Novel Approaches to Geometry of Moduli Spaces

长期以来，代数几何一直在数学中占据核心地位，提供了一种复杂的语言来描述被称为代数族的几何形状--应用范围从物理学中的配置空间到统计学中的参数模型。这种通用的语言在整个代数中被使用，并推动了最近的多个进展，不仅在代数几何本身，而且在表示论、数论、辛几何和其他领域。代数簇通常被赋予额外的结构，例如向量丛。矢丛的局部部分是物理中场的数学抽象，使得代数几何对于研究镜像对称和其他对偶等物理现象是不可或缺的。模理论中一个反复出现的主题是向量丛的模空间和代数簇范畴之间的相互作用，向量丛的模空间将向量丛进行几何参数化，并且可以用解析的方法进行研究，而代数簇的派生范畴编码向量丛的代数性质和同调性质。派生范畴提供了从代数几何到非交换几何这一新兴领域的桥梁。事实上，派生范畴之间的函子和等价与代数簇的二元(局部)几何密切相关。这一项目将进一步深化派生范畴的研究。PI将为研究生提供研究生级别的迷你课程，并在会议、专业发展活动和暑期学校为研究生提供讲座。很多子课题适合作为研究生的毕业论文题目。此外，在国际学生联合会组织的代数几何研究和培训计划中，有几个问题是专门为本科生设计的。更详细地说，拟议的研究围绕两个主题展开。第一个是更广泛地研究模空间和Fano簇的派生范畴。Fano簇的派生范畴与Calabi-Yau或大多数典范极化簇不同，它允许半正交分解；从非对易几何的角度来看，Fano簇是从更基本的块构造的。一幅美丽的图画出现了，各种法诺品种的分解，通过两栖变换联系在一起，经历了重新排列，我们称之为编织图案。它们的构造是由镜像对称性、量子上同调、最小模型程序的消失定理和量子化的思想推动的。PI将在各种不同的空间中推进这一程序：曲线上的矢丛、抛物丛和Higgs丛的模空间、环丛、旗簇、K3曲面上具有一维支撑的层的模以及射影Hyperkahler簇上的反辛对合的不动点轨迹。第二个主题是继续研究用于奇异代数簇变形的范畴Milnor纤维，描述其镜像对称性解释，并将其应用于几何亏格为零的代数曲面的模数，包括Dolgachev曲面和伪del Pezzo曲面。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。

Algebraic geometry has long occupied a central role in mathematics, providing a sophisticated language to describe geometric shapes known as algebraic varieties - with applications ranging from configuration spaces in physics to parametric models in statistics. This versatile language is used throughout algebra and has fueled multiple recent advances, not only in algebraic geometry itself but also in representation theory, number theory, symplectic geometry and other fields. Algebraic varieties are typically endowed with additional structures, such as vector bundles. Local sections of vector bundles are mathematical abstractions of fields in physics, making algebraic geometry indispensable for the study of physical phenomena like mirror symmetry and other dualities. A recurring theme in moduli theory is the interplay between moduli spaces of vector bundles, which parametrize them geometrically and can be studied analytically, and the derived categories of algebraic varieties, which encode algebraic and homological properties of vector bundles. Derived categories provide a bridge from algebraic geometry to the emerging field of non-commutative geometry. Indeed, functors and equivalences between derived categories are deeply related to the birational (local) geometry of algebraic varieties. This project will further the study of derived categories. The PI will deliver graduate-level mini-courses and lectures at conferences, professional development events, and summer schools for graduate students. Many sub-projects are suitable as thesis topics for graduate students. Furthermore, several problems are designed specifically for undergraduate participants in the research and training program in algebraic geometry organized by the PI.In more detail, the proposed reserch is centered around two main themes. The first is the study of derived categories of moduli spaces and Fano varieties more broadly. The derived categories of Fano varieties, unlike Calabi-Yau or most canonically polarized varieties, admit semi-orthogonal decompositions; from the perspective of non-commutative geometry, Fano varieties are built from more elementary blocks. A beautiful picture emerges, where the decompositions of various Fano varieties, related by birational transformations, undergo rearrangements, which we call weaving patterns. Their construction is motivated by ideas of mirror symmetry, quantum cohomology, vanishing theorems of the minimal model program, and quantization. The PI will advance this program for a wide variety of spaces: moduli spaces of vector bundles, parabolic bundles and Higgs bundles on curves, toric varieties, flag varieties, moduli of sheaves with one-dimensional support on K3 surfaces, and fixed-point loci of anti-symplectic involutions on projective hyperkahler varieties. The second theme is to continue the study of the categorical Milnor fiber for deformations of singular algebraic varieties, describe its mirror symmetry interpretation, and find applications to moduli of algebraic surfaces of geometric genus zero, including Dolgachev surfaces and fake del Pezzo surfaces.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.