2015 Summer School on Moduli Problems in Symplectic Geometry

ABSTRACTS

Mini-courses

Wendl (3 lectures): Classical transversality methods in SFT. For the purposes of this minicourse, a transversality result is called "classical" if it is of the form, "for a certain class of (domain-independent) almost complex structures J, the moduli space of J-holomorphic curves of a certain type is smooth and has the correct dimension." I will discuss results of this kind for finite-energy punctured J-holomorphic curves in symplectic cobordisms, i.e. the objects that are meant to be counted in Symplectic Field Theory. The focus on classical methods means that we exclude a variety of fancier notions such as domain-dependent/inhomogeneous/multivalued/abstract perturbations, thus it is in the nature of things that classical methods will not always suffice for a given application. But classical methods are generally what you should try first, because when they do work, they'll often make your life easier and reveal geometric information that can be harder to access in more abstract approaches.

Wehrheim/Fish/Hofer (9 lectures): Polyfolds and the construction of Symplectic Field Theory. (mini-course page) Polyfold theory provides a language and a large body of results in an area, which is a mixture of a generalized differential geometry, nonlinear functional analysis, and category theory. It includes a very general concept of a Fredholm section accompanied by Sard-Smale type transversality theory over the rational numbers, i.e. perturbing through rationally weighted multi-sections. As such, it is a powerful tool to address complicated transversality issues as they arise in constructing compact moduli spaces in symplectic geometry. In the first week, Fish and Wehrheim will introduce the analytic foundations of polyfold theory, including the notions of sc-calculus, retracts, M-polyfolds, strong bundles, Fredholm sections and regularization theorems. In the second week, Hofer will build on these foundations, aiming to construct the moduli spaces needed for general Symplectic Field Theory.

Hutchings (3 lectures): Obstruction Bundle Gluing. Obstruction bundle gluing is a method of calculating the number of ways of gluing certain configurations in which transversality fails, but not too badly. We will introduce this technique and show how it works in simple examples from Morse theory, contact homology, and embedded contact homology. (There might not be enough time to cover all of these examples.)

Ekholm (3 lectures): Knot contact homology. We define knot contact homology as the Legendrian differential graded algebra of the unit conormal lift of a knot and show how to compute it in terms of flow trees. Next, we show how to relate knot contact homology to a version of string topology. We conclude the mini course with how knot contact homology is related to the physical theories, Chern-Simons theory and topological string theory. Throughout we will use the 3-sphere as a guiding example.

• References:
  • Ekholm, Tobias; Etnyre, John B.; Ng, Lenhard; Sullivan, Michael G. Knot contact homology. Geom. Topol. 17 (2013), no. 2, 975–1112.
  • Ng, Lenhard Framed knot contact homology. Duke Math. J. 141 (2008), no. 2, 365–406.
  • Aganagic, Mina; Ekholm, Tobias; Ng, Lenhard; Vafa, Cumrun Topological strings, D-model, and knot contact homology. Adv. Theor. Math. Phys. 18 (2014), no. 4, 827–956.
  • Ekholm, Tobias Notes on topological strings and knot contact homology. Proceedings of the Gökova Geometry-Topology Conference 2013, 1–32, Gökova Geometry/Topology Conference (GGT), Gökova, 2014.

Abouzaid (3 lectures): Lagrangian Floer cohomology in families. We will begin with a brief overview of Lagrangian Floer cohomology, in a setting designed to minimise technical difficulties (i.e. no bubbling). Then we will ponder the question of what happens to Floer theory when we vary Lagrangians in families, which we will not require to be Hamiltonian. We will see rigid analytic spaces naturally arise from such families; these spaces are the analogue of complex analytic manifolds over the Novikov field. In order to be faithful to the theme of the conference, we will end by constructing lots of moduli spaces in order to see that Floer complexes give rise to analytic coherent sheaves. In preparation, for the course, reading arXiv:1404.2659 is recommended, while reading arXiv:1408.6794 is definitely not recommended. A revised version of the second reference is in preparation, and should be available by the beginning of the summer school.

 
Abstracts:
Monday July 6

• McDuff Introduction to Regularization Problems. Moduli spaces of pseudoholomorphic curves arise as the zero set of a Fredholm section of a suitable bundle, and one expects and hopes that they can be regularized in order to define invariants that are stable under perturbations. This lecture provides an overview of some of the analytic difficulties that must be solved in order to construct such a regularization, and briefly explains some traditional approaches to their solution, namely via geometric regularizations and finite dimensional reductions.

 

• Wendl Classical transversality methods in SFT (1 of 3). I will give a quick review of the Sard-Smale theorem and the universal moduli space approach to transversality, discuss the relative merits of classical vs. inhomogeneous perturbations, and sketch proofs of the standard theorems stating that the moduli space of regular J-holomorphic curves in symplectic cobordisms is a smooth orbifold, and that all somewhere injective curves are regular for generic J.

 

• Wehrheim Introduction to Polyfold Regularization. This lecture will discuss the overall ideas and challenges in regularizing moduli spaces, and introduce the two basic ideas behind polyfold theory: Making reparametrization actions "smooth" and making pregluing a "chart map". [related literature: Sections 2.1 and 3.3 of Polyfolds: A First and Second Look. Related videos: Lecture 8 and Lecture 20 from Wehrheim's special topics course.]

 

• Fish sc-Banach spaces and the sc-calculus (1 of 3). Polyfold theory is built on two new fundamental analysis concepts, and this talk is focused of the first: the sc-calculus. We discuss how in general the action of a finite dimensional smooth reparametrization group on typical Banach spaces of maps is not smooth (in fact, not even differentiable), and then introduce sc-Banach spaces and the notion of sc-differentiability. Two key results of this talk are that the action of reparametrization is sc-smooth, and for sc-differentiable functions the chain rule holds, so that many constructions in classical differential geometry functorially extend to sc-differentiable geometry. [Related literature: Sections 2.2 and 4.2 of Polyfolds: A First and Second Look. Related videos: Lecture 20 from Wehrheim's special topics course.

Tuesday July 7

• Biran Lagrangian Cobordisms, Dehn-twists and Real Algebraic Geometry. We will explain the relevance of Lagrangian cobordisms in Lefschetz fibrations to the study of the (derived) Fukaya category of the fiber. In particular we will give a cobordism interpretation of Seidel's long exact sequence, introduce cobordism groups and also outline how to study real algebraic structures using cobordisms. The talk is based on joint work with Octav Cornea.

 

• Wendl Classical transversality methods in SFT (2 of 3). In this talk I will discuss two transversality results that are standard but perhaps not so widely understood: (1) Dragnev's theorem that somewhere injective curves in symplectizations are regular for generic translation-invariant J, and (2) my theorem on automatic transversality in 4-dimensional symplectic cobordisms (which generalizes earlier results for closed curves by Gromov, Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin). The common feature of these two theorems is that both can be proved by considering the restriction of the usual linearized Cauchy-Riemann operator to the "generalized normal bundle" of a (not necessarily immersed) holomorphic curve.

 

• Fish The rise of sc-retracts (2 of 3). In this talk, we discuss the second of two fundamental analysis concepts polyfold theory is built on: sc-retracts. In particular, we discuss how they arise naturally as a means of using pre-gluing maps to parametrize a neighborhood of nodal and non-nodal (or broken and unbroken) maps near a nodal (or broken) map. Despite locally varying dimensions, such retracts support a version of the sc-calculus on which the chain rule holds, and we define M-polyfolds (manifold-like polyfolds) to be those topological spaces locally modeled on such retracts. [Related literature: Sections 2.3 and 5.1 of Polyfolds: A First and Second Look. Related videos: Lecture 22 from Wehrheim's special topics course.
•  
• Wendl (discussion)

Wednesday July 8

• Fish Boundary, corners, strong bundles, and implicit function theorems (3 of 3). In this talk, we generalize the notion of sc-retracts to include cases with boundary and corner structure. In addition, we develop the notion of a strong bundle (of which the Cauchy-Riemann operator is a section) and state an implicit function theorem for transverse Fredholm sections with compact zero-set, which guarantees the zero set of the section is a manifold with boundary and corners, with boundary/corner structure induced from the ambient M-polyfold. [Related literature: Sections 5.2, 5.3, and 6.1 of Polyfolds: A First and Second Look. Related videos: Lecture 23 and Lecture 24 from Wehrheim's special topics course.
• Eliashberg Midsummer Bures Dreams. I will discuss some questions and conjectures concerning symplectic topology of Weinstein manifolds.

 

• Bottman (discussion)

 

• Hutchings Obstruction Bundle Gluing (1 of 3). (See mini-course description above for abstract.)

Thursday July 9

• Viterbo TBA

 

• Wendl Classical transversality methods in SFT (3 of 3). There are easy examples showing that classical transversality methods cannot always succeed for multiply covered holomorphic curves, but the situation is not hopeless. In this talk I will describe two approaches that sometimes lead to interesting results: (1) analytic perturbation theory, and (2) splitting the normal Cauchy-Riemann operator of a curve along irreducible representations of its automorphism group. Both were pioneered by Taubes in his work on the Gromov invariant and Seiberg-Witten theory in the 1990's, and I will illustrate them by sketching two proofs that the multiply covered holomorphic tori counted by the Gromov invariant are regular for generic J. If time permits, I will discuss some ideas as to how both methods can be applied more generally.

 

• Wehrheim Regularization theorem for Fredholm sections of M-polyfold bundles (2 of 2). This lecture will state a rigorous version of this theorem, and explain the notion of a (sc-)Fredholm section. [related literature: Sections 6.2 and 6.3 of Polyfolds: A First and Second Look. Related videos: Lecture 24 and Lecture 25 from wehrheim's special topics course.]

