Fall 2020 Redbud Topology Conference

The talks are on Zoom, and Gathertown is available for socializing between talks.

Zoom link:

Meeting ID: 918 4731 4850
Passcode: C7DMd7VL

Gathertown links:
You may need to use Chrome or Firefox. There is a 25 person limit per room.

Schedule:

Time	Speaker	Talk
8:30 − 9:00	virtual coffee + bagels
9:00 − 9:45	Neil Hoffman	Conjectures related to knot complement commensurability
10:00 − 10:45	Nick Castro	Isotopy classes of relatively trisected 4-manifolds with boundary
11:00 − 11:45	Derrick Wigglesworth	The Farrell-Jones Conjecture for (most) hyperbolic-by-cyclic groups
	lunch break
1:30 − 2:15	Hitesh Gakhar	Künneth formulae in persistent homology
2:30 − 3:15	Wenwen Li	Multi-persistent homology of restricted configuration spaces of metric graphs
3:30 − 4:15	Henry Segerman	Raytracing and raymarching simulations of non-euclidean geometries
4:30 − ?	social hour

Abstracts:

Nick Castro, University of Arkansas
Title: Isotopy classes of relatively trisected 4-manifolds with boundary
Abstract: A relative trisection of a smooth, compact, oriented 4-manifold with boundary \(X\) is a decomposition of \(X\) into three diffeomorphic pieces which have "nice" intersection properties. The trisection induces an open book decomposition on the boundary, which is a surface bundle over \(S^1\) in the compliment of a link in \(\partial X\). It is known that every such 4-manifold admits a trisection and that any two trisections can be made isotopic after suitable "stabilization" operations. In this talk, I will show that any two diffeomorphic relative trisections of the 4-ball which induce isotopic open books on the boundary 3-sphere are in fact isotopic trisections. An interesting feature of the argument is that we do not show that the original diffeomorphism is isotopic to the identity! I will give a good deal of background on trisections, trisection diagrams, and open books. If time permits, I will discuss some practical features of relative trisections which allow us to classify low-"complexity" relative trisections. This work is joint with Patrick Naylor.

Hitesh Gakhar, University of Oklahoma
Title: Künneth formulae in persistent homology
Abstract: The classical Künneth formula in algebraic topology provides a relationship between the homology of a product space and that of its factors. In this talk, we will give similar results for persistent homology---an algebraic computational tool used for feature detection in topological data analysis. That is, we will provide relationships between persistent homology of two different notions of product filtrations and that of their factor filtered spaces. We will also present an application in topological time series analysis.

Neil Hoffman, Oklahoma State University
Title: Conjectures related to knot complement commensurability
Abstract: Two manifolds \(M_1\) and \(M_2\) are commensurable if there is third manifold \(M_3\) that is a finite sheeted cover of \(M_1\) and \(M_2\). Neumann and Reid conjecture that at most 3 hyperbolic knot complements can be commensurable with each other. I will discuss what is known about the conjecture and open questions surrounding commensurable knot complements.

Wenwen Li, University of Oklahoma
Title: Multi-persistent homology of restricted configuration spaces of metric graphs
Abstract: As a continuation of the previous work by James Dover and Murad Özaydın, we consider a finite connected metric graph \(X\). We identify \(X\) with its geometric realization, which is a metric space via the path metric. The \(n\)-th restricted configuration space of a metric space \(X\) with restraint parameter \(\mathbf{r}\) is \(X_{\mathbf{r}}^{n}:=\{(x_{1},\dots, x_{n})\in X^{n}\mid d(x_{i},x_{j})\geq r_{ij}\}\) where \(\mathbf{r} = (r_{ij})_{i\lt j}\) and \(r_{ij}>0\). If \(r_{ij}=2a\) for all \(i,j\) then this is also called the configuration space of thick particles or hard disks (of radius \(a\)). These spaces come up in the area of topological robotics, states of matter, etc. In this ongoing project, we seek to understand how the topology of \(X_{\mathbf{r}}^{n}\) varies with \(n\), \(\mathbf{r}\), and the edge lengths of \(X\) as measured by the multi-persistent homology of \(X_{ \mathbf{r}}^{n}\).

Henry Segerman, Oklahoma State University
Title: Raytracing and raymarching simulations of non-euclidean geometries
Abstract: I'll talk about two related projects, with two different groups, both aiming to see three-dimensional manifolds "from the inside". That is, we generate images assuming that light travels along geodesics in the geometry of the manifold. The first project, with Rémi Coulon, Sabetta Matsumoto, and Steve Trettel, uses ray-marching to generate the inside-view in all eight Thurston geometries. I'll explain the ray-marching technique, and some aspects of our implementation. The second project, with David Bachman, Matthias Goerner, and Saul Schleimer, visualizes "cohomology fractals" in hyperbolic three-manifolds. These images come from cohomology classes in the manifold, and are closely related to the sphere-filling curves discovered by Cannon and Thurston.

Derrick Wigglesworth, University of Arkansas
Title: The Farrell-Jones Conjecture for (most) hyperbolic-by-cyclic groups
Abstract: The Farrell-Jones Conjecture for a group G has its origins in the study of high dimensional manifolds, and provides a method to understand the spectrum of manifolds whose fundamental group is G. Further, FJC provides a method to study the algebraic K- and L- groups of many group rings. After providing some background and motivation, we will discuss some geometric-group-theoretic techniques that have been developed to prove FJC for certain groups and classes of groups. Building on previous work, I will then discuss a proof of the Farrell-Jones conjecture for (torsion-free hyperbolic)-by-cyclic groups. This is joint work with Mladen Bestvina and Koji Fujiwara.