Workshop on Symplectic Field Theory VIII

Workshop on Symplectic Field Theory VIII - Program

schedule precourse outlines minicourse outlines additional talks

Schedule

All talks will take place in Lecture Hall 201 at the Humboldt University's Wirtschaftswissenschaftliche Fakultät (School of Business and Economics), located at Spandauer Strasse 1, near Hackescher Markt. See the practical information page for more details.

Changes from the printed version of the schedule are marked in red.

Precourse

Saturday, August 6, 2016
- 9:30 - 10:00: arrival / registration
- 10:00 - 11:30: Courte precourse, part 1
  lunch
- 13:00 - 14:30: Courte precourse, part 2
  coffee break
- 15:00 - 16:30: Nelson precourse, part 1
Sunday, August 7, 2016
- 10:00 - 11:30: Nelson precourse, part 2
  lunch
- 13:00 - 14:30: Fauck precourse, part 1
  coffee break
- 15:00 - 16:30: Fauck precourse, part 2

Main workshop

Monday, August 8, 2016
- 9:00 - 9:30: arrival / registration
- 9:30 - 11:00: Oancea minicourse, part 1
  coffee break
- 11:30 - 12:30: Dimitroglou Rizell
  lunch
- 14:30 - 16:00: Ekholm minicourse, part 1
  coffee break
- 16:30 - 18:00: McLean minicourse, part 1
- 18:00 - 19:30: reception
Tuesday, August 9, 2016
- 9:30 - 11:00: Oancea minicourse, part 2
  coffee break
- 11:30 - 12:30: Lekili
  lunch
- 14:30 - 16:00: Ekholm minicourse, part 2
  coffee break
- 16:30 - 17:30: Gutt
Wednesday, August 10, 2016
- 9:00 - 10:30: McLean minicourse, part 2
  coffee break
- 11:00 - 12:30: Oancea minicourse, part 3
  afternoon free
Thursday, August 11, 2016
- 9:30 - 11:00: Ekholm minicourse, part 3
  coffee break
- 11:30 - 12:30: Abouzaid
  lunch
- 14:30 - 16:00: McLean minicourse, part 3
  coffee break
- 16:30 - 18:00: Oancea minicourse, part 4
Friday, August 12, 2016
- 9:30 - 11:00: Ekholm minicourse, part 4
  coffee break
- 11:30 - 12:30: Siegel
  lunch
- 14:30 - 16:00: McLean minicourse, part 4

Precourse outlines

Sylvain Courte: Liouville and Weinstein manifolds
Symplectic convexity, handle attaching, contact boundaries, flexibility, Lefschetz fibrations and open books, vanishing thimbles.
lecture notes
References:
- Geiges, An introduction to contact topology, Cambridge University Press (2008)
- Cieliebak and Eliashberg, From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds (2012)
- Cieliebak and Eliashberg, Flexible Weinstein manifolds, AMS (2013)
- Seidel, A long exact sequence for symplectic Floer cohomology (2003)
Jo Nelson: Hamiltonian and Lagrangian Floer homology
Motivation via Morse homology, definitions of Hamiltonian and Lagrangian Floer homology (with Novikov coefficients) for closed manifolds and basic properties, pair-of-pants product, A_∞ relations.
lecture notes
References:
- Salamon, Lectures on Floer homology; electronic version (1997)
- Audin and Damian, Morse theory and Floer homology, Springer (2014)
- Hofer and Salamon, Floer homology and Novikov rings (1995)
- Piunikhin, Salamon and Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology; electronic version (1996)
- Seidel, Fukaya categories and Picard-Lefschetz theory, EMS (2008)
- Auroux, A beginner's guide to Fukaya categories (2013)
- Sheridan, notes from minicourse on the Fukaya category at Jussieu summer school, June 2016
Alexander Fauck: Introduction to symplectic homology
Maximum principle, definition of symplectic homology as a direct limit, some computation results (the ball, subcritical case, cotangent bundles — without proof), Viterbo functoriality, Morse-Bott formalism for autonomous Hamiltonians, definition of wrapped Floer homology.
lecture notes
References:
- Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology; arXiv version (2004)
- Seidel, A biased view of symplectic cohomology (2006); arXiv version (2010)
- Wendl, A beginner's overview of symplectic homology (2010)
- Fauck, Handle attaching in symplectic topology - a second glance (2016)

