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.