Seminars and Colloquia by Series

Monday, March 2, 2009 - 13:00 , Location: Skiles 269 , Keiko Kawamuro , IAS , Organizer: John Etnyre
We introduce a construction of an immersed surface for a null-homologous braid in an annulus open book decomposition. This is hinted by the so called Bennequin surface for a braid in R^3. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. Then we compute a self-linking number formula using this embedded surface and observe that the Bennequin inequality is satisfied if and only the contact structure is tight. We also prove that our self-linking formula is invariant (changes by 2) under a positive (negative) braid stabilization which preserves (changes) the transverse knot class.
Monday, February 23, 2009 - 13:00 , Location: Skiles 269 , Stavros Garoufalidis , School of Mathematics, Georgia Tech , , Organizer: Stavros Garoufalidis
A cubic graph is a graph with all vertices of valency 3. We will show how to assign two numerical invariants to a cubic graph: its spectral radius, and a number field. These invariants appear in asymptotics of classical spin networks, and are notoriously hard to compute. They are known for the Theta graph, the Tetrahedron, but already unknown for the Cube and the K_{3,3} graph. This is joint work with Roland van der Veen: arXiv:0902.3113.
Monday, February 16, 2009 - 13:00 , Location: Skiles 269 , John Etnyre , School of Mathematics, Georgia Tech , Organizer: John Etnyre
I will discuss a couple of applications of transverse knot theory to the classification of contact structures and braid theory. In particular I will make the statement "transverse knots classify contact structures" precise and then prove it (if we have time). I will also discuss how progress on two of Orevkov's questions concerning quasi-positive knots that have implications for Hilbert's 16th problem.
Monday, February 2, 2009 - 13:00 , Location: Skiles 269 , John Etnyre , School of Mathematics, Georgia Tech , Organizer: John Etnyre
I will discuss a "duality" among the linearized contact homology groups of a Legendrian submanifold in certain contact manifolds (in particular in Euclidean (2n+1)-space). This duality is expressed in a long exact sequence relating the linearized contact homology, linearized contact cohomology and the ordinary homology of the Legendrian submanifold. One can use this structure to ease difficult computations of linearized contact homology in high dimensions and further illuminate the proof of cases of the Arnold Conjecture for the double points of an exact Lagrangian in complex n- space.
Monday, December 1, 2008 - 16:00 , Location: Skiles 269 , Yanki Lekili , MIT , Organizer: John Etnyre
A broken fibration is a map from a smooth 4-manifold to S^2 with isolated Lefschetz singularities and isolated fold singularities along circles. These structures provide a new framework for studying the topology of 4-manifolds and a new way of studying Floer theoretical invariants of low dimensional manifolds. In this talk, we will first talk about topological constructions of broken Lefschetz fibrations. Then, we will describe Perutz's 4-manifold invariants associated with broken fibrations and a TQFT-like structure corresponding to these invariants. The main goal of this talk is to sketch a program for relating these invariants to Ozsváth-Szabó invariants.
Monday, December 1, 2008 - 14:30 , Location: Skiles 269 , Sandra Ritz , University of South Carolina , Organizer: John Etnyre
We will begin with an overview of the Burau representation of the braid group. This will be followed by an introduction to a contact category on 3-manifolds, with a brief discussion of its relation to the braid group.
Monday, November 24, 2008 - 14:00 , Location: Skiles 269 , Sa'ar Hersonsky , University of Georgia , Organizer: John Etnyre
Cannon: "A f.g. negatively curved group with boundary homeomorphic to the round two sphere is Kleinian". We shall outline a combinatorial (complex analysis motivated) approach to this interesting conjecture (following Cannon, Cannon-Floyd-Parry). If time allows we will hint on another approach (Bonk-Kleiner) (as well as ours). The talk should be accessible to graduate students with solid background in: complex analysis, group theory and basic topology.
Friday, November 21, 2008 - 14:00 , Location: Skiles 269 , Ken Baker , University of Miami , Organizer: John Etnyre
Lickorish observed a simple way to make two knots in S^3 that produced the same manifold by the same surgery. Many have extended this result with the most dramatic being Osoinach's method (and Teragaito's adaptation) of creating infinitely many distinct knots in S^3 with the same surgery yielding the same manifold. We will turn this line of inquiry around and examine relationships within such families of corresponding knots in the resulting surgered manifold.
Friday, November 7, 2008 - 14:00 , Location: Skiles 269 , Igor Belegradek , School of Mathematics, Georgia Tech , Organizer: Igor Belegradek
In the 1980s Gromov showed that curvature (in the triangle comparison sense) decreases under branched covers. In this expository talk I shall prove Gromov's result, and then discuss its generalization (due to Allcock) that helps show that some moduli spaces arising in algebraic geometry have contractible universal covers. The talk should be accessible to those interested in geometry/topology.
Monday, October 27, 2008 - 14:00 , Location: Skiles 269 , Mohammad Ghomi , School of Mathematics, Georgia Tech , Organizer: John Etnyre
We prove that every metric of constant curvature on a compact 2-manifold M with boundary bdM induces (at least) four vertices, i.e., local extrema of geodesic curvature, on bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two vertices on bdM. Furthermore, we characterize the sphere as the only closed orientable Riemannian 2-manifold M which has the four-vertex-property, i.e., the boundary of every compact surface immersed in M has 4 vertices.