CAREER: Geometry of moduli spaces of curves

Polynomial equations, which are essentially made by adding and multiplying together variables, are among the most fundamental equations in mathematics and arise in many areas of science and engineering. The field of algebraic geometry seeks to classify the shapes defined by polynomial equations. The 1-dimensional shapes, called algebraic curves, have important applications in cryptography and string theory. The starting point for the classification of algebraic curves is Riemann's work in the 1850s, which introduced the concept of their moduli space -- a space in which each point corresponds to a different algebraic curve. Moreover, curves with certain geometric properties correspond to subsets of the moduli space. In order to understand how these different properties interact with each other, one must understand how different subsets of the moduli space intersect each other. The project's main goal is to develop novel tools in intersection theory to shed light on different aspects of the geometry of the moduli space of curves. This research will be complemented by educational activities for a range of students, including providing enrichment for elementary and middle school students at local math circles, mentoring undergraduate research projects, and organizing a summer school in algebraic geometry for graduate students. More precisely, the research will have three main directions. First, the PI will pioneer new approaches to the intersection theory of moduli spaces of curves of low genus, including connecting them to, and studying, other closely-related moduli spaces. Second, the PI will apply her expertise in intersection theory to study the cohomology and point counts of moduli spaces of curves over finite fields. Finally, a given algebraic curve can map to different spaces in different ways. The study of these different concrete realizations, known as Brill-Noether theory, is essential to understanding curves. While the Brill-Noether theory of general curves is well understood, the theory breaks down for special curves. Building upon the PI's earlier work for curves of low gonality, the third research direction is to further develop the Brill-Noether theory of special curves. 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.