Combinatorial Approaches to Algebraic Varieties and Moduli Problems

代数几何是研究多项式方程组的解集的学科。这样的集合被称为代数族，出现在从理论物理到计算机科学的各个领域。我的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变体来获得新的数学见解。 我提出的研究的第一个主题领域是对Fano品种的研究。这些特殊的品种正是那些具有正曲率的品种，并构成了其他品种的一种积木。它们出现在许多语境中，从镜像对称到所有种类的分类。一个主要的悬而未决的问题是Fano变种的所有族的分类。我建议通过使用退化和变形技术来洞察这个问题，将Fano簇与更多被称为Toric簇的组合对象联系起来。 我建议研究的第二个领域是形变理论，即对代数变种家族的系统研究。这个代数几何的中心课题与分类和模问题有关。我的目的是通过研究组合结构的特殊变形问题，更好地理解变形理论中的一般现象。这类问题的具体例子包括环簇的变形理论的研究，以及有理齐性空间的余切上同调的计算。 我提出的第三个研究领域是代数簇的线性子空间的研究。嵌入变种的许多几何可以通过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特殊变种的线性子空间，找出它们在结构上的质的差异。特别地，我打算研究环面簇的线性子空间，以及永久和行列式超曲面的线性子空间。后两个变种的线性子空间与代数复杂性理论特别相关。 这一研究计划将提供纯数学方面的基本见解，特别是代数几何。拟议的研究目标直接涉及该领域的核心问题。我的研究成果将对科学家研究各种各样的问题具有重要意义，从镜像对称到极端度量学再到复杂性理论。此外，我的研究计划将有助于培养加拿大的新一代数学家。

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.