Brandeis Topology Seminar, Spring 2024

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Ruth Charney (charney@brandeis.edu), Kiyoshi Igusa (igusa@brandeis.edu), Thomas Ng (thomasng@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

January 16: Tanushree Shah (University of Vienna)

Note: Location for January 16 is 116 Goldsmith
Title: Tight contact structures on Seifert fibered 3-manifolds.  

Abstract: I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.

January 23: Sam Hughes (Oxford University)
Title: Finite quotients of Coxeter groups

Abstract: In this talk I will present recent work with Samuel Corson, Phillip Möller and Olga Varghese on the question of profinite rigidity of Coxeter groups.  Here a group is profinitely rigid if it is determined up to isomorphism by its finite quotients amongst all finitely generated residually finite groups.  I will discuss two results, first that there are right-angled Coxeter groups (RACGs) which have infinitely many non-isomorphic subgroups with the same finite quotients, and second, that amongst Coxeter groups, RACGs are determined by their finite quotients.

January 30: 

February 06: 

February 13: Hokuto Konno (University of Tokyo)
Title: Families Frøyshov invariant

Abstract: The Frøyshov invariant of a rational homology 3-sphere is an important numerical invariant out of Floer theory, which allows us to generalize Donaldson's diagonalization theorem to 4-manifolds with boundary. We introduce a family version of the Frøyshov invariant. Namely, given a smooth family of rational homology 3-spheres, we define numerical invariants out of families (Seiberg–Witten) Floer homotopy types, which give constrains on smooth families of 4-manifolds bounded by the family of 3-manifolds. This is joint work with Hirofumi Sasahira.

February 20:  February break (No seminar)

February 27:  Dev Sinha (University of Oregon)
Title: Lie coalgebras and linking of letters - from geometry to homotopy theory to computational group theory

Abstract: How do cochains measure the fundamental group?  In the theory of geometric cochains,  a 1-cochain is a codimension one submanifold such as a surface in a three-manifold, which evaluates by signed count of intersection.  If one has multiple such submanifolds, one can track combinatorics of intersections, looking for invariant counts.

Over the rational numbers, such invariant counts are governed by the Harrison complex of the cochains of any space, which computes their derived indecomposibles.  Thus this non-abelian intersection theory is strengthened by the formal apparatus of rational homotopy theory, and enriched with the structure of Lie coalgebras.    The theory is quite flexible, with any space with a given fundamental group being a suitable “presentation” of the group. 

In algebra, consider two homomorphisms f, g from a free group to the rational numbers and define a homomorphism {f}g on the commutator subgroup of the free group by  {f}g ( [v,w] ) = f(v) g(w) - f(w) g(v).  This generalizes to the notion of Lie coalgebraic dual of a group, a framework for all groups which encompasses Magnus expansion and Fox derivatives for free groups.  One of our main theorems is that the Harrison complex presents the Lie coalgebraic dual of a group.   It explicitly pairs with the Malcev Lie algebra, giving a complete set of computable functionals over the rational numbers for any fundamental group and a perfect duality in the finitely generated setting.

Presentation of a group as given by generators and relations, or equivalently as the fundamental group of a two-complex, leads to counting algorithms based on combinatorics of words which we call linking of letters.  These algorithms have been coded, so that one can enter a group presentation in Sage and produce functionals which determine whether some power of a word is a commutator of a fixed depth.

March 05: Vivian He (University of Toronto/Fields Institute)
Title: Random walks on groups and superlinear divergent geodesics

Abstract: The central limit theorem of random walks answers the question "how quickly does the random walk drift away from the origin". Historically, it has been proven (under some assumptions) for free groups, hyperbolic groups, and various generalizations of hyperbolic groups. We proved this for one generalization of hyperbolic groups: groups containing superlinear divergent quasi-geodesics. The advantage of this setting compared to previous versions of CLT is that it is invariant under quasi-isometry.
In this talk, I will delve into the superlinear divergence property, as well as its geometric consequences that led to the theory of random walks on groups containing superlinear divergent quasi-geodesics. This talk is based on joint work with Kunal Chawla, Inhyeok Choi, and Kasra Rafi.

March 12:  Chaitanya Tappu (Cornell University)
Title: A Moduli Space of Marked Hyperbolic Structures for Big Surfaces.

Abstract: We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichm\"uller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.

March 19: CANCELLED

March 26: Yulan Qing (University of Tennessee Knoxville)
Title: The Quasi-redirecting Boundary

Abstract: In this talk we construct a Gromov-like boundary set from first principles for all proper geodesic spaces. We show that, if X is an asymptotically tree-graded space with special subsets, we show that the Quasi-redirecting boundary is a compact, metrizable and  quasi-isometry invariant topological space.  We show that in this setting the sublinearly Morse boundaries are topologically subspaces of the Quasi-redirecting boundary. Furthermore, we show that the Quasi-redirecting boundary naturally identify with the Bowditch boundary when the space X has cocompact action by a finitely generated group. We also compute the Quasi-redirecting boundary of the Croke-Kleiner space and discover a surprising new set of QI-invariant directions. This is based on joint work with Kasra Rafi.

April 02: Inhyeok Choi (KIAS/Fields Institute)
Title: Genericity of pseudo-Anosov mapping classes

Abstract: In the mapping class group G of a finite-type surface, each mapping class is either periodic, reducible or pseudo-Anosov. One can ask which category has the largest proportion in G. For example, when you generate a length-n word using a finite generating set of Gas an alphabet, what will be the typical outcome? Or, in a large metric ball on the Teichmüller space or the Cayley graph of G, what is the number of mapping classes in each category? In this talk, I will explain some results and methods about this question.

April 09: Caglar Uyanik (University of Wisconsin Madison)
Title: Hyperbolicity in free group extensions

Abstract: Similar to the surface group extensions and their relations to mapping class groups, the coarse geometry of free group extensions is determined by the monodromy homomorphism into the outer automorphism group of the free group. I will talk about our recent work with M. Clay toward a complete understanding of free group extensions, using the dynamics of subgroups of Out(F) on various combinatorial and metric objects.

April 16:  Nicholas Vlamis (CUNY - Queens College)
Title: The virtual Rokhlin property for topological groups

Abstract: A topological group has the Rokhlin property if it contains a dense conjugacy class; it has the virtual Rokhlin property if it contains a closed finite-index subgroup with the Rokhlin property.  I will discuss how this topological property can have algebraic consequences (e.g., obstructing the existence of abstract homomorphisms) and geometric consequences (e.g., implying boundedness properties for the group).  I will also present joint work with Justin Lanier in which we classify the surfaces whose homeomorphism groups have the virtual Rokhlin property.