Titles and abstracts of plenary talks

Christin Bibby
Title: Supersolvable posets and fiber-type arrangements
Abstract: We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell–Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. We obtain a combinatorially determined class of K(π,1) spaces, and under a stronger combinatorial condition prove a factorization of the Poincar ́e polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk–Randell’s formula relating the Poincar ́e polynomial to the lower central series of the funda- mental group. This is joint work with Emanuele Delucchi.

Christy Hazel
Title: The Bredon cohomology of equivariant configuration spaces
Abstract: The configuration space of points in a Euclidean space has rich structure. In this talk we explore an equivariant analog, considering configurations of points in a G-representation where G is a finite group. The configuration space inherits an action of the group G, and we compute the RO(G)-graded Bredon cohomology. We’ll review the classical singular cohomology computations by Arnold and Cohen, discuss the basics of Bredon cohomology, and then present new techniques used to compute the equivariant cohomology of ordered configurations of points in G-representations. This is joint work with Dan Dugger.

Neil Hoffman
Title: Complexity and decision problems for 3-manifolds
Abstract: With the resolution of the Geometrization Conjecture, it is a solvable problem to identify a 3-manifold given one of its triangulations. The question now becomes how long such a procedure can take. After giving the necessary background, I will discuss what is currently known about some of the most fundamental problems in the field and give some families of ex- amples that serve to show how complexity grows with the number of tetrahedra of the input triangulation. This talk draws on joint work with Paul Fili, Robert Haraway, Kate Petersen, and Eric Samperton.

Morgan Opie
Title: Applications of higher real K-theory to vector bundle enumeration
Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking bundle-theoretic questions. However, in general, there are many non- equivalent bundles that represent the same class in K0. Bridging the gap between K-theory and actual bundles is challenging even for the simplest CW complexes.
For example, given a fixed rank r, the number of rank r bundles on CPn that are stably trivial is, in general, unknown. In this talk, we give lower bounds for the number of rank r bundles on CPn in infinitely many cases. For example, we can show that there are at least p rank p − 1, stably-trivial bundles on CP2p−1 for all primes p.
Our story will start with classical results of Atiyah and Rees for rank 2 bundles on CP3, before taking a detour through Weiss calculus. Building on work of Hu, we will use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real K-theory of particular simple spectra. The result will involve actual numbers! This is joint work with Hood Chatham and Yang Hu.

Katherine Raoux
Title: A 4-dimensional rational genus bound
Abstract: The minimal genus question asks: “What is the minimum genus of a surface repre- senting a particular 2-dimensional homology class?” Historically, minimal genus questions have focused on 2-dimensional homology with integer coefficients. In this talk, we study a minimal genus question for classes with Q mod Z coefficients. We define the rational 4-genus of knots and present a lower bound in terms of Heegaard Floer tau invariants. Our bound also leads to PL slice genus bounds. This is joint work with Matthew Hedden.

Noelle Sawyer
Title: The Boundary at Infinity and Geodesic Currents
Abstract: I plan to spend time talking about the boundary at infinity, geodesic currents (a measure on the space of geodesics), and why they’re both interesting tools to help to understand geometric notions. Afterwards, I will go more in depth about how I use these tools: Given a sur- face, the marked length spectrum (MLS) is the collection of the lengths of the closed geodesics, with each length marked by the free homotopy class it belongs to. Under certain conditions, information on most of the MLS is enough to completely determine the metric of the surface; I will give some intuition on how and why geodesic currents tie into the proof. If time allows, I will also give examples of some interesting length spectrum results.

Titles and abstracts of rodeo talks

Arka Banerjee
Title: Thickening finite complexes into manifolds
Abstract: Given an aspherical simplicial complex, what is the minimum dimension of a mani- fold that is homotopy equivalent to that complex? Does this number remain the same if we take finite index cover of the complex? I will describe a class of finite aspherical complexes where the answer to the second question is no.

Omar Dennaoui
Title: Contractibility of the orbit space of a fusion system after Steinberg
Abstract: Recently, Steinberg proved that the orbit space of the p-subgroup complex of a finite group is contractible using Brown-Foreman discrete Morse theory. We extend Steinberg’s argument to show the contractibility of the orbit space of the p-subgroup complex of a group that is pseudo finite at the prime p. We then apply the result to show that the orbit space of a saturated fusion system is contractible.
Authors: Omar Dennaoui and Jonathon Villareal

Hannah Housden
Title: Diagram Actions on Topological Spaces
Abstract: The notion of group action on a topological space can be generalized to the notion of an action by a category. This talk will explore the basic notions and discuss what happens in the case of a category with two objects with one non-identity morphism. As it turns out, there are highly nontrivial homotopical structures here, despite the category being very simple.

Khanh Le
Title: Totally geodesic surfaces in knot complements
Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In this talk, I will present examples of infinitely many non- commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly k totally geodesic surfaces for every positive integer k. This is a joint work with Rebekah Palmer.

Chloe Lewis
Title: Algebraic structures in Real topological Hochschild homology
Abstract: In the trace methods approach to algebraic K-theory, we work with more computa- tionally accessible invariants of rings and their topological analogues. Previously, it was shown that one important invariant in this trace methods story - topological Hochschild homology - can be endowed with the structure of a Hopf algebra. Knowledge of this structure is useful in calculations which approximate algebraic K-theory. In this talk, we’ll investigate an equivariant generalization of THH called Real topological Hochschild homology which encodes the C2-action of involution and give a description of the algebraic structures that are present.

Keith Mills
Title: The Structure of the Spinh Bordism Spectrum
Abstract: Spinh manifolds are the quaternionic analogue to spinc manifolds. The spinh bordism groups are isomorphic to the homotopy groups of the spinh bordism spectrum MSpinh. At the prime 2, these homotopy groups are determined by the structure of M = H∗(MSpinh;Z/2Z) as a module over the mod 2 Steenrod algebra. In this talk, we will discuss a structure theorem for M that decomposes it into well-known pieces, its consequences for the homotopy groups of the spinh bordism spectrum, and describe an algorithmic counting process to obtain any particular homotopy group of interest.

Prayagdeep Parija
Title: Random quotients of hyperbolic groups and property (T)
Abstract: What does a typical quotient of a group look like? Gromov looked at the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d<1/2, the typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov’s result to show that for d<1/2, the typical quotient of many hyperbolic groups is also non-elementary hyperbolic.
Z ̇uk and Kotowski–Kotowski proved that for d > 1/3, a typical quotient of a free group has Property (T). We show that (in a closely related density model) for 1/3<d<1/2, the typical quotient of a large class of hyperbolic groups is non-elementary hyperbolic and has Property (T). This provides an answer to a question of Gromov (and Ollivier).

Hanh Vo
Title: Short geodesics with self-intersections
Abstract: We consider the set of closed geodesics on a hyperbolic surface. Given any non- negative integer k, we are interested in the set of primitive essential closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.