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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654235
• 关键词: Combinatorial; Approaches; Algebraic; Varieties; Moduli; 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.**

代数几何形状是对多项式方程系统溶液集的研究。这种称为代数品种的集合出现在与从理论物理学到计算机科学的领域有关。我的代数几何研究计划的总体目标是通过研究表现出组合结构的代数品种来获得新的数学见解。这些特殊的品种恰好是具有正曲率的那些品种，并形成了其他品种的基础。它们出现在许多情况下，从镜像对称到各种品种的分类。一个主要的开放问题是所有Fano品种的家族分类。我建议通过使用变性和变形技术来深入了解此问题，将Fano品种与称为感谢您的品种的更多组合物体有关。代数几何形状的中心主题与分类和模量问题有关。我的目标是通过研究与组合结构的特殊变形问题，以更好地理解变形理论中发生的一般现象。此类问题的特定例子包括研究曲折品种的变形理论，以及用于合理均匀空间的辅助同种学的计算。可以从包含的线性子空间来理解嵌入式品种的许多几何形状。我的长期目标之一是通过比较线性子空间来找到特殊品种结构的质量差异。特别是，我打算研究复曲面品种的线性子空间，以及永久性和决定性的超曲面。后两个品种的线性子空间与代数复杂性理论特别相关。****该研究计划将提供纯数学的基本见解，特别是代数几何。拟议的研究目标直接解决了该领域核心的重要问题。我的研究结果将与研究各种问题的科学家有关，从镜像对称到极端指标再到复杂性理论。此外，我的研究计划将有助于培训加拿大的新一代数学家。**