byu algebraic geometry seminar

I'll give a very broad overview of Beauville's proof of the Yau—Zaslow formula for counting rational curves of genus g on a K3 surface. The goal is to get you excited about K3 surfaces and some of the topics we might cover along the way.

Section 1.1 of these notes on projective varieties.

Section 1.1 of these notes on projective varieties.

Section 5 of these notes on sheaves.

Sections 3 and 13 of these notes on algebraic geometry.

Section 15 of these notes on algebraic geometry.

Section 16 of these notes on algebraic geometry.

This talk will be a friendly introduction to using topological invariants in enumerative geometry and how one might use equivariant homotopy theory to answer enumerative questions under the presence of a finite group action. Recent work with Kirsten Wickelgren (Duke) defines a global and local degree in stable equivariant homotopy theory that can be used to compute the equivariant Euler characteristic and Euler number. I will discuss an application to counting orbits of rational plane cubics through an invariant set of 8 points in general position under a finite group action on $\mathbb{C}\mathbb{P}^2$, valued in the representation ring and Burnside ring. This recovers a signed count of real rational cubics when $\mathbb{Z}/2$ acts on $\mathbb{C}\mathbb{P}^2$ by complex conjugation.

In this talk, I will introduce certain cut-and-paste invariants of algebraic varieties called motivic measures, or group homomorphisms from the Grothendieck ring of varieties, K_0(Var). With Zakharevich’s construction of the higher K groups of the category of varieties, there is an ongoing project to understand K_i(Var) by lifting these motivic measures (on the level of K_0) to derived motivic measures on the level of K_i for all i. I will describe some of this ongoing work, hopefully showing that there is interesting information recorded in K_i(Var), and it is understandable when i=1.

Dagger categories are categories equipped with an involution on morphisms and a notion of positivity. The most famous example is the category of Hilbert spaces, which is made out of complex vector spaces, linear maps, and the involution of taking adjoint operators. I will discuss how dagger categories let us define unitarity for functorial field theories as well as prove mathematical versions of the spin-statistics theorem in physics, which says that unitarity places constraints on the behavior of particles. This talk is inspired by joint work with Lukas Müller and Luuk Stehouwer, where we prove an extended spin statistics theorem.