Brandeis Topology Seminar, Spring 2026

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)

January 15: Seraphina Lee 

Title: Infinitely many Lefschetz pencils on ruled surfaces

Abstract: Works of Donaldson and Gompf show that a closed, oriented 4-manifold admits a symplectic structure if and only if it admits the structure of a Lefschetz pencil. However, the question of how many Lefschetz pencils (or fibrations) a given symplectic 4-manifold admits remains open. Works of Park--Yun and Baykur construct 4-manifolds admitting arbitrarily large (but finite) numbers of Lefschetz pencils or fibrations of the same genus. In this talk, we will construct infinitely many non-isomorphic Lefschetz pencils of the same genus on ruled surfaces of negative Euler characteristic. In fact, our construction gives the first example of infinitely many non-isomorphic but diffeomorphic Lefschetz pencils and fibrations of the same genus. This is joint work in progress with Carlos A. Serván.

January 22: No seminar 

January 29: Isabella Khan (MIT) 

Title: A Heegaard Floer perspective on the Z-hat invariant

Abstract:  Heegaard Floer homology is a powerful and computable 3-manifold invariant which is known to be isomorphic to Némethi’s lattice homology construction for all plumbed 3-manifolds. In this talk, I will discuss progress towards using this isomorphism to express the Z-hat invariant from quantum topology in terms of Heegaard Floer generators, and the possible applications of such a result.

February 05: Arya Vadnere (Buffalo) 

Title: Gromov Boundary of the Grand Arc Graph

Abstract: In 1999, E. Klarreich found a very intriguing correspondence between the Gromov boundary of the curve graph for closed surfaces (a very GGT object) with the space of ending laminations on the surface (a very geometric object). Since then, Hamendstädt, Schleimer and Pho-On have thought about various proofs for this result, and generalizations to the arc graph / the arc-and-curve graph for finite-type surfaces. The grand arc graph is a type of arc graph associated with certain infinite-type surfaces, which is also an infinite-diameter hyperbolic graph. In this talk, we shall talk about a couple of ways to define  “laminations that should correspond to points on the Gromov boundary of the grand arc graph”. This work is joint with Carolyn Abbott and Assaf Bar-Natan.

February 12: Edu Fernandez (Georgia Tech)

Title: The h-principle fails for prelegendrians in corank 2 fat distributions

Abstract:  It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.

In this talk I will discuss rigidity phenomena for fat distributions, which can be viewed as higher corank generalizations of contact structures. These admit natural symplectizations and contactizations. I will introduce a natural class of submanifolds in fat manifolds, called prelegendrians, which admit canonical Legendrian lifts to the contactization. The main result of the talk is that these submanifolds exhibit rigidity: in the “standard corank-2 fat manifold” there exists an infinite family of prelegendrian tori, all of them formally equivalent but pairwise not prelegendrian isotopic. In other words, the h-principle fails for prelegendrians. The talk is based on joint work with Álvaro del Pino and Wei Zhou.

February 26: Joe Boninger (Boston College)

Title: Knot Floer Homology, the Burau Representation, and Quantum gl(1|1)

Abstract: In the past two decades, homology theories associated to knots and three-manifolds have led to remarkable advances in low-dimensional topology. These theories are developed from a few different viewpoints, including symplectic geometry and the representation theory of quantum groups, and it is an ongoing project to unify these perspectives. We contribute to this goal by demonstrating a geometric correspondence between knot Floer homology and the gl(1|1) quantum tangle invariant. This connection relies on the classical Burau representation of the braid group, and as an application show that a large number of matrices related to the Burau representation can be categorified by appropriate Heegaard Floer theories.

March 05: No seminar

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Abstract: 

March 12: Daniel Levitin

Title: Groups Acting on Horocyclic Products

Abstract: Horocyclic products are a well-studied class of metric spaces that provide models for various Lie groups, Baumslag-Solitargroups, and Lamplighter groups. In this talk, I will discuss my recent work with Noah Caplinger concerning groups that act onhorocyclic products of CAT(-1) spaces. In it, we show that the topologies of the boundaries of the CAT(-1) spaces impose strongalgebraic conditions on the group that acts on their horocyclic product; every such group is either an ascending HNN extension of a finitely-generated virtually nilpotent group, or else is not finitely presented.

March 19: Xiaoying He (Brandeis) 

Title: Dimension of singularities of Lagrangian maps

Abstract: The singularities of a generic smooth map between smooth manifolds admit a natural stratification by corank, called the Boardman stratification. Boardman gave a combinatorial formula which computes the expected codimensions of these strata for generic smooth maps. However, for generic Lagrangian maps, a special kind of smooth map between manifolds of the same dimension, this formula is not correct. We determine the correct combinatorial formula for the codimensions of the Boardman strata of a generic Lagrangian map.

March 26: Shaked Bader (Oxford University) 

Title: Quasi-isometric embeddings of RAAGs

Abstract: CAT(0) cube complexes are a well-studied and well-behaved class of spaces. We can get special examples of such spaces by looking at right angled Artin groups (RAAGs), which are a hybrid between free abelian and free groups – all the generators have infinite order and the relators are some commutators of generators. Many RAAGs are known to be quasiisometrically rigid. In this talk, I will discuss rigidity of quasi-isometric embeddings between RAAGs of the same rank. In the introduction talk, I will mention examples of exotic quasiisometric embeddings between them. This talk is based on joint work with Oussama Bensaid and Harry Petyt.

April 16: Hyeran Cho (Tufts)

Title:  Small Cancellation for Random Branched Covers of Groups

Abstract: In this talk, I introduce a random model for an n-fold branched cover of a finite 2-complex X with mild hypotheses and investigate its structural and probabilistic properties. In particular, we show that as n goes to infinity, a random branched cover is asymptotically almost surely homotopy equivalent to a 2-complex satisfying geometric small cancellation. The research presented in this talk is joint work with Jean-Francois Lafont and Rachel Skipper. 

April 23: Gage Martin (Harvard)

Title:  4-dimensional skein modules and handle attachments

Abstract: 4-dimensional skein modules are a recent tool developed for the study of 4-manifolds. Applications of these skein modules sometimes rely on an understanding standing of how these change under handle attachments. In this talk we will review an introduction to these modules as well as stating and proving general formula for how these modules change under 1-, 2-, and 3-handle attachments. These generalize existing formula of Chen, Manolescu-Neithalath, Manolescu-Walker-Wedrich, and Ren-Willis. These formula are derived from a complete description of the gluing homomorphism on skein modules. This description was also introduced independently by Blackwell-Krushkal-Luo.

This is joint work with Mary Stelow and Mira Wattal

TUESDAY, April 28 (Brandeis Thursday): Mike Sullivan (UMass Amherst)

Title: Some quantitative results for Legendrian knots and submanifolds

Abstract: Legendrians submanifolds of contact manifolds lie on both sides of the flexible/rigid divide in contact topology/geometry. I'll discuss results on the rigid side, which mostly rely on the barcodes (persistence) of various Floer-type pseudo-holomorphic curve theories associated to Legendrians. This includes: non-degeneracy of the Shelukhin-Hofer metric, $C^0$-closure, displacement energy bounds, existence of Reeb chords, flats in the group of contactomoprhisms, etc. I'll start with definitions and a survey of the well-known analogous symplectic ridigity results. This is old and new work, all joint with Georgios Dimitroglou Rizell.