# Live from Budapest

From July 9-20, we’re at the CAST (Contact and Symplectic Topology) 2012 summer school and conference at the Alfred Renyi Institute of Mathematics in Budapest, Hungary.  The format is two mini-courses per week along with two research talks every afternoon.

First Week:

Robert Lipshitz – Bordered Heegaard Floer Homology (lecture notes)

This week, Robert lectured on a Heegaard Floer-like structure associated to bordered 3-manifolds, which was constructed by himself, Peter Ozsvath and Dylan Thurston.  This associates two modules, $\hat{CFA}(Y), \hat{CFD}(Y)$ to a 3-manifold with connected boundary, depending upon whether we think of the boundary as positively or negatively oriented.  There are also more complicated modules associated to the nonconnected boundary case that mixes -A and -D structure.  The main theorem is the Pairing Theorem (LOT), which states that for $Y = Y_1 \cup Y_2$ then $\hat{CF}(Y) = \hat{CFA}(Y_1) \otimes \hat{CFD}(Y_2)$ where $\hat{CF}$ is the chain complex  for the hat version of Heegaard Floer.  For the first three days, he developed the terminology and combinatorics of parametrizations of the boundary $\partial Y$, defined the differential and discussed the relevant moduli of holomorphic curves.  One key aspect of this construction is that it relies on his cylindrical reformulation of HF to bring it more in line analytically with Symplectic Field Theory.  The fourth day, Jen Hom went over how to compute $\hat{CFA}(Y)$ for knot complements and the fifth day Robert discussed computing $\hat{CFA}(Y)$ for mapping tori and that it may be possible to simplify computations of $\hat{HF}(Y)$ by computing the bordered version for generators of the mapping class group and composition of these elementary cobordisms.

Kai Cieliebak – Stein Structures: Existence and Flexibility (lecture notes)

The subject of this talk was building Stein structures and classifying them up to the appropriate homotopy equivalence.  Most of the content comes from an upcoming textbook on Stein structures written by Cieliebak and Yasha Eliashberg.  A Stein manifold is a complex manifold that embeds properly, holomorphically into some $\mathbb{C}^n$; equivalently, it admits an exhausting (proper, bounded below), J-convex (or strictly plurisubharmonic) function $\phi$, which can be used to set up the embedding into affine space.  An important result, which is due to Milnor and can be verified easily, is that the Morse index of a nondegenerate critical point of $\phi$ must be less than or equal to $n$.  The goal of the lecture series is to prove the converse, that if an open, smooth, oriented manifold $M$ of dimension $2n > 4$ admits an almost complex structure $J$ and a generic Morse function $\phi$ with critical points of index less than $n$, then there is a homotopy of almost-complex structures from $J$ to some $J'$ so that $\phi$ is $J'$-convex, giving a Stein structure.  The basic idea is to use a Stein h-cobordism theorem to simplify the set of critical points and attaching spheres of handles, build a model J-convex structure on a handles and extend the standard Stein structure on the unit ball as we add on handles.  The last issue is flexibility and classifying Stein structures up to Stein homotopy.  In the subcritical case, it follows from the h-principle for isotropic embeddings that if two Stein structures $(V,J), (V',J')$ have homotopic almost-complex structures then they are Stein homotopic.  This is enough for the subcritical case, when $M$ has the homotopy type of an $n-1$-dimensional manifold.  The critical case relies on Max Murphy’s notion of loose legendrians, which do satisfy an h-principle.   A Stein structure is flexible if its critical attaching spheres are loose, and flexible Stein structures with homotopic almost-complex structures are in fact Stein homotopic.

Second Week:

Michael Hutchings – Embedded Contact Homology

Lenny Ng – Knot Contact Homology