Topology Festival

Given a Riemann surface X, we can equip it with a holomorphic differential w. The pair (X,w) is called a translation surface. Translation surfaces are equivalent to a collection of polygons in the complex plane with sides identified pairwise by translation (defined up to cut-and-paste operations). This induces a flat cone metric on X, and the singularities of w coincide with the cone points of (X,w). A stratum is the space of all translation surfaces with a prescribed number of cone points and list of cone point angles. In this introductory talk, we will study examples of translation surfaces and describe properties of strata such as their local coordinates, their components, and their natural GL(2,R) action.

An enumerative problem naturally induces a generically finite and generically étale cover $Y \rightarrow X$, where $Y$ is the moduli space of solutions and $X$ is the moduli space of geometric conditions. The degree of this map gives the classical solution count, but the deeper structural relationships between these solutions are controlled by the cover’s monodromy (or Galois) group. In particular, it determines whether the equations defining the solutions can be explicitly written down and solved by radicals in terms of the base parameters.

In this talk, I will introduce monodromy groups in enumerative geometry, explaining how they are defined and why they are essential for understanding the intrinsic geometry and splitting behavior of these problems. I will present several illustrative examples and conclude by outlining how upgrading our moduli spaces from schemes to stacks serves as a valuable tool for studying monodromy problems.

The moduli space M_{g,n} parametrises genus-g, n-marked Riemann surfaces. It is not compact, but by allowing degenerate nodal Riemann surfaces — obtained by pinching loops to nodes — we obtain the Deligne-Mumford compactification. The compactification decomposes into strata corresponding to the different combinatorial types of degenerations. We present this story from a topological perspective, and explain how it lifts to the universal cover of M_{g,n}, the Teichmüller space T_{g,n}, resulting in the augmented Teichmüller space introduced by Bers.

Geometrization is a powerful principle which has shaped our understanding of irreducible 3-manifolds and, more recently, the structure of their diffeomorphism groups. However, diffeomorphism groups of reducible 3-manifolds remain somewhat elusive. Inspired by an approach of Hatcher, we construct a “splitting map” from the classifying space of the diffeomorphism group of a reducible 3-manifold to that of its irreducible prime factors. This results in a “prime decomposition fiber sequence” whose fiber we describe explicitly as a homotopy colimit over a finite category of graphs.  We also develop a framework for computation, which we apply to calculate the rational cohomology ring of the classifying space in specific examples. This is joint work with Rachael Boyd and Jan Steinebrunner.

Quadratic Gromov--Witten invariants allow one to obtain an arithmetically meaningful count of genus 0 curves satisfying constraints over a field k without assuming that k is the field of complex or real numbers. They were developed in joint work with Kass, Levine, and Solomon. They take values in the Hermitian K-theory of k. In this talk we will compute structures on these invariants, which for example compute all the invariants for the projective plane. This is joint work with Erwan Brugallé and Johannes Rau.

The moduli space M_{g,n}^{trop} of metric graphs is a polyhedral object. It is “the tropicalization of” the moduli space M_{g,n} of Riemann surfaces, as I will explain. Teichmuller space T_{g,n} is the space of Riemann surfaces together with a topological marking. It is the universal cover of M_{g,n}. Culler-Vogmann Outer space CV_{g,n} is the space of metric graphs together with a topological marking, and lives over M_{g,n}^{trop}. I will discuss the extent to which “tropicalization” makes sense in the context of spaces like T_{g,n} and CV_{g,n}. This talk is based on joint work with Rob Silversmith, Karen Vogtmann, and Rebecca Winarski.

I'll sketch the proof of the theorem with Kent on the existence of atoroidal surface bundles over surfaces, and if there is time, describe our conjecture regarding the coarse geometry of such bundles.