Seminars and Colloquia by Series

HOMFLY-PT polynomial and Legendrian links in the solid torus

Series
Geometry Topology Seminar
Time
Monday, September 27, 2010 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dan Rutherford – Duke University

Please Note: This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will follow at 5.

A smooth knot in a contact 3-manifold is called Legendrian if it is always tangent to the contact planes. In this talk, I will discuss Legendrian knots in R^3 and the solid torus where knots can be conveniently viewed using their `front projections'. In particular, I will describe how certain decompositions of front projections known as `normal rulings' (introduced by Fuchs and Chekanov-Pushkar) can be used to give combinatorial descriptions for parts of the HOMFLY-PT and Kauffman polynomials. I will conclude by discussing recent generalizations to Legendrian solid torus links. It is usual to identify the `HOMFLY-PT skein module' of the solid torus with the ring of symmetric functions. In this context, normal rulings can be used to give a knot theory description of the standard scalar product determined by taking the Schur functions to form an orthonormal basis.

A monodromy invariant in the space of knots

Series
Geometry Topology Seminar
Time
Monday, September 13, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Jason McGibbon – University of Massachusetts
Knot contact homology (KCH) is a combinatorially defined topological invariant of smooth knots introduced by Ng. Work of Ekholm, Etnyre, Ng and Sullivan shows that KCH is the contact homology of the unit conormal lift of the knot. In this talk we describe a monodromy result for knot contact homology,namely that associated to a path of knots there is a connecting homomorphism which is invariant under homotopy. The proof of this result suggests a conjectural interpretation for KCH via open strings, which we will describe.

Spherical images of hypersurfaces

Series
Geometry Topology Seminar
Time
Monday, August 30, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Mohammad Ghomi – Ga Tech
We discuss necessary and sufficient conditions of a subset X of the sphere S^n to be the image of the unit normal vector field (or Gauss map) of a closed orientable hypersurface immersed in Euclidean space R^{n+1}.

Knots in overtwisted contact structures

Series
Geometry Topology Seminar
Time
Monday, August 23, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
John Etnyre – Ga Tech
The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.

Isotopies of links carried by Matsuda branched surfaces

Series
Geometry Topology Seminar
Time
Monday, August 16, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Bill Menasco – University of Buffalo
We introduce two related sets of topological objects in the 3-sphere, namely a set of two-component exchangable links termed "iterated doubling pairs", and a see of associated branched surfaces called "Matsuda branched surfaces". Together these two sets possess a rich internal structure, and allow us to present two theorems that provide a new characterization of topological isotopy of braids, as well as a new characterization of transversal isotopy of braids in the 3-sphere endowed with the standard contact structure. This is joint work with Doug Lafountain, and builds upon previous seminal work of Hiroshi Matsuda.

Positivity of monodromies of open book decompositions

Series
Geometry Topology Seminar
Time
Tuesday, June 15, 2010 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Andy Wand – Berkeley and Max Planck Institute
I will describe some results concerning factorizations ofdiffeomorphisms of compact surfaces with boundary. In particular, Iwill describe a refinement of the well-known \emph{right-veering}property, and discuss some applications to the problem ofcharacterization of geometric properties of contact structures interms of monodromies of supporting open book decompositions.

On the categorification of the quantum Casimir

Series
Geometry Topology Seminar
Time
Monday, April 26, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
A. Beliakova – University of Zurich
In the talk, I will gently introduce the Lauda-Khovanov 2-category, categorifying the idempotent form of the quantum sl(2). Then I will define a complex, whose Euler characteristic is the quantum Casimir. Finally, I will show that this complex naturally belongs to the center of the 2-category. The talk is based on the joint work with Aaron Lauda and Mikhail Khovanov.

The topology at infinity of real algebraic manifolds

Series
Geometry Topology Seminar
Time
Friday, April 2, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Clint McCrory – UGA –
A noncompact smooth manifold X has a real algebraic structure if and only if X is tame at infinity, i.e. X is the interior of a compact manifold with boundary. Different algebraic structures on X can be detected by the topology of an algebraic compactification with normal crossings at infinity. The resulting filtration of the homology of X is analogous to Deligne's weight filtration for nonsingular complex algebraic varieties.

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