Differential geometry and symplectic topology seminar (Fall 2023)

Abstracts

• Christopher Kuo

  • Perverse sheaves are topological invariants that classify objects from several different fields. For example, they are equivalent to regular holonomic D-modules on the same complex manifold and holomorphic Lagrangians in infinitesimal Fukaya category of its cotangent bundle. In this talk, I will discuss a joint work with Côté, Nadler and Shende, where we perform a construction on complex contact manifolds, globalizing the previously defined perverse microsheaves on coprojective bundles.

• Guangbo Xu

  • (joint with Shaoyun Bai) While the Arnold conjecture provides a (topological) lower bound on the number of fixed points of Hamiltonian diffeomorphisms, it has been proved that in many cases Hamiltonian diffeomorphisms have infinitely many periodic points (i.e. fixed points of iterations), such as the proof of the Conley conjecture in various cases. On the other hand, manifolds such as projective spaces admit Hamiltonian diffeomorphisms which have only finitely many simple periodic orbits. The Hofer-Zehnder conjecture stated that even for these manifolds, if the number of fixed points is strictly greater than the Arnold lower bound, i.e., there are “redundant” fixed points, then there should be infinitely many periodic points. A recent major breakthrough towards the Hofer-Zehnder conjecture was the work of Shelukhin, who proved that for a monotone symplectic manifold, when the quantum cohomology is semisimple (over a certain field), then the Hofer-Zehnder conjecture is true. Inspired by Shelukhin’s work and the picture of mirror symmetry, we prove the Hofer-Zehnder conjecture for ALL compact toric manifolds. There are a few key ingredients in the proof. First, following Givental, Hori-Vafa, Fukaya-Oh-Ohta-Ono etc. toric manifolds are mirror to Landau-Ginzburg models and hence their quantum cohomology are “generically” semisimple. Second, as toric manifolds are GIT quotients of vector spaces, we use the gauged linear sigma model (GLSM) to do Floer theory over integers without the need of virtual technique.

• Luya Wang

  • Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with $\mathbb{CP}^1$ with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan.
• Sunrose Shrestha
  • The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations) has been well studied due to its connections to polygonal billiards and Teichmüller theory. However, less is known in general regarding straight-line flows on non-compact infinite area translation surfaces. In this talk, we will consider straight line trajectories on the Mucube – an infinite Z^3 periodic half-translation surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions (aka cylinder directions) on the Mucube in terms of an infinitely generated infinite index subgroup of SL(2,Z). Using the characterization, we show that the characterizing group is in fact the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.
• Mohan Swaminathan
  • Gromov’s compactness theorem (later refined by Kontsevich) shows that any sequence of pseudo-holomorphic maps from compact (non-singular) Riemann surfaces to a compact almost Kahler manifold (with fixed genus and homology class) has a subsequence converging to a “stable map”, i.e., a pseudo-holomorphic map defined on a compact, but possibly nodal, Riemann surface of the same arithmetic genus (satisfying a stability condition). Even in the nicest case (e.g., degree d curves of genus g with d»g in CP^N), a dimension count shows that most stable maps which have “ghosts” (i.e., irreducible components on which the map is constant) can never appear as the limit of any sequence of maps with non-singular domains. Thus, we may ask: (a) which stable maps actually do appear as limits & (b) which ones may be discarded, to obtain a more optimal compactification. In this talk, I will describe recent work (joint with Fatemeh Rezaee) where we use a gluing construction to provide a partial answer to (a), in all genera, for the case of smooth projective varieties. The key geometric input is a new class of explicit model solutions which governs the gluing construction near the ghosts.
• Martin Pinsonnault
  • Existence of symplectic embeddings of $k$ disjoint balls of given capacites $c_1,\ldots, c_k$ into a given symplectic manifold is a central problem in symplectic topology. However, beside a few examples, very little is known about the space of all such embeddings. In this talk, I will discuss the case of rational $4$-manifolds of small Euler numbers, with a special attention to the minimal manifolds $\mathbb{C}P^2$ and $S^2\times S^2$. For rational manifolds, a very rich and intricate picture emerges that blends symplectic topology, complex geometry, and algebraic topology.
• Yoosik Kim
  • In this talk, we explain polygon spaces and bending completely integrable systems. We then explain how to understand the topology of fibers of completely integrable systems. Based on the structural result of the monotone Fukaya category and toric degenerations, we compute the disk potential function of the monotone torus fiber of the caterpillar bending system and derive a Floer theoretical SYZ mirror for an equilateral and generic polygon space. This talk is based on joint work with Siu-Cheong Lau and Xiao Zheng.
• Fabio Gironella
  • The main objects of the talk will be symplectic foliations, and more precisely a subclass of them called “strong”. These are one of the possible generalizations of taut foliations to high dimensions, and indeed have quite a rigid nature, with techniques such as pseudo-holomorphic curves à la Gromov and asymptotically holomorphic sequences of sections à la Donaldson working well in this setting. I will present a joint work (in progress) with Klaus Niederkrüger and Lauran Toussaint that aims at giving a new obstruction for a symplectic foliation to be strong, that comes in the form of a symplectic high-dimensional version of vanishing cycles for smooth codimension 1 foliations on 3-manifolds. The proof relies on pseudo-holomorphic curve techniques, in a way which is parallel to the case of Plastikstufe introduced by Niederkrüger ‘06 in the contact case.
• Yandi Wu
  • The marked length spectrum of a negatively curved metric space can be thought of as a length assignment to every closed geodesic in the space. A celebrated result by Otal says that metrics on negatively curved closed surfaces are determined completely by their marked length spectra. In my talk, I will discuss my work towards extending Otal’s result to a large class of surface amalgams, which are natural generalizations of surfaces.
• Austin Christian
  • Morse functions are generic among real-valued functions on a smooth manifold, and distinct Morse functions on a fixed manifold are related to each other in well-understood ways. In the contact and symplectic categories, Morse theory is more delicate and certainly less well-understood. In this talk we will address the question of whether an exact symplectic manifold-with-boundary can be made compatible with a Morse function — that is, whether a Liouville domain can be made Weinstein. We provide some explicit constructions for realizing this compatibility, and use these to demonstrate that a famous family of Liouville-but-not-Weinstein domains first studied by Geiges and Mitsumatsu is stably Weinstein. This talk is based on joint work with Joseph Breen.