Seminars and Colloquia by Series

Monday, October 25, 2010 - 14:00 , Location: Skiles 269 , Joan Birman , Barnard College-Columbia University , jb@math.columbia.edu , Organizer: Dan Margalit
Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980's.  We will discuss the 1995 Bestvina-Handel algorithmic proof of Thurston's theorem, and in particular the "transition matrix" T that their algorithm computes.  We study the Bestvina-Handel proof carefully, and show that the dilatation is the largest real root of a particular polynomial divisor P(x) of the characteristic polynomial C(x) = | xI-T |.  While C(x) is in general not an invariant of the mapping class, we prove that  P(x) is.  The polynomial P(x) contains the minimum polynomial M(x) of the dilatation as a divisor, however it does not in general coincide with M(x).In this talk we will review the background and describe the mathematics that underlies the new invariant.  This represents joint work with Peter Brinkmann and Keiko Kawamuro.
Monday, October 18, 2010 - 14:00 , Location: Skiles , No Speaker , Ga Tech , Organizer: John Etnyre
Monday, October 11, 2010 - 15:05 , Location: Skiles 269 , Stavros Garoufalidis , Georgia Tech , stavros@math.gatech.edu , Organizer: Stavros Garoufalidis
Given a knot, a simple Lie algebra L and an irreducible representation V of L one can construct a one-variable polynomial with integer coefficients. When L is the simplest simple Lie algebra (sl_2) this gives a sequence of polynomials, whose sequence of degrees is a quadratic quasi-polynomial. We will discuss a conjecture for the degree of the colored Jones polynomial for an arbitrary simple Lie algebra, and we will give evidence for sl_3. This is joint work with Thao Vuong.
Monday, October 4, 2010 - 14:00 , Location: Skiles 269 , Joan Licata , Stanford University , Organizer: John Etnyre
In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. I'll describe a rigorous combinatorial formulation of Legendrian contact homology for Legendrian knots in these manifolds. This work is joint with J. Sabloff.
Monday, September 27, 2010 - 17:00 , Location: Skiles 269 , Evan Fink , University of Georgia , Organizer: John Etnyre

This is the second talk in the Emory-Ga Tech-UGA joint seminar. The first talk will begin at 3:45. 

 There are many conjectured connections between Heegaard Floer homology and the various homologies appearing in low dimensional topology and symplectic geometry.  One of these conjectures states, roughly, that if \phi is a diffeomorphism of a closed Riemann surface, a certain portion of the Heegaard Floer homology of the mapping torus of \phi should be equal to the Symplectic Floer homology of \phi.  I will discuss how this can be confirmed when \phi is periodic (i.e., when some iterate of \phi is the identity map).  I will recall how a mapping torus can be realized via Dehn surgery; then, I will sketch how the surgery long exact triangles of Heegaard Floer homology can be distilled into more direct surgery formulas involving knot Floer homology.  Finally, I'll say a few words about what actually happens when you use these formulas for the aforementioned Dehn surgeries: a "really big game of tic-tac-toe".
Monday, September 27, 2010 - 15:45 , Location: Skiles 269 , Dan Rutherford , Duke University , Organizer: John Etnyre

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.
Monday, September 20, 2010 - 14:00 , Location: Skiles 114 , Xavier Fernandes , Emory University , Organizer: John Etnyre
We adapt techniques derived from the study of quasi-flats in Right Angled Artin Groups, and apply them to 2-dimensional Graph Braid Groups to show that the groups B_2(K_n) are quasi-isometrically distinct for all n.
Monday, September 13, 2010 - 14:00 , Location: Skiles 114 , Jason McGibbon , University of Massachusetts , Organizer: John Etnyre
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.
Monday, September 6, 2010 - 14:00 , Location: Skiles , No Speaker , --- , Organizer: John Etnyre
Monday, August 30, 2010 - 14:00 , Location: Skiles 114 , Mohammad Ghomi , Ga Tech , Organizer: John Etnyre
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}.

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