IAS-Princeton Symplectic seminar, Spring 2016

IAS and Princeton Symplectic Geometry Seminar

THURSDAY 10.30-11.30, Room Simonyi 101 (when at IAS),

THURSDAY 10.30-11.30, Fine 214 (when at Princeton)

Organized by the IAS and Princeton Symplectic group-

Octav Cornea; Helmut Hofer; Gang Tian; Jo Nelson; Nick Sheridan; Egor Shelukhin

Date | Time/Location notes | Speaker | Title |

1/21/16 | IAS | Dan Cristofaro-Gardiner (Harvard) | From symplectic geometry to combinatorics and back |

1/28/16 | NO SEMINAR | ||

2/4/16 | IAS | Mohammed Abouzaid (Columbia) | Floer theory revisited |

2/11/16 | ALIAS, 10.40 S-101 | Baptiste Chantraine (Nantes) | Restrictions on the fundamental group of some Lagrangian cobordisms. |

ALIAS, 1-2pm S-101 | Lenny Ng (Duke) | Toward a contact Fukaya category | |

ALIAS, 2.30-3.30 S-101 | John Etnyre (Georgia Tech) | Satellite operations and Legendrian knot theory. | |

2/12/16 | ALIAS, 10.40 S-101 | Emmy Murphy (MIT) | A frontal view on Lefschetz fibrations I |

ALIAS, 1-2pm S-101 | Roger Casals (MIT) | A frontal view on Lefschetz fibrations II | |

ALIAS, 2.30-3.30 S-101 | Lisa Traynor (Bryn Mawr) | A Quantitative Look at Lagrangian Cobordisms | |

2/18/16 | IAS | Frol Zapolsky (Haifa) | Spectral invariants for contactomorphisms of prequantization bundles and applications |

2/25/16 | IAS | Peter Albers (Muenster) | Positive loops - on a question by Eliashberg-Polterovich and a contact systolic inequality |

3/3/16 | IAS | Kyler Siegel (Stanford) | Subflexible symplectic manifolds |

3/10/16 | IAS | Ailsa Keating (Columbia) | Homological Mirror Symmetry for singularities of type T_{pqr}. |

3/17/16 | NO SEMINAR DUSA FEST at CU | ||

3/21/16 | ***Monday, 3.10-4.10pm, IAS S-101*** | Thomas Kragh (Uppsala) | Stable homotopy theory and Floer theory. |

3/31/16 | ***AT 4.30pm*** Princeton Fine 314 (TOPOLOGY seminar) | Michael Sullivan (Amherst) | Cellular homology, augmentations and generating families for Legendrian surfaces |

4/1/16 (FRIDAY) | Joint Seminar with CU at CU | Mark McLean (Stony Brook) | Log canonical threshold and Floer homology of the monodromy |

4/1/16 (FRIDAY) | Joint Seminar with CU at CU | Renato Vianna (Cambridge) | Infinitely many monotone Lagrangian tori in Del Pezzo surfaces |

4/7/16 | IAS ***AT 9.45am**** | Jeremy van Horn Morris | Stein fillings of cotangent bundles of surfaces |

4/7/16 | ***AT 11am**** IAS | Georgios Dimitroglou-Rizell (Cambridge) | Classification results for two-dimensional Lagrangian tori |

4/7/16 | ***AT 4.30pm*** Princeton Fine Hall 314 (TOPOLOGY seminar) | Gordana Matic (Georgia) | Filtering the Heegaard-Floer Contact Invariant |

4/15/16 | ***ON FRIDAY**** 10.30am IAS | Ana Rita Pires (Fordham) | Symplectic embeddings and infinite staircases |

4/21/16 | IAS | Cagatay Kutluhan (Buffalo and IAS) | A Heegaard Floer analog of algebraic torsion |

4/28/16 | Princeton Fine 1201 | R. Inanc Baykur (U Mass Amherst) | Small symplectic and exotic 4-manifolds via positive factorizations |

Abstracts:

1/21: Cristofaro-Gardiner Abstract: A much studied combinatorial object associated to a polytope is its "Ehrhart function". This is typically studied for integral or rational polytopes. The first part of the talk will be about joint work with Li and Stanley extending this theory to certain irrational polytopes; the idea for this was inspired by work of McDuff and Schlenk on symplectic embeddings of four-dimensional ellipsoids. In the second part of the talk, I will introduce recent joint work with Holm, Mandini, and Pires, which aims to apply this "irrational" Ehrhart theory to study embeddings of ellipsoids into four-dimensional symplectic toric manifolds.

