Research Seminar: Symplectic Geometry (WS 2022/23)

We will give an introduction to Engel structures and discuss recent progress in the classification of Engel manifolds and their submanifolds.

I will review the necessary prerequisites for my talk next week.
Firstly, I'll show that simple holomorphic curves are generically Fredholm regular; i.e. "transversality holds for simple curves". The proof is standard, but will become much more interesting in the equivariant case. Secondly, we will review the slice theorem for smooth Lie group actions on Banach manifolds --- with an eye towards infinite-dimensional applications. If time permits, we might see how these are related.

Consider a closed symplectic manifold with a compact Lie group G acting symplectically. What can we say about the moduli space of holomorphic curves on M, w.r.t. a generic G-equivariant almost complex structure? We should not expect it to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into disjoint strata which are smooth manifolds of explicitly computable dimension. I will focus on the case of simple curves (which is already surprisingly interesting): I'll explain how to decompose the moduli space into iso-symmetric strata and prove that each stratum is smooth.

In this talk, we will discuss the construction and properties of Khovanov homology, a link invariant introduced by Khovanov in 1999. We will discuss some of its applications in low dimensional topology, symplectic and contact geometry.

In this talk, we will discuss properties of fibered-positive links and prove a vanishing theorem of Khovanov homology for such links and cables obtained from them. Fibered-positive links are a proper subclass of links whose associated open book decomposition induces the unique tight contact structure on S³.

Khovanov homology was introduced by Mikhail Khovanov at the end of last century. This link invariant, which categorifies the Jones polynomial, provides geometric and topological information about the link. In particular, it is an unknot detector. Some years later, Lipshitz and Sarkar introduced the Khovanov spectrum, a refinement of Khovanov homology: given a link diagram D, they provide a method to obtain a space whose stable homotopy type is a link invariant, and whose cohomology is isomorphic to the Khovanov homology of the link represented by D. In this talk we will present some simplifications of the Khovanov spectrum for some specific gradings. As a consequence, we will obtain properties of the Khovanov homology of several families of links.

In standard Morse theory on a compact manifold M, for every Morse function f on M a generic Riemannian metric g satisfies that (f,g) is Morse-Smale, i.e. stable and unstable manifolds intersect transversally for every pair of critical points. If we impose an additional finite symmetry on M, i.e. if a finite group G acts on M by diffeomorphisms such that f and g are G-invariant, then this statement fails: there are Morse functions f such that (f,g) is never Morse-Smale, for any g. We will show that nontheless there exist for any manifold M and any group G always G-invariant Morse functions f such that generic G-invariant Riemannian metrics give Morse-Smale pairs (f,g), and we will show that whenever the action by G is not free, then there also always exist G-invariant Morse functions such that (f,g) is for no G-invariant metric Morse-Smale.

I will talk about some algebraic structures arising on Lagrangian Floer homology and Fukaya categories in the presence of a Hamiltonian action of a compact Lie group. On the one hand, this should be related to a conjecture of Teleman, motivated by Homological Mirror Symmetry. On the other hand, our motivation comes from gauge theory in low dimensions, and is part of a program aimed at recasting Donaldson-Floer theory into an extended field theory. More speculatively, we also expect these structures to arise in higher dimensional gauge theory, following the Donaldson-Segal program, and in relation with the Moore-Tachikawa field theories. This is based on two joint works in progress, one with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and Thibaut Mazuir.

I will begin by describing two pieces of the context for this talk: first, the symplectic (A-infinity,2)-category (Symp), which is the natural setting for building functors between Fukaya categories; and second, Lagrangian torus fibrations, which are the central geometric objects in SYZ mirror symmetry. Next, I will explain my construction with Daria Poliakova of a family of polytopes called constrainahedra, which we introduced in order to define the notion of a monoidal A-infinity category. Finally, I will describe work-in-progress that aims to equip the Fukaya category of a Lagrangian torus fibration with a monoidal A-infinity structure, which should be mirror to the tensor product of sheaves. This is based on past and ongoing work with Daria Poliakova and Mohammed Abouzaid, including arXiv:2208.14529 and arXiv:2210.11159.

In this talk, we explain some interesting properties of the symplectic capacities that appeared in the paper by McDuff and Siegel, Symplectic capacities, unperturbed curves, and convex toric domains.

The Lagrangian capacity is a symplectic capacity defined by Cieliebak and Mohnke. In this talk, via a neck-stretching argument of Cieliebak-Mohnke, we compare the Lagrangian and McDuff-Seigel capacities of Liouville domains. As a consequence, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. This positively settles a conjecture of Cieliebak and Mohnke for the Lagrangian capacity of the 4-dimensional ellipsoids.

References:

• Miguel Pereira, On the Lagrangian capacity of convex or concave toric domains
• Kai Cieliebak and Klaus Mohnke, Punctured holomorphic curves and Lagrangian embeddings