Novel Approaches to Geometry of Moduli Spaces

代数几何长期以来在数学中占据着中心地位，它提供了一种复杂的语言来描述被称为代数变异的几何形状，其应用范围从物理学中的构型空间到统计学中的参数模型。这种多用途的语言在整个代数中使用，并推动了许多最近的进展，不仅在代数几何本身，而且在表示理论，数论，辛几何和其他领域。代数变量通常被赋予额外的结构，如向量束。向量束的局部截面是物理学中场的数学抽象，使得代数几何在研究镜像对称和其他对偶性等物理现象中不可或缺。模理论中反复出现的主题是矢量束的模空间之间的相互作用，矢量束的模空间可以几何参数化并可以解析研究，而代数变体的派生范畴则编码了矢量束的代数和同调性质。派生范畴提供了从代数几何到新兴的非交换几何领域的桥梁。实际上，函子和派生范畴之间的等价与代数变体的双族（局部）几何密切相关。该项目将进一步研究衍生类别。PI将在会议、专业发展活动和研究生暑期学校为研究生提供研究生水平的迷你课程和讲座。许多子项目适合作为研究生的论文题目。此外，有几个问题是专门为参加PI组织的代数几何研究和训练计划的本科生设计的。更详细地说，拟议的研究主要围绕两个主题。第一部分是更广泛地研究模空间和Fano变的派生范畴。与Calabi-Yau或大多数正则极化品种不同，Fano品种的派生范畴允许半正交分解；从非交换几何的角度来看，Fano变体是由更基本的块构成的。一幅美丽的图画出现了，在那里，各种法诺品种的分解，与民族转换有关，经历了重新排列，我们称之为编织模式。它们的构造是由镜像对称、量子上同调、最小模型程序的消失定理和量子化的思想所激发的。PI将把这个程序推进到各种各样的空间：向量束的模空间，曲线上的抛物束和希格斯束，环型簇，旗型簇，K3曲面上具有一维支撑的束的模空间，以及射影超kahler簇上的反辛对合的不动点轨迹。第二个主题是继续研究范畴Milnor纤维对奇异代数变体的变形，描述其镜像对称解释，并找到几何属零的代数曲面的模的应用，包括Dolgachev曲面和fake del Pezzo曲面。该奖项反映了美国国家科学基金会的法定使命，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。

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.