Combinatorial Approaches to Algebraic Varieties and Moduli Problems

代数几何是对多项式方程组解集的研究。这些集合被称为代数簇，出现在从理论物理到计算机科学的各个领域中。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簇来获得新的数学见解。 我提出的研究的第一个主题领域是法诺品种的研究。这些特殊品种正是具有正曲率的品种，为其他品种提供了一种积木。它们出现在许多上下文中，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。我建议通过使用退化和变形技术来深入了解这个问题，将 Fano 簇与称为环面簇的更多组合对象联系起来。 我提出的研究的第二个领域涉及变形理论，即代数簇族的系统研究。代数几何的这一中心主题与分类和模问题有关。我的目标是通过研究组合结构的特殊变形问题来更好地理解变形理论中发生的一般现象。此类问题的具体例子包括环曲面簇变形理论的研究以及有理齐次空间的余切上同调的计算。 我提出的研究的第三个领域是代数簇的线性子空间的研究。嵌入簇的大部分几何形状可以根据它包含的线性子空间来理解。我的长期目标之一是通过比较特殊品种的线性子空间来发现其结构的质的差异。特别是，我打算研究复曲面簇的线性子空间，以及永久和行列式超曲面。后两种类型的线性子空间与代数复杂性理论特别相关。 该研究计划将为纯数学，特别是代数几何提供基本见解。提出的研究目标直接解决该领域的核心问题。我的研究成果将与研究各种问题的科学家相关，从镜像对称到极值度量再到复杂性理论。此外，我的研究计划将有助于培养加拿大新一代数学家。

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.