Mirror Symmetry, Birational Geometry, and Moduli.

弦理论认为，小尺度下基本粒子的相互作用可以用我们宇宙的隐藏维度来解释，这些维度被包裹起来，在每一点上形成一个微小的几何空间，称为卡拉比-丘流形。镜像对称现象断言卡拉比-丘流形以镜像对X和Y出现，它们决定了相同的物理性质，这意味着X和Y的几何性质之间存在令人惊讶的关系。双有理几何是研究空间的外科手术，通过切割低维的子空间并在其位置上粘合另一个子空间而获得。空间Y的模空间参数化了通过变形Y而得到的所有可能的空间。如果X和Y是镜像对，则X的双有理几何决定了Y的模空间在某个极限点附近的结构。基于这个启发式，莫里森证明了一个卡拉比-丘流形X只允许X的100多种可能的变换直到X的对称性。PI的目的是表明猜想是假的一般情况下，但一个较弱的声明足以适用于模持有，建议最近的工作PI与研究生的无界卡拉比-丘流形。PI还将研究称为Fano流形的正弯曲空间以及通过镜像对称在Calabi-Yau流形模空间的极限点处出现的奇点。这些项目将与赠款支持的研究生一起进行。PI还将组织研讨会和会议，重点是培养研究生。PI将研究莫里森的锥猜想和应用到双有理几何和模，与合作者联合。莫里森证明了Calabi-Yau流形的自同构群作用在其具有有理多面体基本整环的nef锥上，使得镜像流形的模空间的尖点的邻域通过构造Looijenga而允许由该数据确定的紧化.最近的工作PI与研究生的日志卡拉比-丘流形和镜像变形的奇点表明，猜想并不持有一般，但一个较弱的版本足以适用于模量应持有。PI将研究Q-Fano 3-folds的镜像对称性，并与研究生一起研究K3表面模量的分类和非算术曲线的应用。Q-Fano 3-folds作为最小模型程序的最终产品出现，因此是我们理解3-folds的基础。镜像对称理论表明，Q-Fano 3-fold的镜像是在无穷远处具有单值性的仿射线上的K3纤维化，在有限基变化后是最大幂幺的。该镜对应于极化K3曲面模空间上的刚性有理曲线。 计算表明，这些曲线不是志村曲线，但均匀化的非算术组。PI将研究表面奇点的Milnor纤维的镜像对称性以及在表面的辛态群和模量中的应用，这是一个与其他项目联合的项目。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

String theory posits that interactions of fundamental particles at small scales are explained by hidden dimensions of our universe which are wrapped up to form a tiny geometric space called a Calabi--Yau manifold at each point. The mirror symmetry phenomenon asserts that Calabi--Yau manifolds come in mirror pairs X and Y which determine the same physics, implying surprising relations between geometric properties of X and Y. Birational geometry is the study of surgeries of spaces obtained by cutting out a subspace of lower dimension and gluing another in its place. The moduli space of a space Y parametrizes all possible spaces obtained by deforming Y. If X and Y are a mirror pair, then the birational geometry of X determines the structure of the moduli space of Y near a certain limit point. Based on this heuristic, Morrison conjectured that a Calabi--Yau manifold X admits only finitely many possible surgeries up to symmetries of X. The PI aims to show the conjecture is false in general, but a weaker statement sufficient for applications to moduli holds, as suggested by recent work of the PI with graduate students on unbounded Calabi--Yau manifolds. The PI will also study positively curved spaces called Fano manifolds and singularities that arise at limit points of the moduli space of Calabi--Yau manifolds via mirror symmetry. These projects will be pursued together with graduate students supported by the grant. The PI will also organize seminars and a conference focused on training of graduate students.The PI will study Morrison's cone conjecture and applications to birational geometry and moduli, joint with a collaborator. Morrison conjectured that the automorphism group of a Calabi--Yau manifold acts on its nef cone with rational polyhedral fundamental domain, so that a neighborhood of a cusp of the moduli space of the mirror manifold admits a compactification determined by this data via a construction of Looijenga. Recent work of the PI with graduate students on log Calabi--Yau manifolds and mirror deformations of singularities suggests that the conjecture does not hold in general, but a weaker version sufficient for applications to moduli should hold. The PI will study mirror symmetry for Q-Fano 3-folds and applications to classification and non-arithmetic curves on moduli of K3 surfaces, joint with a graduate student. Q-Fano 3-folds arise as end products of the minimal model program and so are basic to our understanding of 3-folds. Mirror symmetry heuristics suggest that the mirror of a Q-Fano 3-fold is a K3 fibration over the affine line with monodromy at infinity that is maximally unipotent after a finite base change. The mirror corresponds to a rigid rational curve on a moduli space of polarized K3 surfaces. Computations suggest that these curves are not Shimura curves but are uniformized by non-arithmetic groups. The PI will study mirror symmetry for Milnor fibers of surface singularities and applications to symplectomorphism groups and moduli of surfaces, a project that joint with others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.