GWU Topology Seminar
Spring 2008
January 31, 2008,
Thursday, 5:45- 7:00pm
Speaker: Marko Stosic (CEMAT,
Title: Properties of Khovanov homology of positive braid
knots and torus knots I
Place: Monroe Hall,
February 1, 2008, Friday,
11:10-12:25 pm
Speaker: Marko Stosic (CEMAT,
Title: Properties of Khovanov homology of positive braid
knots and torus knots II
Place: Monroe Hall,
Abstract
In this talk, I will present
a collection of results on the
properties of Khovanov homology of positive braid knots and
torus
knots. They will also include proofs of various
conjectures: the
existence of stable homology for torus knots, triviality of
the first
homology group of positive braid knots, the determination of
all
almost-alternating torus knots.
Moreover, our tools do not
rely heavily on the particular definitions
of the Khovanov homology, and consequently we obtain
majority of the
analogous results for Khovanov-Rozansky
homology, as well.
Topology/Geometry Seminar
Speaker: Mohammad Ghomi (Georgia
Tech)
Title: Topology of Riemannian submanifolds
with prescribed boundary
Place: Monroe Hall,
Abstract
In this talk we show that a
closed submanifold of codimension
2 in
Euclidean space bounds at most finitely many topological types of
complete hypersurfaces with nonnegative curvature.
This settles a
question of Guan and Spruck related to a problem of Yau. Further we
discuss analogous results for arbitrary Riemannian submanifolds.
On
the other hand, we show that these finiteness theorems may not hold
if the codimension is too high or the boundary is not
sufficiently
regular. The proofs employ, among other methods, the Gromov-Perelman
theory of Alexandrov spaces with curvature bounded
below, and a
relative version of Nash's isometric embedding theorem. These results
include joint works with Stephanie Alexander, Robert Greene, and
Jeremy Wong.
GWU Topology Seminar, Fall 2007
September
13, Thursday. 2:45 –
3:45pm
Speaker: Maciek
Mroczkowski (
Title: Homflypt
and Kauffman skein modules of twisted I bundles of unoriented
surfaces
Place: GWU, Mathematics
Department, Monroe Hall, Seminar Room
Contact person: Jozef Przytycki (przytyck@gwu.edu)
Speaker: Oleg Viro (Stony Brook and
Title: Khovanov homology of
framed and signed chord diagrams
Place: Monroe Hall,
Speaker: John Armstrong,
Title: Categorification of Quandle Coloring Numbers by Anafunctors*
(abstract below)
Place: Monroe Hall,
Speaker: Alexander
Shumakovitch,
Title: Rasmussen invariant
and sliceness of knots * (abstract below)
Place: Monroe Hall,
Speaker: Hao
Wu,
Title: Bennequin
inequalities from the Khovanov-Rozansky homology
(a.k.a. How to play Alex's game for n>2)* (abstract below)
Place: Monroe Hall,
Speaker: Milena
Pabiniak,
Abstract: see below.
Place: Monroe Hall,
Speakers: Dan Silver and
Susan Williams,
Title: On applications of
Dynamical systems to Knot Theory I - Introduction
Place: Monroe Hall,
November 30, 2007,
Friday, 12:45 - 2:00
Speakers: Radmila Sazdanovic, GWU.
Title: Torsion in Khovanov
homology of semi-adequate links
Place: Monroe Hall,
December 5, 2007,
Wednesday, 3:45 - 5:00
Speakers: Dan Silver and
Susan Williams,
Title: On applications of
Dynamical systems to Knot Theory II - Algebraic dynamics
Place: Monroe Hall,
December 6, 2007,
Thursday, 11:10 – 12:25
Speakers: Dan Silver and
Susan Williams,
Title: On applications of
Dynamical systems to Knot Theory III - Twisting
Place: Monroe Hall,
KNOTS IN
The 25th Conference on
Knot Theory and its Ramifications
December 7-9, 2007
http://home.gwu.edu/~przytyck/knots/knotsinwashington25.htm
The conference will start at
1.30 PM on Friday, December 7th at the MPA building Room 310
with the Colloquium type talk:
Speakers: Dan Silver and
Susan Williams
Title: From Nuts to Knots:
An irreverent look at the origins of Knot Theory
==============================================================================
ABSTRACTS OF TALKS
October 5, 2007, John
Armstrong, Categorification of Quandle Coloring
Numbers by Anafunctors
The number of colorings of a
link by a given quandle is a classical
invariant of links up to ambient isotopy. We would
like to categorify
and extend this invariant to the category $\mathcal{T}ang$ of tangles.
Here, we show how to associate, functorially, to each
tangle an
anafunctor between two comma categories of quandles. When we restrict
this assignment to knots and links and specify a quandle
$Q$ of
colors we recover $Q$-coloring invariant. If we first decategorify
and specify a quandle $Q$ of colors we recover the
$Q$-coloring
matrix of a given tangle.
October 25, 2007,
Alexander Shumakovitch, Rasmussen invariant and sliceness
of knots
We use Rasmussen
invariant to find knots that are topologically
locally-flatly slice but not smoothly slice. We also note that this
invariant can be used to give a combinatorial proof of the
slice-Bennequin inequality for links. Finally,
we compute the Rasmussen
invariant for quasipositive knots and show that
most of our examples of
non-slice knots are not quasipositive and were
to the best of our knowledge
unknown before.
November 15, 2007, Hao Wu, Bennequin inequalities
from the Khovanov-Rozansky homology (a.k.a. How to
play Alex's game for n>2)
This talk is a continuation
of Alex's recent talk. I will explain three things.
(i)
How to establish a strong Bennequin type inequality
using the Khovanov-Rozansky homology.
(ii) How to generalize the
Rasmussen invariant and the Alex-Olga inequality.
(iii) How to generalize
Olga's invariant for transversal links. I will also talk about relations
between these objects.
Skein modules were defined
by J.H.Przytycki to generalize various skein
relations of polynomial
invariants in S^3 to
arbitrary 3-manifolds. Soon it was noticed that for some class of manifolds,
trivial I-bundles, Fx I, over orientable surface F, one can easily obtain an algebra, not
only module,
structure. We will start with definition and basic properties
of Kauffman Bracket Skein Module, present
results for few basic 3-manifolds and examples of algebra
structures for I-bundles. Then we will
concentrate on twisted I -bundles. Our goal was to define a
structure of algebra or similar to algebra for
skein module of twisted I-bundles over non orientable surfaces F. It appears that for simple non-orientable
surfaces we can still define commutative algebra structure
Some previous Topology
Seminars