Titles & Abstracts

 Let S be a simply-connected rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of S is at most three. In this talk, we leverage results from the study of smooth 4-manifolds, such as the Donaldson diagonalization theorem, to establish additional conditions for S. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. We also identify infinite families of singularities that satisfy properties in algebraic geometry, including the orbifold BMY inequality, but are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss experimental results related to this problem. This is joint work with Jongil Park and Woohyeok Jo.

To construct a finite dimensional model for Dirac operator, one method is to use finite direct sum of  the eigenspaces. This method is effectively used in some aspects of gauge theory.  Another method is the Finite Element Method. In this latter case  the naive approximation cannot capture the information of the index of the original Dirac operator. In this talk we explain that if we add "Wilson term" to the naive approximation, then it captures the information, at least for the regular lattice approximation of torus in any dimension. For the integer valued index this was known in physics and mathematically established by D.H. Adams around 2000. We give a new approach which is valid for mod 2 index, family index and equivariant index. Joint work with Hidenori Fukaya, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi and Mayuko Yamashita.

 In this talk a  symplectic divisor in a symplectic 4-manifold refers to a normal crossing configuration of embedded symplectic surfaces that intersect positively. Such divisors in rational surfaces were used by Jongil Park and collaborators to construct small exotic symplectic surfaces and new algebraic surfaces. We survey some aspects of symplectic divisors.

 A normal projective surface with at worst quotient singularities is called a rational homology projective plane if it has the same Betti numbers as the complex projective plane. In this talk, I will explain examples, then discuss the current state of the art on the classification of such surfaces. This problem contains the Montgomery-Yang problem from differential topology, originated from the study of circle actions on the 5-sphere.

In this presentation, we explore deformations of sandwiched surface singularities through an anti-MMP approach. We focus on the Kollár Conjecture. This conjecture suggests that deformations of rational surface singularities are induced by their P-modifications. We apply this idea to prove the conjecture for sandwiched surface singularities that have only big nodes.

 In this lecture, we first introduce the new nonlinear elliptic system of bordered contact instantons  with  Legendrian boundary condition in the quantitative study of contact topology. Then we explain how the study of compactified moduli spaces of contact instantons in combination with the contact Hamiltonian geometry can be used to prove the Shelukhin's conjecture which reads that any contactomorphism has a translated (fixed) point whenever its oscillation norm is smaller than twice the period gap of the contact manifold. The relevant contact geometric construction involves the Legendrianization of contact diffeomorphisms and the usage of the $\Z_2$-symmetry of anti-contact involution.

I will discuss Lagrangian Floer theory on symplectic orbifolds.  The notion of twisted sectors or inertia orbifolds plays an important role in orbifold Gromov-Witten theory.  We introduced the notion of dihedral twisted sectors associated with a Lagrangian in a symplectic orbifold.  I will explain how it is used in the construction of Lagrangian Floer theory on a symplectic orbifold and discuss some related topics.  It is based on a joint work with Bohui Chen (Sichuan University) and Bai-Ling Wang (Australian National University). 

 One of the most beautiful results of the Gromov-Witten theory is the celebrated GW/H correspondence discovered by Okounkov and Pandharipande. The correspondence relates GW invariants of curves and Hurwitz numbers. In this talk, I will discuss a spin analog of GW/H correspondence that relates GW invariants of Kähler surfaces and spin Hurwitz numbers – Giacchetto et al. recently proved the genus zero case.

We show that if a topological four-manifold with fundamental group Z/2Z and with non-spin universal cover admits a smooth structure, then — with possibly finitely many exceptions — it admits infinitely many.

Given a closed 3-manifold,  it is natural to ask what is the "simplest" 4-manifold into which it embeds. One version of this question asks which 3-manifolds embed smoothly in 4-dimensional Euclidean space. While a complete answer to this seems out of reach for the moment, joint work with Bulent Tosun (2021) showed that if one asks for an embedding whose image bounds a symplectically convex region of 4-space, then there is an obstruction coming from Floer theory. In particular, we used this to show that no Brieskorn homology sphere admits such a symplectically convex embedding. I will review these ideas and results, and how they fit in with other notions of convexity (e.g., holomorphic convexity and rational convexity), and describe how recent progress in complex geometry allows extensions some of these non-embedding results to 4-manifolds other than Euclidean space, such as rational complex surfaces.

