Timetable

One of the central topics in the study of derived categories are semi-orthogonal decompositions – ways of presenting the category as glued from smaller pieces. The simplest case are exceptional sequences, where each piece is equivalent to the derived category of vector spaces. Fukaya-Seidel categories provide a geometric interpretation of exceptional sequences in the context of symplectic topology. The first part of the talk will be an introduction to these ideas. In the second part I will report on recent joint work with Chang and Schroll (arXiv:2301.04398) which provides a fairly complete picture in the lowest-dimensional case of surfaces, showing that all exceptional sequences come from geometry.

In my previous work with Peter Albers, we proved that the Rabinowitz Floer homology(RFH) for a circle subbundle of a negative line bundle vanishes if the radius of the circle is small. Later Sara Venkatesh showed that, in this setting, the RFH is not always invariant under the change of radius by computing this for some examples. In this talk, I will explain how one can study the RFH of negative line bundles using the filtration given by the intersection number of Floer cylinders and the zero-section and then revisit the above computations. This is joint work with Peter Albers.

It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the (wrapped) Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018). The latter authors also showed that this equivalence holds for a more general class of symplectic manifolds, called Weinstein manifolds. Namely, they introduced a way to compute wrapped Fukaya categories of Weinstein manifolds by taking the homotopy colimit of wrapped Fukaya categories of their sectorial coverings. I will talk about the relevant definitions and results, and categorical/computational aspects of Fukaya categories. In particular, I will explain our formula computing homotopy colimits of dg categories, and its applications including expressing wrapped Fukaya categories of plumbing spaces as perfect modules over Ginzburg dg algebras (work in progress with Sangjin Lee).

Thanks to the affirmative answer to the symplectic Thom conjecture, all connected symplectic surfaces representing a fixed homology class in a closed symplectic 4- manifold are diffeomorphic. In this talk, I will present a family of connected symplectic submanifolds of codimension $2$ in a closed symplectic 6-manifold which are mutually homotopy inequivalent but homologous. This shows that the uniqueness of symplectic representatives fails in higher dimensions. A key ingredient of the construction is hyperelliptic Lefschetz fibrations.

Frobenius manifolds are manifolds with a rich structure on the tangent bundle, a multiplication and a metric which harmonize in the most natural way. They were defined by Dubrovin in 1991, motivated by the work of Witten, Dijkgraaf, E. Verlinde, and H. Verlinde on topological field theory. Originally coming from physics, the first example of Frobenius manifolds were discovered by K. Saito and M. Saito (around 1980 even before the name was given) as a base manifold of a universal unfolding of an isolated hypersurface singularity. Later Y. Manin introduced (2005) the notion of F-manifolds with a compatible flat structure (flat F-manifolds) which are weaker variants of Frobenius manifolds but still shares some of the deeper properties of Frobenius manifolds. In this talk, we construct a dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra associated to an IHS (X,0) and formulate a system of differential equations (coming from the Gauss-Manin connection), whose solution gives rise to the structure of a formal flat F-manifold for a universal unfolding of (X,0). Then we provide an explicit computer algorithm to solve this differential system.

The Coherent-Constructible Correspondence(CCC) is a version of Homological Mirror Symmetry for toric varieties: an equivalence between the derived category of a toric variety and a category of certain constructible sheaves on a torus. Later Harder and Katzarkov stated and proved a version of CCC for toric P^1-bundles and claimed that there should be a version of CCC for general toric fiber bundles, not only for P^1. In this talk, I will explain how we can formulate and prove a version of CCC for toric fiber bundles that Harder-Katzarkov expected. This is joint work with Yuxuan Hu.

Motivated by the geometry of Milnor fiber and vanishing cycles, Kyoji Saito introduced the notion of generalized root systems. A generalized root system consists of a lattice, a set of roots and an element of the Weyl group called a Coxeter element. Due to the choice of a Coxeter element, the notion is finer than the usual one. There are indeed several "finite generalized root systems of type D", however, little was known due to lack of their geometric construction. From a Laurent polynomial in one variable, we deduce a finite generalized root system of type D and construct a "natural" Frobenius manifold whose Frobenius potential is a rational function, which support the Dubrovin's conjecture on algebraic Frobenius manifolds. This is a joint work with Akishi Ikeda, Takumi Otani and Yuuki Shiraishi.

1st talk. Homological mirror symmetry and elliptic fibrations 

Abstract: We discuss compatibility of homological mirror symmetry for a singular fiber of type I of an elliptic surface and its neighborhood in the total space. This is related to Karube's work on stability conditions on singular fibers of elliptic surfaces and Arai's work on autoequivalences of derived categories of elliptic surfaces.

2nd talk. Stable Fukaya categories of Milnor fibers

Abstract: We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

Let $X$ be a closed monotone symplectic manifold and let $\Sigma \subset X$ be a smooth Donaldson-type symplectic hypersurface. In joint work with Jungsoo Kang and Sungho Kim, we construct a Gysin-type long exact sequence connecting the Rabinowitz Floer homology of the complement $X \setminus \Sigma$ and the quantum homology of $\Sigma$. I will explain its construction and present a few applications.

Fukaya category of finite type surface is well-understood by now. However, not much has been said about the Fukaya category of infinite type surfaces and current attempts involve gluing constructions. We want to directly construct a Fukaya category of any infinite type surface whose objects are gradient sectorial Lagrangians. In this talk, we will introduce end structure and symplectomorphism classification of surfaces and gradient sectorial Lagrangians. After that, we will define Fukaya category of infinite type surface and present split generators of this category in good cases.

Inspired by the work of Cineli-Ginzburg-Gürel, we introduce a Floertheoretic entropy for Reeb flows. The barcode entropy of symplectic homology is defined as the exponential growth rate of the number of not-too-short bars in the barcode of the associated Floer complex. In this talk, we will show that this barcode entropy is bounded above by the topological entropy for Reeb flows. The talk is based on ongoing joint work with Elijah Fender and Sangjin Lee.