Mirror Symmetry, Birational Geometry, and Moduli.

弦理论假设，基本粒子在小尺度上的相互作用可以通过我们宇宙中隐藏的维度来解释，这些维度被包裹起来，形成一个微小的几何空间，称为Calabi-Yau流形。镜像对称现象认为，Calabi-Yau流形是由确定相同物理的镜像对X和Y组成的，这意味着X和Y的几何性质之间存在惊人的关系。出生几何学是研究空间的外科手术，这些空间是通过切割出一个较低维度的子空间并将另一个子空间粘在它的位置上而获得的。空间Y的模空间将所有可能的空间形变为参数。如果X和Y是镜像对，则X的双曲几何决定了Y的模空间在某一极限点附近的结构。基于这个启发式，Morison猜想Calabi-Yau流形X只允许有限多个可能的手术直到X的对称性。PI的目的是证明这个猜想在一般情况下是错误的，但一个足够应用于模的较弱的陈述成立，正如最近PI在无界Calabi-Yau流形上与研究生所做的工作所表明的那样。PI还将研究被称为Fano流形的正弯曲空间，以及通过镜像对称出现在Calabi-Yau流形的模空间的极限点上的奇点。这些项目将与获得助学金资助的研究生一起进行。国际学生联合会还将组织以研究生培训为重点的研讨会和会议。国际学生联合会将与合作者一起研究莫里森圆锥猜想及其在二次几何和模数中的应用。Morison猜想，Calabi-Yau流形的自同构群作用在具有有理多面体基本域的nef锥上，使得镜像流形的模空间的尖点的邻域通过构造Looijenga而允许由这一数据确定的紧化.最近PI与研究生在对数Calabi-Yau流形和奇点的镜像变形方面的工作表明，这个猜想不是普遍成立的，而是一个足够应用于模的较弱的版本应该成立。PI将学习Q-Fano 3-折叠的镜像对称性以及在分类和K3曲面的模上的非算术曲线上的应用，与一名研究生合作。Q-FANO 3-折叠作为最小模型程序的最终产品出现，因此是我们理解3-折叠的基础。镜像对称性启发式认为，q-Fano 3重的镜像是仿射直线上的K3纤维，在有限的基改变后，它在无穷远处具有极大的单幂等性质。该镜对应于偏振K3曲面的模空间上的一条刚性有理曲线。计算表明，这些曲线不是Shimura曲线，而是由非算术群统一的。PI将研究米尔诺纤维表面奇点的镜像对称性，并将其应用于辛同构群和表面模数，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智力优势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.