VIRGINIA TECH GEOMETRY & TOPOLOGY SEMINAR (FALL 2026)

(Joint with Peter Haine) Many geometric structures can be described as \emph{algebras} for a suitable monad. E.g., compact Hausdorff spaces are exactly the algebras for the ultrafilter monad on sets. I will explain this point of view through several concrete examples, starting with free abelian groups and rings, and then discussing ultrafilters, Eilenberg--Moore categories, and Kleisli categories. The emphasis will be on examples rather than category theory. The main application is in logarithmic geometry. I will explain how logarithmic maps can be realized as Kleisli morphisms for a natural monad built from Olsson's logarithmic stack. This leads to a functorial tropicalization construction for logarithmic schemes, recovering the usual fans of toric varieties and toroidal embeddings while remaining functorial for arbitrary logarithmic maps. (This talk is meant to be accessible to graduate students)

Brill--Noether theory studies linear systems on an algebraic curve. K3 surfaces have been central in Brill--Noether theory since Lazarfeld's proof of the Brill--Noether theorem, which describes the linear series on a general curve of genus $g$. In this talk, we'll survey some recent results concerning the Brill--Noether theory of curves on K3 surfaces. Specifically, we'll discuss a refined Brill--Noether theory, which aims to understand the linear series on curves with a given Brill--Noether special linear series, and related unstable vector bundles on K3 surfaces with applications to the dimensions of components of Giesker--Petri loci in $\mathcal{M}_g$, parameterizing curves admitting a linear series with non-injective Petri map.

Many counting problems in arithmetic statistics can be formulated as counting points of bounded height on moduli stacks over global fields. Ellenberg, Satriano, and Zureick-Brown introduced a theory of heights on stacks that provides a common framework for studying these problems. In this talk, I will present a moduli-theoretic approach to heights on stacks over function fields, inspired by Abramovich and Vistoli’s theory of twisted stable maps. The idea is to construct moduli spaces parametrizing points of fixed height, translating arithmetic counting problems into questions about the geometry of these spaces. I will describe this construction for a suitable class of stacks and outline an approach to more general stacks that is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.

In [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our examples Y are log Calabi-Yau surfaces, i.e., there is a reduced normal crossing divisor D in Y such that K_Y+D=0. This is joint work with Sebastián Torres.

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