2/4: Abouzaid Abstract: I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of

the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I will explain the consequences of quantum cohomology, and the expected applications of this theory.

2/11:

Chantraine Abstract: In this talk we will describe two methods which shows that, under some rigidity assumptions on the involved Legendrian submanifolds, a Lagrangian cobordism is simply connected. The first one uses the functoriality of the fundamental class in Legendrian contact homology with twisted coefficients. The second uses a L^2-completion of the Floer complex associated to the cobordism. This is joint work with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko.

Ng Abstract: I will describe some work in progress (maybe more accurately, wild speculation) regarding a version of the derived Fukaya category for contact 1-jet spaces J^1(X). This category is built from Legendrian submanifolds equipped with augmentations, and the full subcategory corresponding to a fixed Legendrian submanifold \Lambda is the augmentation category Aug(\Lambda), which I will attempt to review. The derived Fukaya category is generated by unknots, with the corollary that all augmentations ``come from unknot fillings''. I will also describe a potential application to proving that ``augmentations = sheaves''. This is work in progress with Tobias Ekholm and Vivek Shende, building on joint work with Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow.

Etnyre Abstract: Satellite operations are a common way to create interesting knot types in the smooth category. It starts with a knot K, called the companion knot, in some manifold M and another knot P, called the pattern, in S^1\times D^2 and then creates a third knot P(K), called the satellite knot, as the image of P when S^1\times D^2 is identified with a neighborhood of K. In this talk we will discuss the relation between Legendrian knots representing K, P, and P(K). Sometimes the classification of Legendrian representatives for K and P yields a classification for P(K), but other times it does not. We will discuss why this happens and a general framework for studying Legendrian Satellites.

2/12

Casals & Murphy Abstract: In this series of two talks we will discuss Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. The main focus is on Weinstein manifolds which admit a Weinstein Lefschetz fibration with an $A_k$--fibre; this provides a large class of Weinstein structures ranging from flexible Weinstein manifolds to rich rigid examples. In particular, we will describe the computation of their symplectic homologies and discuss its implications to Legendrian submanifolds and their Lagrangian fillings.

Traynor Abstract: Lagrangian cobordisms between Legendrian submanifolds arise in Relative Symplectic Field Theory. In recent years, there has been much progress on answering qualitative questions such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism? I will address two quantitative questions about Lagrangian cobordisms: For a fixed pair of Legendrians, what is the minimal “length” of a Lagrangian cobordism? What is the relative Gromov width of a Lagrangian cobordism? Regarding length, I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that the non-cylindrical region can be arbitrarily short; I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are rigid in that there is a positive lower bound to their length. For the second quantitative measure, I will give some calculations and estimates of the relative Gromov width of particular Lagrangian cobordisms. This is joint work with Joshua M. Sabloff.

2/18: Zapolsky Abstract: I'll outline the construction and computation of a Floer homology theory for contact manifolds which are prequantization spaces over monotone symplectic manifolds, and of the spectral invariants resulting therefrom, and present some applications. These include a quasi-morphism on the universal cover of the identity component of the contactomorphism group of the real projective space with the standard contact structure, the existence of a translated point for any contact form for the standard contact structure on any prequantization space over a monotone symplectic manifold (subject to a noninvertibility condition on the Euler class), and a proof that the Reeb flow of the standard contact form on such a prequantization space gives rise to a noncontractible loop in the contact group. The construction involves Lagrangian Floer theory in a nonconvex symplectic manifold. This is joint work in progress with Peter Albers and Egor Shelukhin.

