Titles and Abstracts
Joan Birman (Columbia Univeristy):
Lorenz Links
Abstracts:
23 years ago Bob Williams and I wrote a paper about Lorenz
links, published in Topology. Recently, there has been new interest in
this family of links because of the work of Etienne Ghys and also W.
Tucker. They are a rich family, with many special properties, and also
many open questions. My talk will discuss their origins, some of their
properties, and also what we do not know at this time.
Elisenda Grigsby (Columbia University): A combinatorial description
of the knot Floer homology of cyclic branched covers
Abstract: Knot Floer homology, developed by Ozsvàth and Szabò and
independently by Rasmussen, associates a sequence of graded abelian groups to a nullhomologous knot in a closed
three-manifold, Y.
When Y is S 3, the Euler characteristic of their invariant is the classical Alexander
polynomial; hence, knot
Floer homology has been able to provide more complete answers to some questions that the Alexander polynomial
addresses only partially. For example, knot Floer homology detects the Seifert genus of a knot, provides a complete
test to determine fiberedness, and yields (non-sharp) bounds on the 4-ball genus.
Furthermore, recent work of Manolescu, Ozsvàth, and Sarkar have shown that knot Floer homology has a completely
combinatorial description when Y=S 3. I will discuss how their methods can be extended to
yield a combinatorial
description of the knot Floer homology of the preimage of a knot inside its cyclic branched covers. The motivation
behind studying these objects is to obtain new concordance invariants for knots.
Richard Hain (Duke University): Relative weight filtrations on completions of mapping class groups
Abstract:
In this talk I will explain how to complete the mapping
class group of a surface S. To each curve system on S (a set of simple
closed curves on S lying in distinct isotopy classes) one can
associate a "relative weight filtration." Even though the only known
constructions of relative weight filtrations use Hodge or Galois
theory these filtrations have a close relationship to topology, which
I shall explain.
Ko Honda (University of Southern California): Reeb vector fields and open book
decompositions
Abstract: According to a theorem of Giroux, there is a 1-1
correspondence between isotopy classes of contact structures and
equivalence classes of open book decompositions. We give partial results
towards calculating the contact homology of a contact structure (M, ξ)
(in dimension 3) which is supported by an open book with pseudo-Anosov
monodromy. This is joint work with Vincent Colin.
Eleny Ionel (Stanford University):
Singular spaces and Gromov-Witten invariants
Abstract:
In this talk we will consider the problem of extending the
notion of relative Gromov-Witten invariants of a pair (X,V) to singular
settings (e.g. normal crossings). These singular invariants
should record enough information so that one can recover for example
the GW invariants of an (appropriate) smoothing of either X or V.
In particular we will describe a definition of the relative Gromov-Witten
invariants of a pair (X,V) where X is a smooth symplectic manifold, but V
is a transverse union of real codimension two almost complex submanifolds.
We will then discuss how the relative Gromov-Witten invariants GW(X,V) behave
when either X or V degenerate into a space with normal crossings.
Mikhail Khovanov (Columbia University):
Braid cobordisms and triangulated categories
Abstract:
We'll survey a variety of braid group actions
on triangulated categories and explain how some of them
extend to representations of the category of braid cobordisms.
Feng Luo (Rutgers University):
Volume and angle structures on 3-manifolds
Abstract:
We introduce a finite-dimensional variational approach to find
constant curvature metrics on triangulated closed 3-manifolds. The
approach is based on the Schlaefli formula for the volumes of tetrahedra.
The main result shows that for a 1-vertex triangulation of a closed
3-manifold if the volume function on the space of all angle structures has
a local maximum point, then either the manifold admits a
constant-curvature Riemannian metric, or the manifold is reducible.
Daniel Ruberman (Brandeis University): Periodic Dirac operators and positive scalar curvature on
4-manifolds
Abstract:
Which smooth manifolds admit a Riemannian metric whose
scalar curvature is positive? This question has been attacked using
minimal surface theory (Schoen-Yau) and the Dirac operator
(Lichnerowicz, Gromov-Lawson, and many others). Using Taubes' theory
of periodic-end operators, we will discuss the Dirac operator on a
non-compact 4-manifold that is an infinite cyclic cover of a compact
spin manifold X. We show that such an operator is Fredholm for a
generic metric, and use this to give a new interpretation of the
Rohlin invariant of X. This new interpretation gives rise to a new
obstruction to the existence of metrics of positive scalar curvature.
This is joint work with Nikolai Saveliev (U. of Miami).
Please send comments to Shelly Harvey - shelly at math.rice.edu.
