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

##### Fixed-point expressions for the Fukaya endomorphism algebra of RP^{2m} and higher genus open invariants

The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps.

##### Lagrangian cobordism: what we know and what is it good for

(Joint seminar with Columbia.) I will describe how the notion of Lagrangian cobordism, introduced by Arnold in 1980, offers a systematic perspective on the study of Lagrangian topology. There are three aspects that will be emphasized: the relations with the triangulated structure of the derived Fukaya category, the connection to Seidel’s representation, cobordism based extensions of the Hofer norm. The talk is based mainly on joint work with Paul Biran (ETH).

##### Non-trivial Hamiltonian fibrations via K-theory quantization

We produce examples of non-trivial Hamiltonian fibrations that are not detected by previous methods (the characteristic classes of Reznikov for example), and improve theorems of Reznikov and Spacil on cohomology-surjectivity to the level of classifying spaces. Joint work with Yasha Savelyev.

##### Real Gromov-Witten theory in all genera

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces. In reasonably regular cases, these invariants can be used to obtain lower bounds for counts of real curves of arbitrary genus. Joint work with A. Zinger.

##### 3d mirror symmetry and symplectic duality

**Please note special location (Fine 224) and time (3:00-4:00)****. **In recent work of Braden, Licata, Proudfoot, and Webster, a "symplectic duality" was described between pairs of module categories O(M), O(M') associated to certain pairs of complex symplectic manifolds (M, M'). The duality generalizes the Koszul duality of Beilinson-Ginzburg-Soergel for categories of modules associated to flag varieties. I will discuss how symplectic duality can be obtained from the physics of boundary conditions in three-dimensional supersymmetric gauge theories, and some new structure that arises from these boundary conditions. (Joint work with M. Bullimore, D. Gaiotto, & J. Hilburn.)

##### Dehn twists exact sequences through Lagrangian cobordism

In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RP^n and CP^n, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a "mirror" of the construction in algebraic geometry, realized by a new surgery and cobordism construction. This is a joint work with Cheuk-Yu Mak.

##### Floer homology for translated points

**PLEASE NOTE: THIS SEMINAR WILL BE HELD AT COLUMBIA UNIVERSITY IN ROOM MATH 520. **A point q in a contact manifold (M,ξ) is said to be a translated point of a contactomorphism ϕ, with respect to a contact form α for ξ, if it is a "fixed point modulo the Reeb flow", i.e. if q and ϕ(q) are in the same Reeb orbit and ϕ preserves α at q. Translated points are key objects to look at when studying contact rigidity phenomena such as contact non-squeezing, orderability of contact manifolds and existence of bi-invariant metrics and quasimorphisms on the contactomorphism group. Based on the notion of translated points, in 2011 I proposed a contact analogue of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms.

##### Functors and relations from Fukaya categories of LG models

**PLEASE NOTE: THIS SEMINAR WILL BE HELD AT COLUMBIA UNIVERSITY IN ROOM MATH 407. **The Fukaya category of a Landau-Ginzburg (LG) model W: E --> C, denoted F(E,W), enlarges the Fukaya category of E to include certain non-compact Lagrangians determined by W (for instance, Lefschetz fibrations and their thimbles). I will describe natural functors associated to the Fukaya categories of (E,W) and the general fibre M, and introduce a new Floer homology ring for (E,W). Using these, I will explain two new results: (a) a generation criterion for F(E,W), in the sense of Abouzaid/AFOOO, and (b) exact triangles of functors, one each in F(E,W) and F(M). First applications include stability and generation results for Fukaya categories and a new proof of exact sequences for fibered twists.

##### Negative and positive results in the intersection between systolic and symplectic geometry

How small is the smallest period of a closed trajectory of a Reeb flow? In this talk I will present recent answers to instances of this question in three-dimensions which reveal connections between systolic and symplectic geometry. I will present results both of a positive and of a negative nature. Namely, in some situations there are sharp bounds for the systolic ratio, which is defined as the ratio between the square of the smallest period and the contact volume, while in other situations the systolic ratio is unbounded. Our results confirm a conjecture of Babenko and Balacheff and disprove a conjecture of Hutchings. All this is joint work with Abbondandolo, Bramham and Salomão.

##### Legendrian Fronts in Contact Topology

In this talk we focus on Legendrian presentations of Weinstein manifolds; in particular, we discuss loose Legendrian embeddings and their absolute analogues, flexible Weinstein manifolds and overtwisted contact structures. The required definitions and necessary results will be provided. In the first part, we present a dictionary between Legendrian fronts and adapted open books decompositions. In the second part, we explore applications to Weinstein cobordisms and biLefschetz fibrations with Am–Milnor fibres. This is joint work with E. Murphy.

##### Low-area Floer theory and non-displaceability

**Please note change of time: 2:30-3:30. **I will introduce a "low-area" version of Floer cohomology of a non-monotone Lagrangian submanifold and prove that a continuous family of Lagrangian tori in CP2, whose Floer cohomology in the usual sense vanishes, is Hamiltonian non-displaceable from the monotone Clifford torus. Joint work with Renato Vianna.

##### Point-like bounding chains in open Gromov-Witten theory

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention. We generalize Welschinger's invariants with boundary and interior constraints to higher odd dimensions using the language of $A_\infty$-algebras and bounding chains. The bounding chains play the role of boundary point constraints. The geometric structure of our invariants is expressed algebraically in a version of the open WDVV equations. These equations give rise to recursive formulae which allow the computation of all invariants for $\mathbb{C}P^n$. This is joint work with Jake Solomon.

##### Absolute vs. Relative Gromov-Witten invariants

We compare absolute and relative Gromov-Witten invariants with the basic contact vector for very positive divisors. For such divisors, one might expect that these invariants are the same up to a natural multiple. We show that this is indeed the case outside of a narrow range of the dimension of the target and the genus of the domain. We provide explicit examples to show that these invariants are generally different inside of this range. This is joint work with A. Zinger.

##### From symplectic geometry to combinatorics and back

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.

##### Floer theory revisited

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.

##### Restrictions on the fundamental group of some Lagrangian cobordisms.

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.

##### Toward a contact Fukaya category

**Please note special time. **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.

##### A frontal view on Lefschetz fibrations II

**Please note special 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.

##### Satellite operations and Legendrian knot theory

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

##### A Quantitative Look at Lagrangian Cobordisms

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