Stacky Curves, GIT Quotients of Representations, and Deformed Euler Classes

For the last 30 years, string theorists and mathematicians have worked together to “count” curves in certain spaces. These “curve-counting numbers’’ are of intrinsic interest in both physics and mathematics. As the two groups simultaneously study these numbers, insights from the string theorists often lead to developments in mathematics, as well as vice versa. The PI will further develop a variety of mathematical techniques for studying these curve-counting numbers. Some of these developments are directly motivated by open questions in string theory, while others are more closely tied to recent advances in algebraic geometry, especially moduli theory. The PI will use these projects to train graduate students and postdocs in conducting mathematical research and in communicating their results. In particular, the PI will make lecture notes on important foundational topics available to both math and physics graduate students. The PI will also participate in activities to improve communication of research within the mathematics community. In a little more detail, the PI has three research objectives. First, the PI will investigate new modular compactifications of the moduli space of stacky curves and their applications to enumerative geometry. These compactifications have direct implications for the study of non-stacky curves (via Hurwitz theory) as well as potential applications to higher-genus, log, and orbifold Gromov-Witten theory. Second, The PI will study the geometry of Geometric Invariant Theory (GIT) quotients of representations of complex reductive groups. GIT quotients of representations are a well-used testing ground for new algebro-geometric theories, and the PI’s research will make it easier to find and use examples. Lastly, the PI will investigate deformed virtual classes in sheaf cohomology, aiming toward a (0, 2) analog of Gromov-Witten theory. The definition of such has been a mystery in string theory for the last decade, but recent advances in derived algebraic geometry make such a definition mathematically feasible. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.