Columbia Geometric Topology Seminar

Fall 2017

The GT seminar meets on Fridays in Math 520, at 2 PM.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

Fall 2017

Date Speaker Title
Sept 8 Conference for Lee Mosher, Princeton Groups explored through geometry and dynamics
Sept 15 Organizational meeting
Sept 22 Kyle Hayden, BC Complex curves through a contact lens
Sept 29 No seminar
Oct 6 Ilya Gekhtman, Yale Counting loxodromics for actions of hyperbolic groups and other automatic groups
Oct 13 Jenny Wilson, Stanford Stability in the homology of configuration spaces
Oct 20 Sam Taylor, Temple U. Veering triangulations and fibered faces of 3--manifolds
Oct 27-29 Conference in honor of Benson Farb in Chicago "No Boundaries: Groups in Algebra, Geometry, and Topology"
Nov 3 Martin Bridson, Oxford Profinite rigidity and 3-manifolds
Nov 10 Ralf Schiffler, U Conn Cluster algebras and knot theory
Nov 17 Eriko Hironaka, AMS Topological vs Geometry Entropy in Flow Equivalence Classes of pseudo-Anosov maps
Dec 8 Effie Kalfagianni, Michigan State U Quantum representations and monodromies of fibered links



Kyle Hayden, Boston College
Sept 22
Title: Complex curves through a contact lens
Abstract: Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

Ilya Gekhtman, Yale
October 6
Title: Counting loxodromics for actions of hyperbolic groups and other automatic groups
Abstract: We show that for arbitrary nonelementary actions $G\curvearrowright X$ of hyperbolic groups on Gromov hyperbolic spaces, translation length on average grows linearly in word length. In particular, the proportion of loxodromic elements in a large ball in the Cayley graph converges to 1.  This holds even when the action is not in any sense alignment preserving: for example a dense free subgroup of $SL_2R$ acting on the hyperbolic plane, or a hyperbolic subgroup of the mapping class group acting on the curve complex. Along the way we described the behavior in the space $X$ of typical word geodesics in the group: for example, with respect to the Patterson-Sullivan measure on the boundary group, the orbit of almost every word geodesic logarithmically tracks a geodesic in $X$. We prove analogous counting results for more general groups, including relatively hyperbolic groups with virtually abelian subgroups and right angled Artin and Coxeter groups.  Our results hold more generally for automatic groups satisfying certain properties: groups parametrized by paths in a finite directed graph. Indeed, the automatic structure is what allows us to reduce the asymptotic geometry of the Cayley graph of $G$ to a certain Markov chain on a finite graph and a family  of random walks on $G$ associated to vertices of the finite graph. This is joint work with Sam Taylor and Giulio Tiozzo.

Jenny Wilson, Stanford
October 13
Title:Stability in the homology of configuration spaces
Abstract: This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_k(M) to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize.  In this talk I will explain these stability patterns, and how they generalize homological stability results proved by McDuff (with a stable range due to Segal) in the 1970s. I will describe higher-order stability phenomena established in recent work joint with Jeremy Miller.

Sam Taylor, Temple U
October 20
Title: Veering triangulations and fibered faces of 3--manifolds
Abstract: Agol's veering triangulation for 3-manifolds that fiber over the circle can be obtained very explicitly, via a construction of Gueritaud, from the stable and unstable laminations of the monodromy. We study the way in which these triangulations interact with the curve complexes of the surface and its subsurfaces. This allows us to examine the “profile” of subsurface projections associated to each fiber in a fibered face of the Thurston norm ball, obtaining some bounds that do not depend on the complexity of the fibers. This is joint work with Yair Minsky.

Martin Bridson, Oxford
November 3
Title:Profinite rigidity and 3-manifolds
Abstract: There has recently been renewed vigour in pursuit of the old question of the extent to which the finite images of a finitely presented group determine the group, with a particular focus on groups that arise in connection with low-dimensional topology. In this talk, I'll sketch what is now known about the extent to which a 3-dimensional manifold is determined by the finite images of its fundamental group, and I shall present recent joint work with McReynolds, Reid and Spitler showing that certain hyperbolic 3-orbifold groups are distingusihed from all other finitely generated groups by their finite quotients.

Ralph Schifler, U Conn
November 10
Title:Cluster algebras and knot theory.
Abstract: A cluster algebra is a commutative algebra with a special combinatorial structure. Its generators, the cluster variables, are Laurent polynomials in several variables. In knot theory, one of the important knot invariants is the Jones polynomial, introduced by Vaughan Jones in 1984. The Jones polynomial is a Laurent polynomial in one variable. In this talk, I present a concrete relation between cluster variables and Jones polynomials. For a certain class of knots, the so-called 2-bridge knots, the Jones polynomials are specializations of cluster variables. This is joint work with Kyungyong Lee.

Eriko Hirnoaka, AMS
Nov 17
TitleTopological vs Geometry Entropy in Flow Equivalence Classes of pseudo-Anosov maps
Abstract We show that the Alexander and Teichm\"uller norms of a fibered face are equal when the associated flow equivalence class contains an oriented fully-punctured element. As an application we show that for fully-punctured pseudo-Anosov braid monodromies, the normalized dilatations are dense in the interval [\phi^4, \infty) where \phi is the golden ratio.

Effie Kalfagianni, Michigan State U
Dec 8
Title Quantum representations and monodromies of fibered links
Abstract Andersen, Masbaum and Ueno conjectured (2005)  that  certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level).  In this talk I will discuss work with Renaud Detcherry (MSU) that makes progress on this conjecture in the case of surfaces with boundary.

Other relevant information.

Previous semesters:

Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

Other area seminars.

Our e-mail list: You can subscribe here for announcements for this seminar, as well as occasional related seminars and events.