2/25: Albers Abstract: In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether C^0-small positive loops exist. We give a negative answer to this question. Moreover we give sharp lower bounds for the size which, in turn, gives rise to a L^\infty-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no L^2-contact systolic inequality exists. The choice of L^2 is motivated by systolic inequalities in Riemannian geometry. This is joint work with U. Fuchs and W. Merry.

3/3: Siegel Abstract: After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called "subflexible", which are not flexible but become so after attaching some Weinstein handles. For example, the standard symplectic ball has a Weinstein subdomain with nontrivial symplectic topology. These are exotic symplectic manifolds with vanishing symplectic cohomology. I will explain how to study them using a deformed version of symplectic cohomology, and how this invariant can computed using the machinery of Fukaya categories and Lefschetz fibrations. This is partly based on joint work with Emmy Murphy.

3/10: Keating Abstract: We present some homological mirror symmetry statements for the singularities of type $T_{p,q,r}$. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E. We will consider some symplectic invariants of the real four-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space $\P^2$, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different ``flavours" of invariants (e.g., versions of the Fukaya category) match up on both sides.

3/21: Kragh Abstract: In this talk I will define and explain some basic notions from stable homotopy theory, and illustrate how it relates to (and refines) the notion of Floer homology in some simple cases. I will also discuss what extra kind of information this refinement contains and describe an explicit situation where it provides a stronger result than the usual homological invariants (joint work with Mohammed Abouzaid).

3/31: Sullivan Abstract: Given a Legendrian surface in a contact one jet space, there is a local combinatorial DGA associated to the cell decomposition of the base projection of the Legendrian. The DGA is quasi-isomorphic to the Legendrian contact DGA. Using this reformulation I show how a generating family for the Legendrian produces an augmentation for the DGA. I will also sketch a partial converse. This is joint work with Dan Rutherford.

4/1: Vianna Abstract: We will describe how to get almost toric fibrations for all del Pezzo surfaces (endowed with monotone symplectic form), in particular for $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ for $4\le k \le 8$, where there is no toric fibrations. From there, we will be able to construct infinitely many monotone Lagrangian tori. We are able to prove that these tori give rise to infinitely many symplectomorphism classes in $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ for $0 \le k \le 8$, $k \neq 2$, and in $\mathbb{CP}^1 \times \mathbb{CP}^1$. Using the recent work of Pascaleff-Tonkonog one can conclude the same for $\mathbb{CP}^2\#2\overline{\mathbb{CP}}^2$. Some Markov like equations appear. These equations also appear in the work of Haking-Porokhorov regarding degeneration of surfaces to weighted projective spaces and on the work of Karpov-Nogin regarding 3-block collection of exceptional sheaves in del Pezzo surfaces.

4/1: McLean Abstract: The log canonical threshold of a hypersurface singularity is an important invariant which appears in many areas of algebraic geometry. For instance it is used in the minimal model program, has been used to prove vanishing theorems, find Kahler Einstein metrics and it is related to the growth of solutions mod p^k. We show how to calculate the log canonical threshold and also the multiplicity of the singularity using Floer homology of iterates of the monodromy map.

4/7: van Horn Morris Abstract: I'll outline recent results with Steven Sivek classifying the Stein fillings, up to topological homotopy equivalence, of the canonical contact structure on the unit cotangent bundle of a surface. The proof begins with Li, Mak and Yasui's technology for Calabi-Yau caps.

4/7: Dimitroglou-Rizell Abstract: We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone S2 x S2. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.

4/7: Matic Abstract: We define an invariant of contact structures in dimension three based on the contact invariant of Ozsvath and Szabo from Heegaard Floer homology. This invariant takes values in $\Z_{\geq0}\cup\{\infty\}$, is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, and non-decreasing under Legendrian surgery. This is joint work with Cagaty Kutluhan, Jeremy Van Horn-Morris and Andy Wand.

4/15: Pires Abstract: McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.

4/21: Kutluhan Abstract: The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology. This is joint work with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand.

4/28: Baykur Abstract: We will discuss new ideas and techniques for producing positive

Dehn twist factorizations of surface mapping classes (joint work with

Mustafa Korkmaz) which yield novel constructions of interesting symplectic

and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and

exotic rational surfaces, via Lefschetz fibrations and pencils.