# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### A frontal view on Lefschetz fibrations I

**Please note special day and time.** 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.

##### Spectral invariants for contactomorphisms of prequantization bundles and applications

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.

##### Positive loops - on a question by Eliashberg-Polterovich and a contact systolic inequality

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.

##### Subflexible symplectic manifolds

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.

##### Homological Mirror Symmetry for singularities of type T_{pqr}

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.

##### Stable homotopy theory and Floer theory

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).

##### Cellular homology, augmentations and generating families for Legendrian surfaces

**This is a joint Topology - Symplectic Geometry seminar. ** 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.

##### Log canonical threshold and Floer homology of the monodromy

**Joint Seminar with CU at CU. ** 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 pkpk. 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.

##### Infinitely many monotone Lagrangian tori in Del Pezzo surfaces

**Joint Seminar with CU at CU. ** 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.

##### Stein fillings of cotangent bundles of surfaces

**Please note special time. ** 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.

##### Classification results for two-dimensional Lagrangian tori

**Please note special time. **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.

##### Filtering the Heegaard-Floer Contact Invariant

**This is a joint Symplectic Geometry - Topology seminar. ** 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.

##### Symplectic embeddings and infinite staircases

**Please note different day (Friday). ** 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.

##### A Heegaard Floer analog of algebraic torsion

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 Grodana Matic, Jeremy Van Horn-Morris, and Andy Wand.

##### Small symplectic and exotic 4-manifolds via positive factorizations

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.