Brandeis Topology Seminar, Fall 2025

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Dani Álvarez-Gavela (dgavela@brandeis.edu) Kiyoshi Igusa (igusa@brandeis.edu), Thomas Ng (thomasng@brandeis.edu), Surena Hozoori (hozoori@brandeis.edu)

September 02: [cancelled]

September 09: Agniva Roy (Boston College)

Title: Legendrian surfaces and Cluster algebras

Abstract: Cluster algebras have recently played a major part in breakthrough results (Casals - Gao, Casals - Weng, Shende - Treumann - Williams - Zaslow) regarding Legendrian links and their exact Lagrangian fillings. This interaction has also given a geometric interpretation to cluster algebras and their properties. We show that some of this correspondence extends to Legendrian surfaces in R^5, and allows for a geometric interpretation of the cluster phenomenon of "folding". These surfaces are built via the twist-spinning construction defined by Ekholm and Kalman. We are able to show "seeds = exact Lagrangian fillings" for these surfaces, make a conjecture on the number of exact fillings of these surfaces, and obstruct fillability in certain cases. We further explore how, in nice cases, the cluster correspondence affects the isotopy characterization of a different class of Legendrian surfaces called Legendrian doubles, also defined by Ekholm. This is joint work with James Hughes (Duke).

September 16: Thomas Hill (Utah)

Title: Automorphisms of the sphere complex of an infinite graph

Abstract:  For a locally finite, connected graph, we consider the group of proper homotopy equivalences of the graph up to proper homotopy. Informally, this group is sometimes referred to as 'big Out(Fn)'. Excluding sporadic cases, we show that 'big Out(Fn)' is isomorphic to the automorphism group of the sphere complex of the double handle body associated to the graph. This is joint work with Michael Kopreski, Rebecca Rechkin, George Shaji, and Brian Udall.

September 23: No seminar

September 30: Stefanie Zbinden (Max Planck)

Title: Using strong contraction to obtain hyperbolicity

Abstract: If a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct an action on a hyperbolic space where said element acts loxodromically. However, the action depends on the chosen element and other strongly contracting elements are not necessarily loxodromic. It raises the questions whether there exists a single action on a hyperbolic space where all strongly contracting elements act loxodromically. In this talk, we answer the above question positively by introducing the contraction space construction.

October 07: No seminar

October 16*: Olu Oloronde (Cornell)

*note Thursday seminar

Title: An Improved Combination Theorem for A/QI Triples

Abstract: Let G be a hyperbolic group. A classical theorem of Rita Gitik states that given subgroups H and K that are quasi-convex in G, where the "short" words belong to their intersection, we have that the subgroup generated by H and K is quasi-convex in G and is isomorphic to the amalgamated free product of H and K over their intersection. Another situation of interest is where a group may not act geometrically on a hyperbolic space, yet it has a "hyperbolic-like" action. In this talk, I will discuss a generalization of a combination theorem of Carolyn Abbott and Jason Manning that covers a broader class of geometrically natural subgroups of such a group. This is still a work in progress.

October 21: Kenny Blakely (MIT)

Title: Stable splittings of free loop spaces via symplectic geometry

Abstract: It is a classical algebro-topological fact that the free loop space of the 2-sphere stably splits as a wedge sum of spheres and suspensions of projective planes. Viterbo's isomorphism relates the topology of a free loop space to the symplectic geometry of a cotangent bundle. Moreover, smooth divisor complements in projective varieties provide nice examples of symplectic manifolds; in particular, we can realize the cotangent bundle of the 2-sphere as the complement of a quadric hypersurface in complex projective 3-space. In this talk,  we will describe how to compute Floer homotopy types of smooth divisor complements, and we will touch on how the obstruction to splitting as a wedge sum is encoded in a spectral Gromov-Witten invariant relative to the divisor; among other examples, this will recover various stable splittings of free loop spaces.

October 28: Chi Cheuk Tsang (Université du Québec à Montréal

Title: Minimum entropies of braids

Abstract: Every braid can be thought of as a homeomorphism of a punctured disc. Morally, the more complicated a braid is, the more dynamics is contained in the corresponding homeomorphism, which one can quantify using topological entropy. On the other end of the spectrum, the minimum entropy of braids (with a fixed number of strands) can be thought of as the smallest amount of mixing one can perform on a disc while still doing something topologically interesting. In this talk, we will present joint work with Xiangzhuo Zeng in finding the value of this minimum entropy.

November 04: John Klein (Wayne State)

Title: Stochastic dynamics on a CW complex

Abstract: Let X be a finite CW complex X of dimension d. I will describe a stochastic process whose states are the cellular (d-1)-cycles in X. A stochastic transition corresponds to hopping across a d-cell. This is a stochastic process without memory, and consequently it is termed a “CW Markov chain.”

Given a CW Markov chain, I’ll associate a homological observable called the average current. The latter measures the average flux of the probability in the process. In the “low temperature, adiabatic limit,” the average current fractionally quantizes, in which the denominators are combinatorial invariants of the CW complex.†

(An effort will be made to make this talk accessible.)

†These invariants are related to Reidemeister torsion.

November 11: Antonio Alfieri (UGA)

November 18: Mona Merling (UPenn)

November 25: No seminar

December 02: Cary Malkiewich (Binghamton)

December 09: Miriam Kuzbary (Amherst College)