Combinatorial Approaches to Algebraic Varieties and Moduli Problems

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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.**