Columbia Geometric Topology Seminar

Spring 2021

 

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Organizer: Nick Salter.
The GT seminar typically meets on Fridays at 2:00pm Eastern time via the Zoom link above. (The password is `math'). 

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Spring 2021

Date Time (Eastern) Speaker Title
February 12 2 pm Alex Sisto Cubulations of hulls and bicombings
February 19      
February 26 11 am Piotr Nowak On property (T) for Aut(F_n)
March 5 2 pm Elia Fioravante

Coarse-median preserving automorphisms

March 12 2 pm Luke Jeffreys  Ratio-optimising pseudo-Anosovs and single-cylinder pillowcase-covers
March 19 2 pm Ben Knudsen Stable and unstable homology of graph braid groups
March 26 2 pm Lorenzo Ruffoni Projective structures, representations, and ODEs on surfaces
April 2 2 pm Genevieve Walsh Virtual algebraic fibrations, conjectures, and excessive homology
April 9 2 pm Ian Biringer Determining the finite subgraphs of the curve graph
April 16 2 pm Sander Kupers The Disc-structure space of a manifold
April 23 2 pm Corey Bregman  
April 30   Alice Kerr  

Abstracts

Sander Kupers, University of Toronto

April 16, 2021
Title:
The Disc-structure space of a manifold 

Abstract:Surgery theory attempts to understand the category of manifolds by mapping it to the category of topological spaces, with the goal of understanding the target and fiber of this map using homotopy-theoretic methods. In this talk I'll propose an alternative inspired by embedding calculus, mapping the category of manifolds to the category of simplicial presheaves on the category of discs, and explore its surprising features. This is joint work with Manuel Krannich.

Ian Biringer, Boston College

April 9, 2021
Title:
Determining the finite subgraphs of the curve graph 

Abstract: The curve graph of an orientable surface S has vertices corresponding to isotopy classes of simple closed curves on S, and edges connect isotopy classes with disjoint representatives. We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph. This is the main result of a 2018 paper with Tarik Aougab and Jonah Gaster.

 

Genevieve Walsh, Tufts

April 2, 2021
Title:
Virtual algebraic fibrations, conjectures, and excessive homology 

Abstract: A group G "algebraically fibers" if G surjects the integers with finitely generated kernel.  Groups which algebraically fiber are natural generalizations of fibered 3-manifolds, and it is an interesting question to determine which groups virtually algebraically fiber.  It is conjectured for example that hyperbolic 4-manifold groups always virtually algebraically fiber. We discuss this question for free-by-free, surface-by-free and surface-by-surface groups.   We show that free-by-free groups do not always virtually algebraically fiber and we relate the question for surface-by-free and surface-by-surface groups to a conjecture of Putnam and Wieland.  This is joint work with Robert Kropholler and Stefano Vidussi.

Lorenzo Ruffoni

March 26, 2021
Title:
Projective structures, representations, and ODEs on surfaces. 

Abstract:In one of its easiest formulations, Hilbert's XXI problem deals with the relationship between linear ODEs on a surface and representations of its fundamental group. When a complex structure on the surface is fixed, a classical theory is available. However, not much is understood in the complementary case, i.e. when the type of the ODE is fixed, while the complex structure is allowed to vary. Projective structures have been known since Poincare's times to be a geometric bridge between the analytic and the algebraic side of this picture. In this talk we will present how their geometric deformation theory can be used to explore the space of ODEs associated with a fixed representation, including some recent results about branched projective structures.

Ben Knudsen, Northeastern

March 19, 2021
Title:
Stable and unstable homology of graph braid groups 

Abstract:The homology of the configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. The degree of this polynomial is captured by a connectivity invariant of the graph, and its leading coefficient may be computed explicitly in terms of cut counts and vertex valences. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin, but exotic non-toric classes abound unstably. These mysterious classes are intimately tied to questions about generation and torsion whose answers remain elusive except in a few special cases. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole. 

Luke Jeffreys, Bristol

March 12, 2021
Title:
Ratio-optimising pseudo-Anosovs and single-cylinder pillowcase-covers 

Abstract:The systole map between Teichmüller space and the curve graph of a surface played a key role in Masur and Minsky’s proof that the curve graph is Gromov-hyperbolic. In particular, they showed that the systole map is coarsely-Lipschitz; that is, it is Lipschitz up to an additive error. Gadre-Hironaka-Kent-Leininger proved that the optimal Lipschitz constant is comparable to 1/log(g), for g the genus of the surface. In their proof of this fact, they considered a class of pseudo-Anosov diffeomorphism now called ‘ratio-optimising’. Aougab-Taylor gave a more general construction of these ratio-optimising pseudo-Anosovs in such a way that they stabilise the Teichmüller disk of a flat structure on the surface. A natural question to ask is which flat structures can be achieved in this construction. In this talk, I will discuss the answer to this question and how it relates to a specific class of flat structure called a single-cylinder pillowcase-cover.

Elia Fioravante

March 5, 2021
Title: Coarse-median preserving automorphisms

Abstract:
 We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If \phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(\phi) is finitely generated and undistorted. Up to replacing \phi with a power, we show that Fix(\phi) is even quasi-convex with respect to the standard word metric. This implies that Fix(\phi) is separable and a special group in the Haglund-Wise sense. Some of our techniques may be applicable in the more general context of Bowditch's coarse median groups. Based on arXiv:2101.04415.

Piotr Nowak, IMPAN

February 26, 2021, 11 am
Title: On property (T) for Aut(F_n)

Abstract:
 The goal of this talk is to present the recent proof that Aut(F_n), the automorphism group of the free group on n generators, has Kazhdan’s property (T) for n\ge 5. This is joint work with Marek Kaluba and Taka Ozawa (n=5) and with Kaluba and Dawid Kielak (n\ge 6). Our proof uses a characterization of property (T) via an algebraic notion of positivity in the group ring, due to Ozawa and computer assistance in the form of semidefinite programming (i.e. convex optimization over positive definite matrices). As applications we confirm the explanation of the effectiveness of the Product Replacement Algorithm predicted by Lubotzky and Pak, as well as obtain new asymptotically optimal estimates of Kazhdan constant for Aut(F_n) and SL_n(Z).

 

Alex Sisto, Heriot-Watt University

February 12, 2021
Title: Cubulations of hulls and bicombings

Abstract: 
It is well-known that the quasi-convex hull of finitely many points in a 
hyperbolic space is quasi-isometric to a tree. I will discuss an analogous fact in the context of hierarchically hyperbolic spaces, a large class of spaces and groups including mapping class groups,Teichmueller space, right-angled Artin and Coxeter groups, and many others. In this context, the approximating tree is replaced by a CAT(0) cube complex. I will also briefly discuss applications, including how this can be used to construct bicombings. Based on joint works with Behrstock-Hagen and Durham-Minsky.

 

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