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Timetable (SYGW05)

Symplectic geometry - celebrating the work of Simon Donaldson

Monday 14th August 2017 to Friday 18th August 2017

Monday 14th 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.
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.
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.
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. 
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 15th August 2017
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.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Mikhail Gromov (IHES)
100 Problems around Scalar Curvature
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.
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.
Wednesday 16th August 2017
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. 
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
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.
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)
Thursday 17th August 2017
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)

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.
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. 

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.
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.
Friday 18th August 2017
10:00 to 11:00 Katrin Wehrheim (University of California, Berkeley)
A polyfold lab report
Co-authors: Ben Filippenko (UC Berkeley), Wolfgang Schmaltz (UC Berkeley), Zhengyi Zhou (UC Berkeley), Joel Fish (UMass Boston), Peter Albers (Uni Heidelberg)

I will survey various results on applications and extensions of Hofer-Wysocki-Zehnder's polyfold theory: - fiber products of polyfold Fredholm sections - equivariant transversality - existence and obstructions - equivariant fundamental class - coherent perturbations - Gromov-Witten axioms - two polyfold proofs of the Arnold conjecture These are joint with or due to Peter Albers, Ben Filippenko, Joel Fish, Wolfgang Schmaltz, and Zhengyi Zhou.

Related Links
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Frances Kirwan (University of Oxford)
Graded linearisations for linear algebraic group actions
Co-authors: Gergely Berczi (ETH Zurich), Brent Doran (ETH Zurich)

In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford's geometric invariant theory (GIT) to construct and study such quotient varieties. The aim of this talk is to describe how Mumford's GIT can be extended effectively to actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way.

The classical examples of moduli spaces which can be constructed using Mumford's GIT are the moduli spaces of stable curves and of (semi)stable bundles over a fixed curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves (with suitable fixed discrete invariants) or unstable bundles (with fixed Harder-Narasimhan type).
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 Thomas Walpuski (Massachusetts Institute of Technology)
On the ADHM Seiberg–Witten equations
Co-authors: Andriy Haydys (Albert-Ludwigs-Universität Freiburg), Aleksander Doan (Stony Brook University)

The ADHM Seiberg–Witten equations are a class of generalized Seiberg–Witten equations associated with the hyperkähler quotient appearing in the Atiyah, Drinfeld, Hitchin, and Manin's construction of the framed moduli space of ASD instantons on R4.  Heuristically, degenerations of solutions to the ADHM Seiberg–Wiitten equation are linked with Fueter sections of bundles of ASD instantons moduli spaces (through the Haydys correspondence).  In joint work with Andriy Haydys, we studied when and how this heuristic can be made rigorous (following work of Taubes on flat PSL(2,C)–connections.)  This work immediately leads to a number of questions.  In particular, whether a given Fueter section can be realized as a limit and whether singular Fueter sections might appear. In joint work with Aleksander Doan (partially in progress), we answer the first question and the second (assuming a conjectural improvement of the work with Haydys).  Time permitting, I will briefly discuss which role we expect the ADHM Seiberg–Witten equation to play in gauge theory on G2–manifolds.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Ivan Smith (University of Cambridge)
Symplectic topology of K3 surfaces via mirror symmetry
Co-Author: Nick Sheridan (Princeton & Cambridge)

We prove that there are symplectic K3 surfaces for which the Torelli group, of symplectic mapping classes
acting trivially on cohomology, is infinitely generated.  The proof combines homological mirror symmetry for
Greene-Plesser mirror pairs with results of Bayer and Bridgeland on autoequivalence groups of derived categories
of K3 surfaces.  Related ideas in mirror symmetry yield a new symplectic viewpoint on Kuznetsov's K3-category
of a cubic fourfold.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons