I will motivate the study of coproducts and describe a new coproduct structure on the symplectic cohomology of Liouville manifolds. Time permitting, I will indicate how to compute it in an example to show that it's not trivial.
Tag - Symplectic geometry
I will discuss how the Deligne-Mumford compactification of curves arises from the uncompactified moduli spaces of curves as a result of some algebraic operations related to (pr)operadic structures on the moduli spaces. I will describe how a variation of this naturally gives rise to another new partial compactification of moduli spaces curves. Time permitting, I will indicate how it is related to secondary operations on symplectic cohomology and discuss some ongoing work in this direction.
Ben Wormleighton: Embedding Obstructions for Non-Toric Rational Surfaces from Newton-Okounkov Bodies
ECH capacities have found many applications to symplectic embedding problems, most of which in the toric setting. I will discuss a new application of ECH to studying optimal embeddings for non-toric rational surfaces. The key convex geometric objects in our story are Newton-Okounkov bodies, which supply embeddings by work of Kaveh and also compute ECH capacities by joint recent work with Julian Chaidez.
We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.
We introduce new invariants to the existence of Lagrangian cobordisms in ℝ4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries.
We develop appropriate sign conventions and results to characterize boundary points of 1-dimensional moduli spaces with boundaries on Lagrangian tangles. We then use these to define (SFT-like) algebraic structures that recover the previously described obstructions.
The ellipsoidal embedding function of a symplectic four manifold M measures how much the symplectic form on M must be dilated in order for it to admit an embedded ellipsoid of some eccentricity. It generalizes the Gromov width and ball packing numbers. In most cases, finitely many obstructions besides the volume determine the function. If there are infinitely many obstructions determining the function, M is said to have an infinite staircase. This talk will give a classification of which Hirzebruch surfaces have infinite staircases. We will focus on explaining the correspondence between the obstructions coming from exceptional classes and the constructions from almost toric fibrations. We define a way to mutate triples of exceptional classes to produce new triples of exceptional classes, which corresponds to mutations in almost toric fibrations. This is based on various joint work with Dusa McDuff, Ana Rita Pires, and Morgan Weiler.
The spectral norm provides a lower bound to the Hofer norm. It is thus natural to ask whether the diameter of the spectral norm is finite or not. During this short talk, I will give a sketch of the proof that, in the case of Liouville domains, the spectral diameter is finite if and only if the symplectic cohomology of the underlying manifold vanishes. With that relationship in hand, we will explore applications to symplecticaly aspherical symplectic manifolds and Hofer geometry.
Persistence modules and barcodes are used in symplectic topology to define new invariants of Hamiltonian diffeomorphisms, however methods that explicitly calculate these barcodes are often unclear. In this talk I will define one such invariant called the GF-barcode of compactly supported Hamiltonian diffeomorphisms of ℝ2n by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e., a finite sequence of explicit calculation steps) that approximates it along with a few computation examples. This is joint work with Pazit Haim-Kislev.
What is the symplectic analogue of being convex? We shall present different ideas to approach this question. Along the way, we shall present recent joint results with J.Dardennes and J.Zhang on monotone toric domains non-symplectomorphic to convex domains and with M.Pereira and V.Ramos on cube-normalized capacities.
An Anosov flow Φ on a closed 3-manifold M gives rise to a non-Weinstein Liouville structure on V = [−1,1] × M. Building upon the work of Hozoori, we establish a homotopy correspondence between Anosov flows and certain pairs of contact forms. Moreover, the symplectic invariants of V only depend on the homotopy class of Φ. We focus on a subcategory W0 of the wrapped Fukaya category of V whose objects are in bijection with the simple closed orbits of Φ. In contrast with the Weinstein case, W0 is not homologically smooth, as it is not finitely split-generated in a maximal way. We expect W0 to be a powerful new invariant of Anosov flows.
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