09:00 to 09:50 Registration 09:50 to 10:00 Welcome from David Abrahams (INI Director) 10:00 to 11:00 Paul Seidel (Massachusetts Institute of Technology)Fields of definition of Fukaya categories of Calabi-Yau hypersurfaces Fukaya categories are algebraic structures (in fact, families of such structures) associated to symplectic manifolds. Kontsevich has emphasized the role of these structures as an intrinsic way of thinking of the "mirror dual" algebraic geometry. If that viewpoint is to be fruitful, Fukaya categories of specific classes of manifolds should exhibit deeper structural features, which reflect aspects of the "mirror geometry". I will explain what one can expect in the case of Calabi-Yau hypersurfaces in a Lefschetz pencil. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Kenji Fukaya (Stony Brook University)Atiyah Floer conjecture Co-author: Aliakbar Daemi (Simons Center for Geometry and Physics) Atiyah Floer conjecture relates the instanton Floer homology (the Floer theory of 3 manifolds based on ASD instanton (Donaldson theory) with Lagrangian Floer theory. I am going to report the status of our project to prove this conjecture. INI 1 12:30 to 13:00 Free time 13:00 to 14:00 Lunch @ Churchill College 14:00 to 14:30 Free time 14:30 to 15:30 Eleny Ionel (Stanford University)The Gopakumar-Vafa conjecture for symplectic manifolds Co-authors: Thomas H Parker (MSU); Penka Georgieva (IMJ-PRG). In the late nineties string theorists Gopakumar and Vafa conjectured that the Gromov-Witten invariants of Calabi-Yau 3-folds have a hidden structure: they are obtained, by a specific transform, from a set of more fundamental "BPS numbers", which are integers. In joint work with Tom Parker, we proved this conjecture by decomposing the GW invariants into contributions of clusters" of curves, deforming the almost complex structure and reducing it to a local calculation. This talk presents some of the background and geometric ingredients of our proof, as well as recent progress, joint with Penka Georgieva, towards proving that a similar structure theorem holds for the real GW invariants of Calabi-Yau 3-folds with an anti-symplectic involution. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Zoltan Szabo (Princeton University)Knot Floer homology and algebraic methods The aim of this talk is to present some recent advances in knot Floer homology. INI 1 17:00 to 18:00 Welcome Wine Reception at INI
 10:00 to 11:00 Peter Kronheimer (Harvard University)An SU(3) variant of instanton homology for webs Let K be a trivalent graph embedded in 3-space (a web). In an earlier talk at this conference, Tom Mrowka outlined how one may define an instanton homology J(K) using gauge theory with structure group SO(3). This invariant is a vector space over Z/2 and has a conjectured relationship to Tait colorings of K when K is planar. In this talk, we will explore a variant of this construction, replacing SO(3) with SU(3). With this modified version, the dimension of the instanton homology is indeed equal to the number of Tait colorings when K is planar. (Without the assumption of planarity, the dimension is sometimes larger, sometimes smaller.) There is a further variant, with rational coefficients, whose dimension is equal to the number of Tait colorings always.Coauthors: Tom Mrowka (MIT) INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Emmy Murphy (Northwestern University)Graph Legendrians and SL2 local systems We will discuss some connections between framed local systems on punctured surfaces and pseudo-holomorphic curves in 5 dimensional contact manifolds. We will also discuss connections with planar graph colorings, representations of dg algebras, Lagrangian cobordisms, loose Legendrians, and maybe some other things. This talk is based on work in progress with Roger Casals. INI 1 12:30 to 13:00 Free time 13:00 to 14:00 Lunch @ Churchill College 14:00 to 15:00 Song Sun (Stony Brook University)Singularities of Hermitian-Yang-Mills connections and the Harder-Narasimhan-Seshadri filtration Co-Author: Xuemiao Chen (Stony Brook)The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connection over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu to the case of reflexive sheaves, and the corresponding connection may have singularities. We study tangent cones around such a singularity, which is defined in the usual geometric analytic way,  and relate it to the Harder-Narasimhan-Seshadri filtration of a suitably defined torsion free sheaf on the projective space, which is a purely algebro-geometric object. INI 1 15:00 to 15:30 Afternoon Tea 15:30 to 16:30 Dusa McDuff (Barnard College)Constructing the virtual fundamental cycle Consider a  space $X$, such as a compact space of $J$-holomorphic stable maps with closed domain, that is the zero set of a Fredholm operator. This note explains how to define the  virtual fundamental class of $X$ starting from a finite dimensional reduction in the form of a Kuranishi atlas, by  representing $X$ as the zero set of a section of a (topological) orbibundle that is constructed from the atlas.     Throughout we assume that the   atlas satisfies Pardon's topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. INI 1 16:30 to 17:00 Free time 17:00 to 18:00 John Pardon (Princeton University)Existence of Lefschetz fibrations on Stein/Weinstein domains I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities).  We also prove an analogous result for Stein domains in the complex analytic setting.  The main tool used to prove these results is Donaldson's quantitative transversality. INI 1