Title: Moduli spaces of surfaces: geometric and arithmetic properties via examples.
Abstract: Unlike moduli spaces of curves, properties of moduli spaces of surfaces are much less understood. I will introduce examples of moduli spaces of surfaces, where we can find not only geometric and arithmetic properties, but also applications outside of moduli spaces of surfaces.
Title: Knot invariants, categorification and representation theory
Abstract: I will give a brief survey talk (geared towards non-experts) about central themes related to my recent research. First, I will recall basic facts about the representation theory of the Lie algebra sl2 and discuss how this relates to the construction of knot invariants such as the well-known Jones polynomial. Secondly, I will introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. The goal of the talk will be to explore how these two topics are related. More precisely, I will discuss how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. Vice versa, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant.
Title: Topological and number-theoretic invariants in algebraic geometry
Abstract: This will be a two part talk. The first half will be the story of conductors and discriminants, which are two closely-related invariants that appear in many deep conjectures in number theory such as the Birch and Swinnerton-Dyer conjecture and Szpiro’s conjecture. The conductor appears in varied guises in related areas of mathematics. In number theory, it shows up as a measure of ramification of natural Galois representations. In algebraic geometry, it can be viewed as an intersection number with the locus of singular curves inside the moduli space of curves. In algebraic topology, it is a sum of Milnor numbers. Despite their close relationship, there are surprisingly few tools to explicitly compare conductors and discriminants of curves. For elliptic curves (genus 1), the Ogg-Saito formula shows that the conductor equals the discriminant. In genus 2, Qing Liu showed that the conductor is less than or equal to the discriminant. In this talk, we will introduce a combinatorial refinement of the discriminant, which allows us to extend Qing Liu's inequality to hyperelliptic curves of higher genus.
In the second half, we will give a brief introduction to A^1 enumerative geometry. A^1 enumerative geometry is a new field that uses modern tools from algebraic topology to study classical algebraic geometry problems over non-algebraically closed fields. Here is one example of such a problem. It is a celebrated 19th century result of Salmon and Cayley that every smooth cubic surface over the complex numbers has exactly 27 lines on it. Over the real numbers, these lines come in two types, the so-called "hyperbolic" and "elliptic" lines. It was observed by Finashin–Kharlamov and Okonek–Teleman that while the number of real lines on a smooth cubic surface depends on the surface, the difference of the two types of lines is always 3. Recently, Kass and Wickelgren give a new proof of this observation using the A^1 enumerative geometry package, which gives an "enriched" count of the lines valued in quadratic forms. (Hint: it has to do with the unique rank 27 non-degenerate quadratic form over the real numbers with signature 3.) In this talk, we will explain a similar enrichment (joint with Wickelgren) for the number of lines meeting four lines in 3-space.