Combinatorial Approaches to Deformation and Degeneration in Algebraic Geometry

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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 numerous fields ranging from theoretical physics and computational biology to computer science and statistics. In general, it is an extremely challenging problem to understand the structure of varieties. A common technique in algebraic geometry is to degenerate or deform a variety to one whose structure is easier to understand. The overarching goal of my program of research is to gain new mathematical insight into algebraic geometry through the study of deformation and degeneration. In this program, I will make extensive use of a special class of geometric objects called toric varieties. They are defined using especially simple equations, making them much easier to understand in detail. The first thematic area of my proposed research is the study of toric degenerations. By degenerating an arbitrary variety to a toric variety, one can use the easily accessible information obtained from the toric variety to gain knowledge about the original variety. However, such a degeneration may not exist in general. I will investigate how to explicitly construct such toric degenerations, and under what circumstances one can expect them to exist. This is a highly active topic, with many subtle questions and features. The second thematic area of my proposed research is combinatorial deformation theory. I will study how toric varieties, and other varieties with similar combinatorial structure, can be deformed into other varieties. This takes the opposite perspective of the first thematic area. Explicit solutions to deformation problems are often difficult to obtain. This motivates the study of deformation problems with extra structure. Such problems are often of significant interest in their own right, due to connections to other problems in algebraic geometry. This program of research will provide fundamental insights in pure mathematics, in particular, algebraic geometry. The proposed research will provide tools to address fundamental problems in mathematics, such as mirror symmetry, classification problems in algebraic geometry, and the existence of extremal metrics. My research will additionally advance the field of applied algebraic geometry, which has applications in a variety of disciplines such as biology, chemistry, and statistics. Finally, the impact of the proposed research will contribute to strengthening Canada's leadership in the study of pure mathematics, and will serve to help train a new generation of mathematicians in Canada.