Minicourse outlines

Tobias Ekholm: Legendrian surgery formulas and applications

We explain how to calculate symplectic homology and wrapped Floer homology from a handle presentation of a Weinstein manifold. More precisely, the theories will be calculated from the Legendrian dga of the attaching spheres. We will also explain how to calculate the multiplicative structures of these theories in terms of the dga. With these basic calculations established we will consider the effect of deformations of the handles (isotopies and handle slides, report on joint work with Shende and Starkston). We will also study "punctured versions" of Lagrangian handle attachments, in particular in the setting of knot contact homology, which leads to Legendrian dga descriptions of the micro-local sheaf approach to computing the Fukaya categorical calculations (report on joint work with Ng and Shende).

Lecture 1: Lagrangian handle attachment and Reeb flow. Legendrian dga.
Lecture 2: Symplectic and wrapped Floer homology without Hamiltonian. Statement of surgery isomorphism theorems for +1 and -1 surgeries.
Lecture 3: Proof of the surgery results. Surgery formula for the pair of pants product on symplectic homology. The A_∞ structure and Legendrian dga of co-cores.
Lecture 4: Handle slides. Surgery formulas for wrapped Floer homology with wrapping stopped by a fixed Lagrangian with applications to knot contact homology and Legendrian dga descriptions of micro-local sheaves.

slides 1 slides 2 slides 3 slides 4

Mark McLean: Computing symplectic homology of affine varieties

Suppose we have a smooth affine variety A which is compactified by a smooth normal crossing divisor. Then we will construct a spectral sequence converging to the (positive) symplectic homology of A whose E₁ page is built from the strata of this compactification (this is just a Morse-Bott spectral sequence). By applying the above result, we can detect Reeb orbits and also show how symplectic homology can detect certain birational invariants. We will also explain how such results can show when cotangent bundles and affine varieties are different.

There are two other very similar spectral sequences that we will mention. The first one (conjecturally) converges to full contact homology of the link of an isolated singularity. The second one converges to Floer homology of iterates of the monodromy of a hypersurface singularity. The E₁ page in both of these cases is built from a log resolution in a similar way. Again both of these sequences have a dynamical application and they show that the respective Floer groups detect certain important algebraic invariants (smoothness, multiplicity, log canonical threshold and minimal discrepancy).

Finally we will show how one can prove the cohomological McKay correspondence using symplectic homology (this is joint work with Alex Ritter). In some sense, there should be an orbifold version of the above spectral sequence lurking in this result.

Lecture 1: Statement of the main spectral sequence along with computations and applications.
Lecture 2: Statement of the other two spectral sequences with computations and applications.
Lecture 3: Discussion of smooth normal crossing divisors from a symplectic perspective, and building an appropriate contact boundary around them and hence constructing the appropriate Morse-Bott spectral sequences.
Lecture 4: Cohomological McKay Correspondence.

slides for the first 2.75 talks

Alexandru Oancea: Symplectic homology and the Eilenberg-Steenrod axioms

The purpose of the minicourse is to explain in what sense symplectic homology and wrapped Floer homology satisfy analogues of the Eilenberg-Steenrod axioms, based on an extension of these theories to pairs of Liouville cobordisms or pairs of Lagrangian cobordisms respectively. I will also explain along the way the relationship to linearised (Legendrian) contact homology and to Rabinowitz-Floer homology. This allows us to fit the various linear homology theories for open symplectic manifolds and their Lagrangians into a unified framework. The minicourse is based on recent joint work with Kai Cieliebak.

