UB Geometry and Topology Seminar

2015-2016

Unless noted, all seminars are on Friday at 4pm, in Mathematics 122. Past seminar listings: 2013-2014, 2014-2015.

Sept 4 Organizational meeting. Adam S. Sikora (Buffalo) Skein Algebras of Surfaces Abstract: For a surface $F$, the space of links in $F \times [0,1]$ modulo Kauffman bracket skein relations is called the skein algebra of $F$, denoted by $S(F).$ It is a non-commutative deformation of the $SL(2,C)$-character variety of $F$, of significant importance to quantum topology. In particular, for $F$ with boundary, it is (almost) the quantum Teichmuller space of $F.$     Except for a few simplest surfaces $F,$ not much is known about the algebraic properties of $S(F)$ for closed $F$. We are going to prove the following two fundamental properties of skein algebras: 1. $S(F)$ has no zero divisors, 2. Away from roots of unity, the center of $S(F)$ is composed of polynomials in knots parallel to boundary components of $F.$ This is joint work with J. H. Przytycki. No Seminar TBA: TBA Abstract: Joel Louwsma Stable commutator length in Baumslag-Solitar groups Abstract: I will discuss results about computing stable commutator length in Baumslag-Solitar groups and about the spectrum of values it takes. In the first direction, we show that, for a certain class of elements, stable commutator length is computable and takes only rational values. We also determine exactly which elements of this class admit extremal surfaces. Our techniques additionally give lower bounds on the stable commutator lengths of all elements. In the second direction, we show that there is a uniform gap in the stable commutator length spectrum: no element of a Baumslag-Solitar group has stable commutator length between 0 and 1/12. Some of the techniques we use to show this apply more generally to groups acting on trees. This is joint work with Matt Clay and Max Forester. TBA: TBA Abstract: Kasra Rafi (U. Toronto) -- postponed TBA Abstract: TBA: TBA Abstract: TBA: TBA Abstract: TBA: TBA Abstract: Tarik Aougab (Brown U.): Local geometry of k-curve graphs Abstract: Let S be a surface of negative Euler characteristic. The k-curve graph of S is the graph whose vertices correspond to isotopy classes of simple closed curves on S, and whose edges correspond to pairs of curves which intersect at most k times. The geometry of these graphs are highly related to the algebra of the mapping class group and the geometry of the Teichmuller space of S. While all of these graphs have roughly the same geometry on the large scale, very little is known about the local combinatorics of these graphs, and how it depends on the parameter k. We prove sharp asymptotic upper bounds on the clique number of the k-curve graph, and upper bounds on the maximum size of the intersection of a pair of links. The proofs use Teichmuller theory. As an application, we give quasi-polynomial upper bounds on the number of simple closed geodesics of length at most L on any flat surface tiled by unit squares. Anastasia Tsvietkova (UC Davis) Hyperbolic structures from link diagrams Abstract: Thurston suggested a method for computing the hyperbolic structure of a 3-manifold that involves triangulating the manifold, and the method was later implemented in the program SnapPea by Jeff Weeks. The talk will briefly introduce an alternative method for computing the structure of a hyperbolic link, based on ideal polygons bounding the regions of a diagram rather than a trianguation (this is a joint work with Morwen Thistlethwaite). We will discuss the ongoing program that uses a blend of the alternative method with various techniques in order to relate diagrammatic properties of hyperbolic links to their intrinsic geometry. Thanksgiving break TBA: TBA Abstract: TBA: TBA Abstract:

Spring 2016

Jan 29       TBA: TBA
Feb 5       TBA: TBA
Feb 12       TBA: TBA
Feb 19       Erkao Bao (UCLA): Semi-global Kuranishi structures and contact homology
Feb 26       Hans Boden (Mcmaster U.) A classical approach to virtual knots
Mar 4       L. Siebenmann (Université de Paris-Sud) Towards a more geometric understanding of planar flows.
Mar 11       TBA: TBA
Mar 18       Spring Break
Mar 25       TBA: TBA
Apr 1,8,15       TBA: TBA
Apr 22       Juanita Pinzon-Caicedo (U. of Georgia) An Overview of Relative Trisections

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