• Speaker: Bin Sun, Oxford
• Title: $L^2$-Betti numbers of fiber bundles
• Date: 09/20/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Vijay B Higgins (higgi231@msu.edu)

We study the $L^2$-Betti numbers of fiber bundles $F \rightarrow E \rightarrow B$ of manifolds. Under certain conditions (e.g., when $F$ is simply connected), $b_*^{(2)}(E)$ can be computed using the twisted $L^2$-Betti numbers of $B.$ We relate the twisted and untwisted $L^2$-Betti numbers of $B$ when $\pi_1(B)$ is locally indicable. As an application, we compute $b_*^{(2)}(E)$ when $B$ is either a surface or a non-positively curved $3-$manifold. This is a joint work with Dawid Kielak.

• Speaker: Daniel Douglas, Yale
• Title: Dimers, webs, and local systems
• Date: 10/11/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Vijay B Higgins (higgi231@msu.edu)

For a planar bipartite graph G equipped with a SLn-local system, we show that the determinant of the associated Kasteleyn matrix counts “n-multiwebs” (generalizations of n-webs) in G, weighted by their web-traces. We use this fact to study random n-multiwebs in graphs on some simple surfaces. Time permitting, we will discuss some relations to Fock-Goncharov theory. This is joint work with Rick Kenyon and Haolin Shi.

• Speaker: Ka Ho Wong, Texas A&M
• Title: On the 1-loop conjecture of fundamental shadow link complements
• Date: 11/01/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Vijay B Higgins (higgi231@msu.edu)

The 1-loop conjecture proposed by Dimofte and Garoufalidis suggests a simple and explicit formula to compute the adjoint twisted Reidemeister torsion of hyperbolic 3-manifolds with toroidal boundary in terms of the shape parameters of any ideal triangulation of the manifolds. In this talk, I will give a brief overview of the conjecture and present our recent result on the 1-loop conjecture for fundamental shadow link complements. This is a joint work with Tushar Pandey.

• Speaker: Calvin McPhail-Snyder , Duke University
• Title: Hyperbolic tensor networks and the volume conjecture
• Date: 11/08/2022
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: Efstratia Kalfagianni (kalfagia@msu.edu)

Quantum invariants of links like the colored Jones polynomial (which arise from the quantum Chern-Simons theory of Witten-Reshetikhin-Turaev) have a purely algebraic construction in terms of the representation theory of quantum groups. Despite this algebraic nature they appear to be connected to geometry: a class of related volume conjectures assert that their semi-classical asymptotics determine geometric invariants like the hyperbolic volume. To better understand these conjectures a number of authors have studied ways to twist quantum invariants by geometric data. In particular, Blanchet, Geer, Patureau-Mirand, and Reshetikhin recently defined quantum holonomy invariants depending on a link in S^3 and a flat 𝔰𝔩₂ connection on its complement. Their construction uses certain unusual cyclic modules of quantum 𝔰𝔩₂. For technical reasons the invariants are quite difficult to compute. In this talk (based on joint work with Nicolai Reshetikhin) I will explain how to effectively compute them using hyperbolic tensor networks constructed from quantum dilogarithms. Our construction reveals deep connections with hyperbolic geometry and suggests a way to break the Kashaev-Murakami-Murakami volume conjecture into two simpler pieces.

• Speaker: Justin Lanier, University of Chicago
• Title: Mapping class groups and dense conjugacy classes
• Date: 11/29/2022
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

I’ll start by introducing infinite-type surfaces—those with infinite genus or infinitely many punctures—and the emerging study of their mapping class groups. One difference from the finite-type setting is that these mapping class groups come with natural non-discrete topologies. I’ll discuss joint work with Nick Vlamis where we fully characterize which surfaces have mapping class groups with dense conjugacy classes, so that there exists an element that well approximates every mapping class, up to conjugacy.