CAREER: Geometry of Curves and Their Moduli Spaces

Polynomials are equations that can be built out of the elementary operations, and are therefore ubiquitous in mathematics and science. In the first nontrivial cases, when their sets of solutions are one-dimensional, these sets are known as algebraic curves; familiar examples include circles and lines. One project that the PI will pursue is the interpolation of algebraic curves through special configurations of points. As an illustration, one can draw a circle through most configurations of three points in the plane --- but the circle "disappears" when the three points specialize onto a line. The goal here is a systematic understanding of what happens, in higher dimensions as the points specialize onto a hyperplane (a higher-dimensional analog of a line in the plane), or as the points collide in various ways. The PI will complement this, and other research projects described below, by educational activities for students at several educational stages, including teaching problem-solving skills to middle and high school students through math competitions, supervising undergraduate research projects, and running a summer school in algebraic geometry for graduate students. In addition to interpolation problems for curves passing through special configurations of points, the PI will also pursue several research projects concerning the geometry of moduli spaces of curves, which are geometric objects whose points classify algebraic curves. First, the PI will study the intersection theory of these moduli spaces, which describes how different conditions one might impose on algebraic curves interact. Second, the PI will use quadratic equations satisfied by realizations of curves in projective space to more deeply understand the birational geometry of these moduli spaces. Finally, the PI will develop higher-rank Brill-Noether theory, which describes how algebraic curves can be realized in Grassmannians. 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.