Combinatorial Approaches to Algebraic Varieties and Moduli Problems

代数簇和模问题的组合方法

• 批准号: RGPIN-2015-03933
• 负责人: Ilten, Nathan
• 金额: $ 1.38万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708454
• 关键词: Combinatorial; Approaches; Algebraic; Varieties; Moduli

Algebraic geometry is the study of solution sets of systems of polynomial equations. Such sets, called algebraic varieties, appear in connection to fields ranging from theoretical physics to computer science. The overarching goal of my program of research in algebraic geometry is to gain new mathematical insight through the investigation of algebraic varieties that exhibit combinatorial structure. The first thematic area of my proposed research is the study of Fano varieties. These special varieties are exactly those with positive curvature, and form a kind of building block for other varieties. They appear in numerous contexts, ranging from mirror symmetry to the classification of all varieties. A major open problem is the classification of all families of Fano varieties. I propose to gain insight into this problem by using degeneration and deformation techniques, relating Fano varieties to more combinatorial objects called toric varieties. The second area of my proposed research concerns deformation theory, the systematic study of families of algebraic varieties. This central subject of algebraic geometry is connected to classification and moduli problems. I aim to better understand general phenomena occurring in deformation theory by studying special deformation problems with combinatorial structure. Particular examples of such problems include the study of the deformation theory of toric varieties, and the calculation of cotangent cohomology for rational homogeneous spaces. The third area of my proposed research is the study of linear subspaces of algebraic varieties. Much of the geometry of an embedded variety can be understood in terms of the linear subspaces it contains. One of my long-term goals is to find qualitative differences in the structure of special varieties through comparison of their linear subspaces. In particular, I intend to study linear subspaces of toric varieties, and of the permanental and determinantal hypersurfaces. Linear subspaces of these latter two varieties are of particular relevance for algebraic complexity theory. This program of research will provide fundamental insights in pure mathematics, specifically, algebraic geometry. The proposed research goals directly address important problems that are central to the field. My research outcomes will be relevant for scientists studying a wide variety of problems, ranging from mirror symmetry to extremal metrics to complexity theory. Furthermore, my research program will serve to help train a new generation of mathematicians in Canada.

代数几何是研究多项式方程组的解集。这样的集合，称为代数簇，出现在从理论物理到计算机科学的各个领域。我在代数几何研究计划的首要目标是通过研究表现出组合结构的代数簇来获得新的数学见解。 我建议的研究的第一个主题领域是Fano品种的研究。这些特殊的变种正是那些具有正曲率的变种，并形成了其他变种的一种构建块。它们出现在许多背景下，从镜像对称到所有品种的分类。一个主要的悬而未决的问题是法诺品种所有科的分类。我建议深入了解这个问题，通过使用退化和变形技术，有关法诺品种更多的组合对象称为复曲面品种。 第二个领域，我提出的研究涉及变形理论，系统研究家庭的代数簇。代数几何的这个中心主题与分类和模问题有关。我的目的是更好地了解一般现象发生在变形理论研究特殊变形问题的组合结构。这类问题的具体例子包括研究复曲面簇的变形理论，以及有理齐性空间的余切上同调的计算。 第三个领域，我提出的研究是研究线性子空间的代数簇。嵌入簇的许多几何可以用它所包含的线性子空间来理解。我的长期目标之一是通过比较它们的线性子空间来发现特殊品种结构中的质的差异。特别是，我打算研究线性子空间的环面品种，永久和行列式超曲面。线性子空间的这后两个品种是特别相关的代数复杂性理论。 该研究计划将提供纯数学，特别是代数几何的基本见解。所提出的研究目标直接解决了该领域的核心问题。我的研究成果将与研究各种问题的科学家有关，从镜像对称到极值度量到复杂性理论。此外，我的研究计划将有助于培养加拿大新一代的数学家。