near-term objective is to complete his joint work on the structures of the moduli spaces of genus-two stable maps; then strengthen the results that they already obtained in high-genus cases; for any genus, they recently obtained the enumerative invariants using derived resolutions over the primary components of the moduli spaces of stable maps; these new invariants were then used to formulate a precise recursive relation for high-genus GW invariants of smooth quintic; these should be useful for verifying physicists' high-genus Mirror Symmetry prediction. Further, the PI plans to push and apply the techniques that they have developed to Gromov-Witten theory, to Mirror Symmetry, and possibly also tobirational geometry. In addition to the above, the PI has introduced a modular compactification of the space of n points in general linear position on the projective plane; this potentially has significant consequences on singularity theory. Lastly, he is also working toward the weighted strong factorization for projective varieties with at worst finite quotient singularities through GIT approach.The field of mirror symmetry of physics has exploded onto the mathematical scene in 1990's. This is a part of extremely rich interaction between mathematics and physics, or more specifically between geometry/analysis and quantum filed theory. Physicists, based on their physical intuition and experiments, proposed that there is a deep duality relation, the mirror symmetry, between two different physical models. Based on this duality, they discovered several astonishing mathematical formulas and statements. The challenge for mathematicians was to understand the mathematics behind the physical theories and prove some of the predications made by the physicists. The moduli spaces of stable maps are the main mathematical devices that play key roles in linking physicists' predications to mathematical proofs. These spaces all together can be quite arbitrary. One of the PI's main goal is to provide local structures of each of these spaces so that some importantgeometric tools can be applied. All the mathematical spaces investigated in the project are connected to different branches of mathematics and are also directly related to the active research in super string theory of theoretical physics. Thus the potential impact of this project will definitely go beyond algebraic geometry and mathematical research.