Feb 19

Erkao Bao (UCLA): Semiglobal Kuranishi structures and contact homology
Abstract: Contact homology was proposed and studied by Eliashbergy, Givental and Hofer 16 years ago. It is a very powerful tool to distinguish different contact structures. However, the rigorous definition did not come out until last year.
In this talk, we will first see that the naive definition does not work because the spaces of "trajectories" that we count to define the differential of contact homology are not transversally cut out. Then we will construct a finite dimensional space K around the spaces of "trajectories" in a systematical way, and inside K we perturb the "trajectories" so that now they are transversally cut out. The space K together with the perturbation is called a semiglobal Kuranishi structure.

Feb 26

Hans Boden (Mcmaster U.) A classical approach to virtual knots
Abstract: Given a virtual knot $K,$ there are various groups naturally associated to $K,$ including the analogue of classical knot group $G_K,$ the virtual knot group $VG_K,$ the extended knot group $EG_K$ (SilverWilliams), and the quandle group $QG_K$ (Manturov). Each group occurs as the quotient of $VG_K$ in a natural way, and we introduce Alexanderline invariants derived from it using elementary ideal theory. For instance, associated to the k=0 ideal of the virtual knot group $VG_K$ is a polynomial $H_K(s,t,q)$ in three variables, and we show how the $q$width of $H_K$ gives information about the virtual crossing number of $K.$ The polynomial $H_K(s,t,q)$ satisfies a skein formula, and one can define a twisted polynomial invariant of virtual knots for any representation from $VG_K$ to $GL_n(R).$

Mar 4

L. Siebenmann (Université de ParisSud) Towards a more geometric understanding of planar flows.
Abstract: Concerning an oriented foliation F of the plane R^2 by the
orbits of a given nowheresingular flow on R^2, there is a
risky conjecture that seems of interest. It involves T the
universal Hausdorff quotient of the (usually nonHausdorff)
leaf space LF of F. This T is known to be in the class of
trees that are coherently oriented, metrizable, separable,
having only countably many branch points, and having no
extremal points. Call such a tree admissible. The preimages in
R^2 of the branch points of T are the socalled Conley
pseudoleaves of F, each of which is a countable union of Reeb
pseudoleaves of F.
The conjecture asserts that there is a natural continuous and
surjective "classifying map" f: D^2 > T+, of a 2 disk D^2
compactifying R^2, onto an end compactification T+ of T, such
that the connected components of the point preimages of the
restriction of f to the interior R^2 of D^2 is the foliation F.
I have studied a class of closed submanifoldswithboundary in
R^2 that are foliated by F. This has led me to a pleasant proof of a
variant of Mather's compactification theorem and also to some
progress on the risky conjecture. In particular, I will
indicate how the conjecture can hopefully be proved in case
the set of branch points in T is locally finite.

Apr 22

Juanita PinzonCaicedo (U. of Georgia) An Overview of Relative Trisections
Abstract: A trisection of closed 4—manifold is a decomposition of a 4—manifold into three copies of $\natural^k S^1\times B^3$, that intersect pairwise in 3dimensional handlebodies, and with triple intersection a 2dimensional surface. Relative trisections are a generalization of a trisection to include 4manifolds with boundary. In the talk I will present the basics of relative trisections starting with their relationship to open book decompositions of the bounding manifolds. I will then introduce a stabilization operation that gives rise to a statement about the uniqueness of relative trisections, thus complementing Gay and Kirby's proof of the existence of relative trisections. Finally, I will introduce the notion of diagrams of relative trisections and describe a method to recover the open book decomposition of the bounding manifold from the trisection diagrams.
