The curve counting theories of local Calabi-Yau surfaces are closely related to that of log Calabi-Yau surfaces via tropical geometry. We discuss the correspondence in the view of Gopakumar-Vafa invariants. We also describe the conjectural relationship between two theories for local curves.

Recently there arose a demand to develop cosection localization in topological settings. In quantum singularity theory, the insertion classes are not algebraic and we are forced to work with topological cycles. Recent developments in the 4-dimensional Donaldson-Thomas theory include the theory of Kuranishi structure where the charts are not algebraic. In this talk, I will discuss possible approaches to topological cosection localization.

The theory of total dual VMRT was firstly introduced by Hwang and Ramanan in the study of Hecke curves on the moduli of vector bundles on curves. Later, Occhetta, Sola Conde and Watanabe generalized it to the case of minimal rational curves, and Hoering, Liu and Shao applied it to study positivity of tangent bundle of del Pezzo manifolds.

In this talk, I will review the theory of total dual VMRT and show how it is used to understand geometric characterization of Fano manifolds with big tangent bundle. This is a joint work with J. Kim and Y. Lee.

Hessenberg varieties are interesting objects in both algebro-geometric and combinatorial perspectives. The Shareshian-Wachs conjecture connects the cohomology of Hessenberg varieties with the chromatic quasi-symmetric functions of the associated graphs, which are refinements of the chromatic polynomials. In this talk, we introduce generalized Hessenberg varieties and study their birational geometry via blowups. As a result, natural maps from Hessenberg varieties to projective spaces or the permutohedral varieties are decomposed into explicit blowups and projective bundle maps. As a byproduct, we also provide an elementary proof of the Shareshian-Wachs conjecture and its natural generalization. This is joint work with Young-Hoon Kiem.

Quantum Lefschetz type properties enables us to relate GW theory complete intersection in an entire space, with the GW theory of the entire space. In this talk, I will introduce various types of quantum Lefschetz type properties in GW theory and quasi-map theory. Also I will introduce recent research on genus two, which is a joint work with Jeongseok Oh and Mu-Lin Li.

TBA

The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will explain the current status of knowledge and how one can show that the derived category of the symmetric product of the base curve can be embedded into the derived category of the moduli space. Key ingredients are geometric representation theory, in particular Schur-Weyl duality, Borel-Weil-Bott-Teleman theory, and a description of the derived category of GIT by Halpern-Leistner. This is joint work in progress with Kyoung-Seog Lee.

We discuss counting surfaces on Calabi-Yau 4-folds. The purpose is twofold: Firstly, we construct three types of moduli spaces of surfaces that are related via GIT wall-crossing and show that they parametrize polynomial Bridgeland stable objects in the derived category. Secondly, we construct reduced Oh-Thomas virtual cycles on these moduli spaces using Kiem-Li's cosection localization and show that they are deformation invariant along Hodge loci. As an application, we prove that having a non-zero reduced virtual cycle implies Grothendieck's variational Hodge conjecture. This is joint work with Younghan Bae and Martijn Kool.

Jun Li's algebraic Donaldson-Floer theory aims to describe the Donaldson invariants of a smooth surface degenerating to a union of two surfaces meeting transversely at a divisor as the pairing of the relative Donaldson invariants. The main tool in his approach is his theory of good degenerations of moduli spaces, which was inspired by Donaldson's analogous program in differential geometry. We introduce the above-mentioned topics, discuss the difficulties appearing in his approach and discuss how those difficulties might be circumvented by relativizing Mochizuki's 'Donaldson = Seiberg-Witten' formula.