THE GEOMETRY-TOPOLOGY SEMINAR
We will meet in Skiles 269 (Skiles is the math bulding in the campus of
Gatech) on Mondays 3:00-4:00.
16 May
Hugh Morton University of Liverpool
Skeins, symmetric functions and algebras
7 April 3pm Room 255
Adam Sikora SUNY Buffalo
Analogies between number theory and 3-dimensional topology
ABSTRACT:
We will discuss similarities between prime numbers and knots
and between 3-manifolds and number fields known under the name
of Arithmetic topology.
4 April 2pm Room 255
Uwe Kaiser Boise State University
From string topology to skein theory of orientable 3-manifolds
ABSTRACT:
I will describe certain extensions and transversal refinements of constructions
due to Chas
and Sullivan, which describe the "interactions" and "self-interactions" of
families of loops in
an oriented 3-manifold M. This approach is naturally motivated by Vassiliev
theory.
The relation of the new structures with the Conway skein theory of M is discussed.
In particular the interaction (intersection) theory of the free loop space of M
is closely
related with the Hoste-Przytycki homotopy skein module. It turns out that
the Hoste-Przytycki module appears as a universal target module of a certain
refined Chas-Sullivan
structure, and also as a module over the 1-dimensional Chas-Sullivan Lie algebra
of M.
Extensions of these ideas to (a) Conway isotopy skein modules, (b) skein modules
based on the homfly
skein relation and (c) a more general theory of "transversal" chains (a skein
theory of
1-cycles and 2-cycles in the free multi-component loop space of M) will be
discussed.
10 March 3pm
John Etnyre University of Pennsylvania
Legendrian knots in R^(2n+1)
ABSTRACT:
In this talk I will discuss ways to visualize Legendrian
submanifolds in R^(2n+1) and describe several constructions of Legendrian
submanifolds. Most of the examples we will consider cannot be distinguished by
any classical invariants (in fact when n is even it was not known if there are
any nontrivial Legendrian submanifolds). Thus to distinguish them we will
describe a new invariant: contact homology. Contact homology of Legendrian knots
in R^3 has been well understood for some time now, but in higher dimensions it
is somewhat more complicated to rigorously define. I will describe joint
work with Ekholm and Sullivan where we give a rigorous definition. If time
permits I will discuss potential applications to classical knot theory and
manifold topology.
17 February
Alexandru Scorpan UC Berkeley
Existence of foliations on 4-manifolds
ABSTRACT:
We will present existence results for certain singular 2-dimensional
foliations on 4-manifolds. The singularities can be chosen to be simple,
e.g. the same as those that appear in Lefschetz pencils. There seems to
be a wealth of such creatures on most 4-manifolds. In certain cases, one
can prescribe surfaces to be transverse or be leaves of these foliations.
21 January, 11:00 am.
Scott Baldridge
Brand new examples of symplectic 4-manifolds with b+=1.
ABSTRACT:
Symplectic 4-manifolds with $b_+=1$ are roughly
classified by the canonical class $K$ and the symplectic form $\omega$.
Dusa McDuff and Dietmar Salamon gave examples for most possible cases,
leaving as an open question the existence of manifolds in one case. In this
talk we will construct examples of this case. Furthermore, we will
show that these manifolds have very special properties --- that their
Seiberg-Witten invariants are independent of the chamber structure,
that they are not complex manifolds, and that they do not have metrics
of positive scalar curvature.
9 December
Jesse Ratzkin University of Utah
Gluing and Moduli for Constant Mean Curvature Surfaces
ABSTRACT:
Recent gluing constructions have shown constant mean curvature
surfaces to be more flexible than was initially thought. In particular, we
have several new examples of noncompact, complete, embedded constant mean
curvature surfaces. I will describe some of these contructions and what
they tell us about the moduli space of such surfaces with fixed topology.
This is joint work with Rafe Mazzeo, Frank Pacard and Dan Pollack.
2 December
Nathan Dunfield Harvard
Surfaces in finite covers of 3-manifolds: The virtual Haken
conjecture.
ABSTRACT:
Haken 3-manifolds are those that contain topologically
essential embedded surfaces. Haken manifolds are very well understood
and most of the important questions in 3-manifold topology and
geometry have been answered in this case. However, there are many
3-manifolds which are not Haken. For a non-Haken 3-manifold with
infinite fundamental group, one can ask if it has a finite cover which
is Haken. That such a manifold should always have a such a cover is
the virtual Haken conjecture. I will broadly survey recent progress
towards, and evidence for, this conjecture, after first giving some
idea of why Haken manifolds are so easy to understand.
11 November
Predrag Cvitanovic Gatech
Group theory, part II
ABSTRACT:
28 October
Marcos Marino Harvard
Chern-Simons theory, matrix models, and
topological strings.
ABSTRACT:
In this talk I present a matrix
model formulation of the Witten invariant
of some rational homology spheres.
This formulation leads to a precise
correspondence between Chern-Simons theory
on lens spaces and topological strings on certain
noncompact Calabi-Yau manifolds.
26 August
Andrew Kricker University of Toronto
Homological algebra in knot theory: Khovanov's 'categorification'
of the Jones polynomial.
ABSTRACT:
Recently, Mikhail Khovanov "categorified" the Jones polynomial of links.
To be precise, Khovanov introduced a procedure which associated a
chain complex of graded vector spaces to a diagram of a link with the
remarkable properties that:
* The cohomology groups of the complex are isotopy invariants.
* The graded Euler characteristic of the complex recovers the Jones
polynomial.
A large part of this talk will be an elementary introduction to Khovanov's
construction. Time permitting, we will also survey some current and
promising
developments, namely:
* An application to constructing isotopy invariants of cobordisms.
* A spectral sequence arising from a certain filtration by spanning trees.