Combinatorics and Geometry of Moduli Spaces

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Long-term goal. Many computational and qualitative problems can be distilled down to mathematical questions about classification and enumeration. The first question asks: what are all the possible curves, surfaces, and other mathematical objects? The second asks: how many of them have a given property, or solve a given problem? In algebraic geometry, we address these questions using moduli spaces: parameter spaces that describe all geometric objects of a given type. My research program focuses on classification and enumeration related to linear spaces and curves, two of the most ubiquitous mathematical objects. The overarching goal of my research is to develop combinatorial tools to count, classify and compute solutions to algebraic and geometric problems involving planes and curves, and to apply these tools to explicitly understand the geometry of the associated moduli spaces. Short-term goals. In the next 5 years, my program will focus on the following: 1. Establish new connections between complex and real geometry involving linear spaces tangent to curves. 2. Develop new combinatorial tools to solve enumerative problems involving moduli of curves. 3. Produce new geometric spaces predicted from combinatorial constructions in representation theory. Approach. Moduli problems are typically approached by focusing on limiting features and by recursively passing to special and degenerate boundary cases. These degeneration techniques usually simplify the geometry while introducing a great deal of combinatorial complexity. As such, tools from combinatorics are essential for the analysis and frequently shed light on the underlying geometry. My research program will establish formal connections between geometry and combinatorics, generally by showing that the geometric processes under examination follow the same recursive patterns as do simpler (discrete) combinatorial models. In most cases, one of either the geometry or the combinatorics is better understood than the other, so the scope of this research also includes developing new combinatorial tools (Objective 2) and constructing and analysing new geometric spaces (Objective 3). Impact. Moduli theory and enumeration are central topics in algebraic geometry, related to deformation theory, birational geometry and broad classification problems. This research program will produce crossover results relating geometry, combinatorics and representation theory. These results will allow researchers to borrow the techniques of other fields, as well as their intuitions, goals and avenues of inquiry. Moreover, the new foundational tools developed for linear spaces and curves will contribute to the development of new techniques for engineering and the natural sciences. This proposal will also support the training of highly qualified personnel in mathematics, particularly algebraic geometry and combinatorics, and will contribute to enhancing Canada's standing in these mathematical fields.