# Timetable (SYGW05)

## Symplectic geometry - celebrating the work of Simon Donaldson

Monday 14th August 2017 to Friday 18th August 2017

 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 Denis Auroux (University of California, Berkeley)Speculations about homological mirror symmetry for affine hypersurfaces The wrapped Fukaya category of an algebraic hypersurface H in (C*)^n is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold whose moment polytope is determined by the tropicalization of H.  In this talk we will first explain the statement, and then discuss a conjectural enhancement to a "relative" version of homological mirror symmetry for the pair ((C*)^n, H). We will illustrate these ideas on simple examples such as pairs of pants. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Mikhail Gromov (IHES)100 Problems around Scalar Curvature 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 Peter Ozsvath (Princeton University)Computing knot Floer homology In this continuation of Zoltan Szabo's talk, I will sketch the identification between knot Floer homology and an algebraically defined knot invariant. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Mina Aganagic (University of California, Berkeley)Mathematical applications of little string theory I will describe applications of a six dimensional string theory to the Geometric Langlands Program and to the Knot Categorification Program. This is based on joint works with Edward Frenkel and Andrei Okounkov. INI 1
 09:00 to 10:00 Nigel Hitchin (University of Oxford)Remarks on Nahm's equations We consider Nahm's equations and associated ones from the point of view of the B-field action on the moduli space of generalized holomorphic bundles on . Particular attention is paid to the fixed points of this action and the associated spectral curves. INI 1 10:00 to 10:30 Morning Coffee 10:30 to 11:30 Michael Atiyah (University of Edinburgh)From Euler to Poincare   I will describe how the 11/8 conjectures of Donaldson theory are related to the Riemann Hypothesis INI 1 11:30 to 12:00 Free time 12:00 to 13:00 Tomasz Mrowka (Massachusetts Institute of Technology)An approach to the four colour theorem via Donaldson- Floer theory This talk will outline an approach to the four colour theorem using a variant of Donaldson-Floer theory. To each trivalent graph embedded in 3-space, we associate an instanton homology group, which is a finite-dimensional Z/2 vector space. Versions of this instanton homology can be constructed based on either SO(3) or SU(3) representations of the fundamental group of the graph complement.  For the SO(3) instanton homology there is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero. It is not unreasonable to conjecture that, if the graph lies in the plane, the Z/2 dimension of the SO(3) homology is also equal to the number of Tait colourings which would imply the four colour theorem.   This is joint work with Peter Kronheimer. INI 1 13:00 to 14:00 Buffet Lunch at INI 14:00 to 16:30 Discussion; tributes to Sir Simon Donaldson 17:00 to 18:00 CONCERT: Music for Simon - Old Library, Pembroke College Oscar Garcia-Prada, countertenor David Mason, piano   Music by Byrd, Caccini, Dowland, Granados, Handel, Lambert, Marin, Monteverdi, Purcell, Quilter and  Vaughan Williams 18:15 to 22:00 Pre-dinner Drinks & Formal Dinner at Pembroke College (by invitation only)
 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