Combinatorial Methods in Algebraic Geometry

University Of California-Davis, Davis CA

Investigators

Abstract

This research project concerns algebraic aspects of a range of surprising recent conjectures due to several authors, relating topics in algebraic geometry, the algebraic theory of knots, topology, difficult combinatorial (counting) problems, number theory, and mathematical physics. Conjectures that relate such a broad range of topics are often especially compelling to mathematicians, and can lead to particularly powerful results. To give one example, mathematicians often find it useful to "count points" on spaces that parametrize mathematical objects. So-called HLV conjectures mentioned below predict not only a formula for doing this for certain spaces commonly called "character varieties," but also generalize them in a way that is connected to their topology. Other conjectures in this family connect closely related spaces with some extremely compelling formulas involving diagrams, called parking functions, that are elementary to test by hand. While this project is based on algebraic methods, a full mathematical understanding of this topic is expected to reveal the geometry behind many deep open problems, some of which have roots in physics. The investigator also plans to involve undergraduate and graduate researchers in the project. This activity will focus on combinatorial methods that require minimal student prerequisites, and on the creation of algebra software for conducting computational experiments, an especially effective approach for student researchers unfamiliar with these topics. The development of general computer software is another impact of this project, which is expected to be useful to researchers in computational fields. Some of the topics this project examines are the cohomology of the affine Springer fiber, Khovanov-Rozanksy knot invariants, some famous conjectures of Hausel, Letellier, and Rodriguez-Villegas (HLV), conjectures relating four-dimensional gauge theory to conformal theory due to Alday, Gaiotto, and Tachikawa (AGT), and related combinatorial extensions of the proof of the shuffle conjecture, such as the nabla-positivity conjecture. On one side of these conjectures, nearly all these topics have in common (conjectured) relationships with sheaves on the Hilbert scheme of points in the complex plane. On the other side, they are connected by the presence of a Riemann surface whose significance is hidden on the Hilbert scheme side, except through formulas. For instance, this Riemann surface would be the punctured disc C^* in the example of the Springer fiber, the punctured surface of genus g defining the character variety in the case of the HLV conjectures, or the two-dimensional surface on which the conformal field theory takes place in the case of AGT. The goal of this project is to make progress towards mathematical proofs of these conjectures, discover new ones, and ultimately understand the general mathematical picture. A major aspect of the approach is to extrapolate from explicit combinatorial formulas when they are available, such as the sort that appear in the shuffle conjecture, often called "nabla formulas" in Macdonald theory. Understanding this connection is of considerable interest to number theory, algebraic geometry, and combinatorics. A second aspect is the creation of sophisticated computer software for testing new conjectures, as well as for generating data to make predictions about the general relationship with geometry. 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.