This talk provides a concise overview of the process for identifying closed simply connected Sasaki-Einstein 5-manifolds derived from K-stable log del Pezzo surfaces. Subsequently, it enumerates the closed simply connected 5-manifolds that have been identified thus far as capable of accommodating Sasaki-Einstein metrics. Additionally, the talk introduces potential candidates for Sasaki-Einstein 5-manifolds, contributing to the comprehensive classification of closed, simply connected Sasaki-Einstein 5-manifolds. 

We review the basics of matrix factorizations of a polynomial, with emphasis on their roles in homological mirror symmetry. Topics include: dg category of matrix factorizations, graded matrix factorizations, localized mirror functors (due to Cho-Hong-Lau), Hochschild invariants etc.

A theorem of Donaldson and Sun asserts that the metric tangent cone of a smoothable Kähler–Einstein Fano variety underlies some algebraic structure, and they conjecture that the metric tangent cone only depends on the algebraic structure of the singularity. Later Li and Xu extend this speculation and conjecture that every klt singularity has a canonical “stable” degeneration induced by the valuation that minimizes the normalized volume. I’ll talk about some joint work with Chenyang Xu on the solution of these conjectures. If time permits, I will also discuss some further implications on the boundedness of singularities. 

Birkar and Hu showed that if a pair (X, Δ) is lc and K_X + Δ admits a birational Zariski decomposition, then (X, Δ) has a minimal model. Analogously, we can prove that if a pair (X, Δ) is pklt and −(K_X + Δ) admits a birational Zariski decomposition, then (X, Δ) has an anticanonical minimal model.

While Fukaya category is a powerful invariant in symplectic topology, its computation is often considered challenging. However, if we consider a Weinstien manifold, i.e., symplectic manifold with nice properties, then the nice properties can simplify the computation of the corresponding Fukaya category. In the first half of the talk, we will discuss why this phenomenon happens, and in the second half, I will present an example computation. The example I will give is a plumbing space of $T^*S^n$. (There will be no prerequisite except the concept of differential forms.)

A cylinder in a normal projective variety is a Zariski open subset that is isomorphic to a product of an affine line and an affine variety. There exists a connection between rationality and cylindricity for a normal variety.  Additionally, a normal projective space has an anti-canonical polar cylinder if and only if its affine cone admits a nontrivial unipotent group action.  Therefore, proving the existence of an anti-canonical polar cylinder is crucial. Using the alpha-invariant suggested by Tian, we can prove the existence of an anti-canonical polar cylinder for Fano varieties. Consequently, there is a connection between K-stability and the existence of an anti-canonical polar cylinder for Fano varieties. In this talk, we study a relationship between K-stability and the existence of an anti-canonical polar cylinder for Fano varieties. 

In 1957, Atiyah introduced the Atiyah class of holomorphic vector bundles which is an obstruction to the existence of holomorphic connections. The Atiyah class is later generalised to differential graded manifolds (dg manifolds) and it takes an important role in the study of dg manifolds.

Recently, Behrend—Liao—Xu showed that dg manifolds of certain type can be thought of as a model for derived intersection of smooth manifolds in the $C^{\infty}$-context. This talk will focus on investigating the Atiyah class of derived intersections in this sense. In particular, we will show that it is related to the cleanness of the intersection.

We study the parabolic Hitchin systems under the Langlands dual, focusing on types B and C. We aim to understand the SYZ and topological mirror symmetries for the Langlands dual parabolic Hitchin systems. We find three levels of dualities/symmetries: 

1. Classical level deals with the parabolic structures, which relate to nilpotent orbits. The duality here is the Springer dual.

2. Local level plays a crucial role; it serves as a bridge between the classical level and the global level. It deals with affine Spaltenstein fiber; the symmetry here is Lusztig's canonical quotient.

3. Global level deals with the moduli space of parabolic Higgs bundles. Mirror symmetries here are SYZ and topological mirror symmetries.

This talk is partly based on the joint work with B. Fu and Y. Ruan (arXiv:2207.10533) and the joint work with B. Wang, and X. Wen (arXiv:2403.07552).