 There are several ways of counting holomorphic curves in Calabi-Yau 3-folds. Counting them as maps gives rise to the Gromov-Witten invariants. This overcounts multiple covers and gives rise to non-integer invariants due to their symmetries. But one can consider instead images of such maps (possibly with multiplicity), regarded either as subsets or as integral currents. This allowed us to prove a structure theorem for the GW invariants of symplectic 6-manifolds and the Gopakumar-Vafa conjecture. The latter states that the GW invariants of CY 3-folds are obtained, by a specific transform, from another set of invariants called BPS states which have better properties: integrality and finiteness. The integrality statement was proved earlier in joint work with Thomas Parker and the finiteness more recently in joint work with Aleksander Doan and Thomas Walpuski. 

 This talk presents some of the the background and the topological ingredients of our proof, as well as recent progress, joint with Penka Georgieva, towards proving that a similar structure theorem holds for the real GW invariants of Calabi-Yau 3-folds with an anti-symplectic involution. 

 We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type A and C. This is a joint work with Qifeng Li.

 The positivity of the (co)tangent bundle of projective manifolds have been studied from various aspects in the last ten years. In this talk, I will focus on some works on the positivity of the (co)tangent bundle of smooth projective surfaces, including a joint work with Jia Jia and Guolei Zhong that classifies non-ruled surfaces with the pseudo-effectiveness of the tangent bundle.

 Since S. Donaldson introduced gauge theory to show that some of topological 4-manifolds do not admit a smooth structure in 1982, there has been a great progress in smooth and symplectic 4-manifolds mainly due to Donaldson invariants, Seiberg-Witten invariants and Gromov-Witten invariants. But the complete un- derstanding of 4-manifolds is far from reach and it is still one of the most active research areas in geometry and topology. Since I started my graduate program in mathematics in 1991, my main research interests are in the study of smooth/symplectic 4-manifolds and complex surfaces using gauge theory.

  In this talk, I briefly review what I have done and what I haven’t done yet in last 30 years. In particular, I’ll focus on the topics such as a rational blow-down surgery, a new family of simply connected symplectic 4-manifolds with $b^+_2 = 1$ and complex surfaces of general type with $p_g = 0$, knot-surgery and Seiberg-Witten theory, symplectic fillings versus Milnor fibers, and a study on 4-manifolds with Euler characteristic with 3.

 The well-known gonality conjecture of smooth curves was raised by Green-Lazarsfeld(1984) and proved by Ein-Lazarsfeld in 2016. In this talk, we’d like to generalize this conjecture to the first non-trivial strand of syzygies in the category of higher secant varieties of curves and further investigate the shape of the Betti table of secant varieties when a curves is embedded in a projective space with degree large enough.

It can be shown that the vanishing and non-vanishing of Betti numbers of secant varieties is completely determined by the gonality sequence of an original curve. It would also be very interesting to get the effective degree bound of the line bundle giving the embedding of the curve. This is a joint work with Junho Choe and Jinhyung Park.

 The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define a new Floer cohomology, called monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special monodromy class in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

 A fundamental problem in low-dimensional topology is to find the minimal genus of embedded surfaces in a 3- or 4-manifold in a given homology class.  In this talk, we will examine the rational Seifert and slice surfaces of rationally null-homologous knots and study the associated minimal genus problems. This is a joint work with Jingling Yang.

 Lefschetz fibration is one of main research tools in the study of  symplectic 4-manifolds. Prof. Park and I studied various aspects of Lefschetz fibration structures in symplectic 4-manifolds including Fintushel-Stern knot surgery 4-manifold. Recently we are interested in a construction of Lefschetz fibrations with positive signature and small Euler characteristic. In the talk, we will briefly review what we did last 20 years and what we are trying to solve recent years.