Seminars and Colloquia by Series

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.

"On the unification of quantum invariants of 3-manifolds" by Qi Chen

Series
Geometry Topology Seminar
Time
Friday, February 26, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Qi Chen – Winston-Salem State University
For every quantum group one can define two invariants of 3-manifolds:the WRT invariant and the Hennings invariant. We will show that theseinvariants are equivalentfor quantum sl_2 when restricted to the rational homology 3-spheres.This relation can be used to solve the integrality problem of the WRT invariant.We will also show that the Hennings invariant produces integral TQFTsin a more natural way than the WRT invariant.

The Jones slopes of a knot

Series
Geometry Topology Seminar
Time
Monday, November 30, 2009 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros Garoufalidis – Georgia Tech –
I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

Geometry, computational complexity and algebraic number fields

Series
Geometry Topology Seminar
Time
Monday, November 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Hong-Van Le – Mathematical Institute of Academy of Sciences of the Czech Republic
In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.

On the Legendrian and transverse classification of cabled knot types

Series
Geometry Topology Seminar
Time
Monday, November 9, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Bulent Tosun – Ga Tech
In 3-dimensional contact topology one of the main problem is classifying Legendrian (transverse) knots in certain knot type up to Legendrian ( transverse) isotopy. In particular we want to decide if two (one in case of transverse knots) classical invariants of this knots are complete set of invariants. If it is, then we call this knot type Legendrian (transversely) simple knot type otherwise it is called Legendrian (transversely) non-simple. In this talk, by tracing the techniques developed by Etnyre and Honda, we will present some results concerning the complete Legendrian and transverse classification of certain cabled knots in the standard tight contact 3-sphere. Moreover we will provide an infinite family of Legendrian and transversely non-simple prime knots.

Schur Weyl duality and the colored Jones polynomial

Series
Geometry Topology Seminar
Time
Wednesday, October 28, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Roland van der Veen – University of Amsterdam –
We recall the Schur Weyl duality from representation theory and show how this can be applied to express the colored Jones polynomial of torus knots in an elegant way. We'll then discuss some applications and further extensions of this method.

Triple linking numbers, Hopf invariants and integral formulas for 3-component links

Series
Geometry Topology Seminar
Time
Monday, October 26, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shea Vela-Vick – Columbia University
To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

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