 

• Hutchings Obstruction Bundle Gluing (2 of 3). (See mini-course description above for abstract.)

Friday July 10

• Bottman Fredholm theory and Deligne-Mumford spaces for witch balls. In work-in-progress with Katrin Wehrheim, we aim to bind together the Fukaya categories of many different symplectic manifolds into a single algebraic object. This object is the "symplectic A-infinity-2-category", whose objects are symplectic manifolds, and where hom(M,N):=Fuk(M-xN). At the core of our project are witch balls -- certain pseudoholomorphic quilts with figure eight singularity. I will discuss recent progress: toward the construction of the moduli space of domains on one hand, and toward establishing the Fredholm property on the other.

 

• Hutchings Obstruction Bundle Gluing (3 of 3). (See mini-course description above for abstract.)

 

• Nelson An Integral lift of contact homology. Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, jointly with Hutchings we have managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S^1-equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.

 

• Hutchings (discussion)

 
Monday July 13

• Hofer Polyfolds and the construction of Symplectic Field Theory (1 of 4). Topics:
  1. Short overview.
  2. The category of stable maps.
  3. I/O structures.
  4. Covering structures.
  5. Additional structural constraints.

• Solomon Holomorphic disks and special Lagrangians. Special Lagrangians in Calabi-Yau manifolds are expected to be plentiful. However, in practice, it is difficult to find special Lagrangian submanifolds in compact Calabi-Yau manifolds except for two special classes: fixed points of anti-symplectic involutions and holomorphic Lagrangians in hyper-Kahler manifolds. Thus it is natural to look for a modified special Lagrangian condition that reduces to the standard one in those two cases. I will describe how moduli spaces of holomorphic disks give rise to such a modification of the special Lagrangian condition. This is joint work with G. Tian

 

• Hofer (discussion)

 

• Ekholm Introduction to knot contact homology (1 of 3). We define knot contact homology as the Legendrian differential graded algebra of the unit conormal lift of a knot. We show how to compute it for knots in the three sphere using flow trees and discuss some of its basic properties. We also introduce the augmentation variety.

 

• Abouzaid Lagrangian Floer cohomology in families (1 of 3). (See mini-course description above for abstract.)

Tuesday July 14
Free Day

Wednesday July 15

• Hofer Polyfolds and the construction of Symplectic Field Theory (2 of 4). Topics:
  1. Polyfold structures.
  2. Consequences of polyfold structures.
  3. Weighted categories and their smooth versions.
  4. The polyfold of stable maps.
  5. Bundle category

• Ekholm Knot contact homology and string topology (2 of 3). We show how to relate knot contact homology to a version of string topology. More precisely we express knot contact homology in terms of strings in the singular space which is the union of the three sphere and the Lagrangian conormal of the knot that split in a certain way when they hit the knot.

 

• Bourgeois A symplectic invariant for contact manifolds. The construction of S^1-equivariant symplectic homology with Alexandru Oancea can be used to define an invariant for a wide class of contact manifolds. This is a substitute for cylindrical contact homology, which often has transversality issues. This symplectic invariant can then be applied to the study of closed Reeb orbits.

 

• Abouzaid Lagrangian Floer cohomology in families (2 of 3). (See mini-course description above for abstract.)

Thursday July 16

• Hofer Polyfolds and the construction of Symplectic Field Theory (3 of 4). Topics:
  1. Strong bundle structure and the CR-section as Fredholm functor.
  2. Polyfold packaging of the SFT problem.
  3. Smooth Multisection functors and smooth weighted subcategories.
  4. Construction of sc^+ multisection functors.
  5. Auxiliary norms and compactness control.

• Ekholm Knot contact homology, Chern-Simons theory, and topological string (3 of 3). We explain how knot contact homology is related to the physical theories mentioned in the title. We report on recent progress developing symplectic field theory beyond genus 0 and how this relates to topological strings and open Gromov-Witten invariants.

 

• Smith A symplectic Khovanov Puzzlebook I will discuss aspects of my joint work with Mohammed Abouzaid on symplectic Khovanov cohomology, focussing on open questions.

 

• Ekholm (discussion)

 

• Abouzaid Lagrangian Floer cohomology in families (3 of 3). (See mini-course description above for abstract.)

Friday July 17

• Hofer Polyfolds and the construction of Symplectic Field Theory (4 of 4). Topics:
  1. Perturbation algorithm in the homogeneous case.
  2. Perturbation algorithm for relating two perturbations.
  3. Remarks on orientations.
  4. Representation theory and SFT.

• Cristofaro-Gardiner Symplectic embeddings of products. McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an “infinite staircase” determined by the odd-index Fibonacci numbers. We show that this result still holds in all higher even dimensions when we "stabilize" the embedding problem. This is joint work with Richard Hind.

 

• Abouzaid (discussion)

 

• Hofer (discussion)