List of Titles and Abstracts

Talks

Alexander Berglund: The Lie graph complex and Poincaré duality fibrations

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to a fibration with Poincaré duality fiber. This higher structure can be used to relate seemingly disparate problems in commutative algebra and differential topology: on one hand, the problem of finding multiplicative structures on free resolutions and, on the other hand, the problem of promoting Poincaré duality fibrations to smooth bundles.

Madeline Brandt: The weight-0 compactly supported Euler characteristic of moduli spaces of marked hyperelliptic curves

Deligne connects the weight-zero compactly supported cohomology of a complex variety to the combinatorics of its compactifications. In this talk, we use this to study the moduli space of n-marked hyperelliptic curves. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for its weight-0 compactly supported Euler characteristic, as a virtual representation of S_n. This is joint work with Melody Chan and Siddarth Kannan.

Francis Brown: Moduli of tropical curves and stable and unstable cohomology of GL_n(Z)

I will talk about several versions of invariant differential forms on symmetric spaces, and their uses for constructing non-vanishing cohomology classes. I will discuss their manifestations as differential forms on moduli spaces of tropical curves, and compare and contrast with the situation for the general linear group. Depending on what has been covered in previous talks, I will cover certain aspects of joint work with Schnetz, with Chan, Galatius and Payne, and with Hu and Panzer.

Benjamin Brück: (Non)-vanishing of high-dimensional group cohomology

A conjecture by Church-Farb-Putman predicts that the rational cohomology of SL_n(Z) vanishes in high degrees. I will talk about the current status of this conjecture and about techniques that have been used to obtain partial solutions to it. I will then give an overview of analogous (non-)vanishing results for the cohomology of related groups and moduli spaces.

Sam Canning: Cycles on moduli spaces of curves and abelian varieties

I will show how the study of non-tautological classes on the moduli space of abelian varieties helps explain the structure of the tautological ring of the moduli space of curves of compact type. On the curves side, this is joint work with Hannah Larson and Johannes Schmitt, and on the abelian varieties side, it is joint with Dragos Oprea and Rahul Pandharipande.

Melody Chan: Hopf structures in the cohomology of moduli spaces of abelian varieties

I will discuss aspects of joint work with Francis Brown, Søren Galatius, Sam Payne, in which we identify a Hopf structure on the weight 0 subspace of the compactly supported cohomology of the moduli space of abelian varieties and deduce a number of consequences.

Danica Kosanovic: Diffeomorphisms of 4-manifolds

Very little is known about homotopy groups of diffeomorphisms of 4-manifolds. The breakthrough work of Watanabe showed that there are many nontrivial classes in the case of S^4, detected by the Kontsevich configuration space integrals, an invariant with values in a graph homology. In this talk I will explain how Watanabe’s classes come from a simple construction involving families of embedded circles.

Shiyue Li: K-rings of wonderful varieties and matroids

I will share some discoveries on the K-rings of wonderful varieties and matroids. The main result is a Hirzebruch—Riemann—Roch-type theorem. I will also discuss applications to the K-rings of moduli spaces of genus zero curves. Joint work with Matt Larson, Sam Payne and Nick Proudfoot.

Florian Naef: Simple homotopy types in string topology

Reidemeister and Whitehead gave a completely algebraic description of finite CW complexes up to cell collapses (aka simple homotopy types). We will see how a weakening of this structure (namely its trace) enters into the construction of an operation on the homology of the free loop space (the loop coproduct). We will also see various ways how to encode and extract such a "trace"-simple homotopy type, one of which is closely related to the notion of a homotopy Frobenius algebra.
This is joint work with Pavel Safronov.

Erik Panzer: The Pfaffian form and the odd commutative graph complex

I will explain how the Pfaffian of an anti-symmetric matrix gives rise to a closed differential form on the space of positive definite symmetric matrices with even rank. Pulling back this form onto the space of metric graphs produces convergent integrals associated to combinatorial graphs. By Stokes’ formula, these integrals define an infinite family of cocycles of the odd commutative graph complex. A calculation of the first of these cocycles shows that it is not exact. This provides a construction of potentially new graph cohomology classes, and gives a geometric description of a twisted differential of Koroshkin-Willwacher-Zivkovic . This is joint work with Francis Brown and Simone Hu.

Sam Payne: Motivic structures in the cohomology of M_g

I will discuss the weight spectral sequence for the Deligne-Mumford compactification of M_g and its interpretation as a graph complex, along with recent work applying this perspective to obtain new results about the cohomology of M_g and its associated Hodge structures and l-adic Galois representations.

Based in part on joint work with Sam Canning, Melody Chan, Soren Galatius, Hannah Larson, and Thomas Willwacher

Dan Petersen: Top weight cohomology of M_g and the handlebody group

Chan-Galatius-Payne have identified an enormous amount of nontrivial unstable cohomology classes on the moduli spaces of curves, via an identification of the "top weight" cohomology of the mapping class group with the cohomology of Kontsevich's graph complex. I will explain that all these classes restrict nontrivially to the handlebody subgroup of the mapping class group, i.e. those mapping classes which extend to a handlebody filling. In the process we obtain a geometrically meaningful classifying space for the handlebody group. (Joint with Louis Hainaut.)

Martin Ulirsch: Boundary complexes, moduli spaces, and tropical geometry: beyond $M_g$

In their recent works, Chan-Galatius-Payne found (and successfully built upon) a connection between the (co-)homology of the graph complex and the top weight cohomology of the moduli space $M_g$ of smooth projective curves of genus $g$. A central ingredient in their work is the identification of the boundary complex of the Deligne--Mumford compactification of $M_g$ and the moduli space of stable tropical curves of genus $g$, observed in work by Abramovich--Caporaso--Payne. This turns out to be a general principle which is applicable to numerous other moduli spaces. A non-exhaustive list includes the following examples:
- moduli of principally polarized abelian varieties
- moduli spaces of weighted stable curves
- moduli spaces of rational and elliptic stable maps to toric varieties
- moduli of spin curves
- moduli of hyperelliptic curves
- moduli spaces of cyclic covers
- universal Jacobians
In this talk I will survey the state of the art in this active area of research and, in particular, describe open problems.

Ric Wade: Handlebody mapping class groups are virtual duality groups

We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems. The proof uses Hainaut-Petersen's description of an orbifold classifying space for the handlebody group to construct a submanifold of Teichmüller space on which the handlebody group acts properly and cocompactly. We use combinatorial methods to identify the boundary of this manifold with the suspension of the complex of non simple disc systems and analyse its topology. Joint work with Dan Petersen.

Nathalie Wahl: Graph complexes in string topology

I’ll give a survey of some of the graph complexes used to describe spaces of operations in string topology, and how they relate to the moduli space of Riemann surfaces.

Ben Ward: Stirling Complexes

I will discuss several examples of graph complexes whose homology can be described, in whole or in part, in terms of the homology of configurations of points in R^3. The main computational tool will be a Koszulity result for a family of modules over the commutative operad.

Thomas Willwacher: The dg dual of BV and cohomology of the handlebody group

I describe a small cyclic homotopy operad model for the dg (Koszul-)dual of the BV operad. As an application, one can show that the Feynman transform of BV and of its dg dual are quasi-isomorphic. As a further application one can compute (to some extent) the top-2-weight cohomology of the handlebody group. The talk is based on arXiv:2311.09037 and arXiv:2308.16845 (joint with Brück and Borinsky).