Seminars and Colloquia by Series

Invariants of Legendrian Knots from Open Book Decompositions

Series
Geometry Topology Seminar
Time
Monday, March 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sinem OnaranSchool of Mathematics, Georgia Tech
Given any contact 3-manifold, Etnyre and Ozbagci defined new invariants of the contact structures in terms of open book decompositions supporting the contact structure. One of the invariants is the support genus of the contact structure which is defined as the minimal genus of a page of an open book that supports the contact structure. In a similar fashion, we define the support genus sg(L) of a Legendrian knot L in a contact manifold M as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framing given by the contact structure and by the page agree. In this talk, we will discuss the support genus of Legendrian knots in contact 3-manifolds. We will show any null-homologous loose knot has support genus zero. To prove this, we observe an interesting topological property of knots and links on the way. We observe any topological knot or link in a 3-manifold sits on a planar page (genus zero page) of an open book decomposition.

Annulus open book decompositions and the self linking number

Series
Geometry Topology Seminar
Time
Monday, March 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Keiko KawamuroIAS
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.

Cubic graphs and number fields

Series
Geometry Topology Seminar
Time
Monday, February 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech
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.

Transverse knots and contact structures

Series
Geometry Topology Seminar
Time
Monday, February 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John EtnyreSchool of Mathematics, Georgia Tech
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.

Duality exact sequences in contact homology

Series
Geometry Topology Seminar
Time
Monday, February 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John EtnyreSchool of Mathematics, Georgia Tech
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.

Broken Lefschetz fibrations and Floer theoretical invariants

Series
Geometry Topology Seminar
Time
Monday, December 1, 2008 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Yanki LekiliMIT
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.

On Cannon's conjecture

Series
Geometry Topology Seminar
Time
Monday, November 24, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sa'ar HersonskyUniversity of Georgia
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.

Multiple knots in a manifold with the same surgeries yielding S^3

Series
Geometry Topology Seminar
Time
Friday, November 21, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ken BakerUniversity of Miami
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.

Branched covers and nonpositive curvature

Series
Geometry Topology Seminar
Time
Friday, November 7, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor BelegradekSchool of Mathematics, Georgia Tech
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.

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