Brandeis Topology Seminar, Fall 2024

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Dani Álvarez-Gavela (dgavela@brandeis.edu) Kiyoshi Igusa (igusa@brandeis.edu), Thomas Ng (thomasng@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

Septemer 03: No seminar

September 10: No seminar

September 17: Dani Álvarez-Gavela (Brandeis)

Title: Flexibility of singularities

Abstract:  I will give a gentle introduction to the world of singularities, explain how to get rid of them in favorable circumstances, and discuss applications. 

September 24: Oleg Lazarev (UMass Boston)

Title: Weinstein domains: a symplectic geometer's handlebodies

Abstract: Weinstein domains are symplectic analogs of smooth handlebodies and come equipped with decompositions with elementary symplectic pieces. As a result, they are easy to construct, have computable invariants, and include classical examples like cotangent bundles. After giving some background, I will survey several questions (and recent answers) about Weinstein domains, many of which are motivated by analogous questions in smooth topology and have categorical interpretations. For example, the minimal number of Weinstein handles in a Weinstein domains is related to the Grothendieck group of its Fukaya category while the minimal number of "elementary" Weinstein sectors needed to cover a Weinstein domain is related to the Rouquier dimension of its Fukaya category; in the case of the cotangent bundle of M, this Rouquier dimension is bounded by the Lusternik-Schnirelmann-category of M. 

October 01: No seminar

October 08:  Catherine Pfaff (Queens University and IAS)

Title: Train track automata for outer automorphisms of free groups and geodesics in outer space

Abstract: The outer automorphism group of the free group Out(F_r) acts as the isometry group on the deformation space of weighted graphs, Culler-Vogtmann Outer space CV_r. The train track theory of Bestvina-Feighn-Handel bridges studying topological representatives of the group elements and geodesics in this space it acts on. We use the asymptotic conjugacy class invariant of the Handel-Mosher ideal Whitehead graph to “stratify” the space of geodesics, and the dynamically minimal “fully irreducible” outer automorphisms, into train track automata for different ideal Whitehead graphs.

October 15:

October 22: No seminar 

October 29:  No seminar

November 05: Kyle Hayden (Rutgers - Newark)

Title: Circling in on exotic aspherical 4-manifolds

Abstract: The Borel conjecture predicts that that the homeomorphism type of a closed, aspherical manifold is determined by its fundamental group. In the smooth category, the analogous question is true in dimension n < 4, false for n > 4, and open in dimension n = 4. In this talk, I'll describe work in progress on the 4-dimensional case, joint with Davis, Huang, Ruberman, and Sunukjian. In particular, we produce pairs of closed, aspherical 4-manifolds X,X' and a homeomorphism between them that is not homotopic to any diffeomorphism. The construction uses the "reflection trick" from geometric group theory to reduce the problem for closed manifolds to a more tractable problem about certain compact manifolds with boundary.

November 12: Emily Stark (Wesleyan)

Title:  Conformal dimension for boundaries of certain Coxeter groups

Abstract: Conformal dimension is a powerful analytic invariant of a metric space that captures the fractal behavior amongst deformations of the metric space. This invariant has made significant impact on geometric group theory and the study of spaces with negative curvature. We study a family of Coxeter groups that are built of Kleinian groups and are hyperbolic relative to virtually abelian subgroups. We give bounds on the conformal dimension for these groups. Our results imply there are infinitely many quasi-isometry classes within the family of Coxeter groups with defining graph a complete graph and edge labels greater than or equal to three. This is joint work with Elizabeth Field, Radhika Gupta, and Robbie Lyman.

November 19: Thomas Massoni (MIT)

Title: Taut foliations through a contact lens

Abstract: In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by both positive and negative contact structures. Additionally, if the foliation is taut then its contact approximations are tight.

In this talk, I will present a converse result on constructing taut foliations from suitable pairs of contact structures. While taut foliations are rather rigid objects, this viewpoint reveals some degree of flexibility and offers a new perspective on the L-space conjecture. A key ingredient is a generalization of a result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, which might be of independent interest.

November 26:  Bena Tshishiku (Brown)

Title: Hyperbolic manifolds, smooth structures, and group actions

Abstract: The Nielsen realization problem for a smooth manifold M asks when a finite group of automorphisms of the fundamental group π_1(M) is induced (or "realized") by an isomorphic group of diffeomorphisms of M. We consider this problem for manifolds that are homeomorphic, but not diffeomorphic, to a hyperbolic manifold. This case was first studied by Farrell-Jones who showed that not every group is realized, answering a question of Schoen-Yau. Our work focuses on determining precisely which groups are realized and classifying the realizations. This is joint work with Mauricio Bustamante.

December 03:  Ethan Dlugie (Brown)

December 10: