Brandeis Topology Seminar, Fall 2026

Organizers: 

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

September 1: Polyana Benk (Brandeis) -- Main talk only; no background talk

Title: Hyperbolic 3-manifold groups

Abstract: In this talk, we study fundamental groups of closed hyperbolic 3-manifolds, with the goal of understanding the result that every closed hyperbolic 3-manifold virtually fibers. To this end, we discuss non-positively curved cube complexes, their hyperplanes, and when a cube complex is special. We then construct the Salvetti complex associated to a graph \Sigma and the corresponding right-angled Artin group (RAAG), which is its fundamental group. Putting this together, we use the typing map to show that every special group is a subgroup of a RAAG, following the result of Haglund–Wise. We then turn to introducing quasi-Fuchsian surface groups and the notion of a dense set of such groups, before stating the Kahn–Markovic theorem, which says that every closed hyperbolic 3-manifold contains a dense set of quasi-Fuchsian surface groups. Finally, we explain how this leads to a compact non-positively curved cube complex and, together with Agol's theorem, to the conclusion that the fundamental group is virtually compact special. We then discuss how this leads to the conclusion that the hyperbolic 3-manifold virtually fibers.  This will serve as my minor exam.

September 8: Shaoyun Bai (MIT) 

Title: Integral counts of holomorphic curves and fixed point rigidity

Abstract: Some years ago, Guangbo Xu and I developed a perturbation scheme that allows one to achieve a suitable form of equivariant transversality, which has found some applications to the study of fixed points of Hamiltonian diffeomorphisms. In this talk, I will present some updates on such a program, including applications to Hamiltonian periodic points and Lagrangian intersection problems.

September 15: Audrey Rosevear (UC Berkeley)

Title: Foliations Forcing Closed Orbits

Abstract: Understanding dynamics transverse to a foliation can yield insights into the geometry of the foliation. In this talk I’ll discuss 2-dimensional foliations on 3-manifolds with the property that every transverse flow has a closed orbit. I’ll show this property is unstable under perturbation of the foliation, and classify finite depth foliations with this property, making connections with the theories of endperiodic maps and dynamical plugs. In the pre-talk, I’ll construct interesting examples of foliations, and draw lots and lots of pictures. This is joint work with Ellis Buckminster.

September 22: Masayuki Asaoka (Doshisha University)

Title: The connected components of the space of 3-dimensional topologically transitive Anosov flows

Abstract: Three-dimensional Anosov flows are important objects at the crossroads of dynamical systems theory and low-dimensional topology. Recently, I proved that if two there-dimensional Anosov flows are topologically equivalent by a homeomorphism homotopic to the identity map then they should be connected by a continuous path in the space of Anosov maps. It allows us to define topological invariants of three-dimensional Anosov flows from their deformation invariants.

In the first half of the seminar, I will review basic results on three-dimensional Anosov flow. In the second half, I will sketch the proof of the above result. In the proof, the key tool is deformation of three-dimensional Anosov flows by reconstructing a smooth structure of the underlying manifold along transverse direction of the invariant foliations of the flow. 

September 29: No seminar

October 6: Yonghwan Kim (MIT)

Title: Simple homotopy theory for Fukaya categories

Abstract: The s-cobordism theorem uses simple homotopy equivalences, a refinement of homotopy equivalences detected by Whitehead torsion, to distinguish h-cobordant manifolds. More recently, it has appeared in symplectic geometry through the work of Abouzaid and Kragh, in connection with the topology of closed Lagrangian submanifolds of cotangent bundles. In this talk, I will present a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. This framework allows us to extend previous known results from cotangent bundles to general Weinstein manifolds.

October 13: No seminar (Brandeis Monday)

October 20: Teddy Weisman (University of Michigan)

Title: Complex hyperbolic Dehn filling spaces

Abstract: Let Gamma be the fundamental group of a finite-volume hyperbolic d-manifold M, which acts on hyperbolic d-space. When d = 2, then Gamma is flexible; it has many discrete faithful deformations, giving the Teichmüller space of M. When d > 3, seminal work of Garland–Raghunathan implies that Gamma is locally rigid, meaning it has no deformations at all. The case d = 3 occupies a fascinating middle ground: the deformations of Gamma form Thurston’s Dehn filling space, and are the source of most of our examples of hyperbolic structures on closed 3-manifolds. In this talk, we discuss upcoming work with Jeff Danciger, showing that one can find "Dehn filling spaces" for certain groups acting on complex hyperbolic space. These Dehn filling spaces give rise to numerous examples of complex hyperbolic manifolds with surprising properties: our constructions show that there are convex cocompact groups in PU(3,1) with "exotic" limit sets, and that global rigidity can dramatically fail for many locally rigid discrete subgroups of PU(3,1).

October 27: TBA

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November 3: Yasin Karacan (UMass Amherst)

Title: Generating Families, Augmentations, and Morse Complex n-Families

Abstract: Legendrian submanifolds are central objects in contact topology, and a basic problem is to tell them apart. The strongest invariants come from counting holomorphic curves, but they are hard to compute in general. Generating families give a Morse-theoretic alternative: a Legendrian is encoded by a family of functions, and invariants come from how the Morse theory of that family changes. For knots and surfaces, these two worlds are known to meet: a generating family produces algebraic data on the holomorphic side. I will explain how to build this bridge for Legendrians of higher dimensions by tracking how gradient flow lines in a family of functions break. I will also describe work in progress toward a combinatorially computable obstruction to Legendrian isotopy in higher dimensions. The first talk will introduce the necessary background, and everything will be defined along the way; some familiarity with Morse theory will be helpful.

November 10: Abigail Ward (Sheffield)

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November 17: George Domat (University of Michigan)

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November 24: Roman Krutowskiy (MIT)

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December 1: Sahana Balasubramanya (Lafayette College)

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