Lecture 1: Symplectic homology for cobordisms. Axioms. Application: obstructions to cobordisms.
Lecture 2: Relationship to linearised contact homology. Relationship to Rabinowitz-Floer homology. Applications: interpretation of the surgery formula, displaceability and vanishing.
Lecture 3: The Viterbo transfer map. Homological algebra. The long exact sequence of a pair.
Lecture 4: Excision. Dualities. Further applications.

scanned notes (provided by Amiel Peiffer-Smadja)

Reference: Cieliebak and Oancea, Symplectic homology and the Eilenberg-Steenrod axioms (2015)

Additional talks

Mohammed Abouzaid: Wrapped Floer cohomology and based loops
Abstract:
The wrapped Floer cohomology of a cotangent fibre is known to be isomorphic to the homology of the based loop space of the zero section, and this is a key ingredient in proving the fact that every closed exact Lagrangian is equivalent to the 0-section (equipped with some local system). I will explain what happens when one drops the conditions of exactness: the basic idea is to replace the homology of the based loop space by a completion.

Georgios Dimitroglou Rizell: An energy-capacity inequality for Legendrians
Abstract:
In joint work with M. Sullivan we establish a result for Legendrian submanifolds analogous to the Energy-Capacity inequality for Lagrangians due to Y. Chekanov. Roughly speaking, this answers the question of how big the contact Hamiltonian and conformal factor must be in order for a contact isotopy to disjoin a given Legendrian submanifold from its image under the Reeb flow. The technique used is Floer homology applied to Lagrangian cobordisms between Legendrian submanifolds inside symplectisations. This can always be defined if one stays below a certain energy level, which is necessary in order to prevent unwanted bubbling.

Jean Gutt: Cube capacities
Abstract:
We study obstructions to symplectically embedding a cube (a polydisk with all factors equal) into another symplectic manifold with boundary of the same dimension. We find sharp obstructions in many cases, including all "convex toric domains" and "concave toric domains" in Cⁿ. The proof uses analogues of the Ekeland-Hofer capacities, which are conjecturally equal to them, but which are defined using S¹-equivariant symplectic homology. This is joint work with Michael Hutchings.

Yankı Lekili: A canonical basis of representations via Floer theory
Abstract:
Let G be a semisimple Lie group. Finite-dimensional representations of G can be realised as the section spaces of equivariant line bundles on the flag variety G/B à la Bott-Borel-Weil. In a previous work, we obtained a mirror dual picture: The Lie algebra g appears as a sub-Lie algebra of the symplectic cohomology of a mirror dual affine variety, and the group action itself is manifested in the action of this Lie algebra on the Floer cohomology groups HF(L,L') for the dual equivariant Lagrangians. An interesting aspect of our theory is that the representations of g on HF(L,L') come equipped with "canonical bases" arising from the geometric intersections of the Lagrangians. This is an additional structure which is not apparent in Bott-Borel-Weil theory. These canonical bases of representations have beautiful properties, including positivity properties, and an intriguing combinatorial structure. We will discuss an explicit realisation of our bases in the cases of sl₂ and sl₃. The latter computation leads to a conjecture relating our bases to Mirković-Vilonen cycles on the affine Grassmannian. This is based on joint work with James Pascaleff and partly also with Nick Sheridan.

Kyler Siegel: Bulk deformed symplectic cohomology
Abstract:
I will describe a refined version of symplectic cohomology, called bulk deformed symplectic cohomology, in which one counts pseudoholomorphic cylinders with interior marked point constraints. This invariant can sometimes be used to distinguish open symplectic manifolds which cannot be distinguished by ordinary symplectic cohomology. For example, I will describe how to construct and analyze a nonflexibile Weinstein subdomain of standard affine space. I will also briefly discuss how to compute this invariant in certain cases using Lefschetz fibration tools.

schedule precourse outlines minicourse outlines additional talks

back to the main workshop page