Singularities in Algebraic Geometry and Symplectic Topology

특이점과 관련된 대수기하학과 사교위상수학의 연구 결과를 공유하고 심도 있는 토론을 진행하는 학회입니다.

연사

• 박경배 _ 강원대학교
  Algebraic Montgomery-Yang problem and smooth obstructions

• 전재관 _ IBS-CGP
  Deformations of weighted homogeneous surface singularities and the anti-MMP

• 정승조 _ 전북대학교
  Hypersurface singularities via $D$-modules

• 좌동욱 _ 고등과학원
  Hamiltonian and Lagrangian Floer theory of isolated singularities

제목과 초록

• 박경배 _ 강원대학교
  
  Algebraic Montgomery-Yang problem and smooth obstructions

Let $S$ be a 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 if its smooth locus is simply-connected. In this talk, we leverage results from the study of smooth 4-manifolds, including the Donaldson diagonalization theorem and Heegaard Floer correction terms, 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. Moreover, we identify large families encompassing infinitely many types of singularities that satisfy the orbifold BMY inequality, a key property in algebraic geometry, yet are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss computational results related to this problem, offering new insights into the algebraic Montgomery-Yang problem. This is joint work with Woohyeok Jo and Jongil Park.

• 전재관 _ IBS-CGP
  
  Deformations of weighted homogeneous surface singularities and the anti-MMP
  
  I will introduce a work in progress on the Koll\'{a}r conjecture. Roughly speaking, the conjecture states that deformations of rational surface singularities can be described by partial resolutions of the given singularities. I and Dongsoo Shin proved the conjecture for weighted homogeneous surface singlarities with big central node by using the deformation theory of sandwiched singularities and the minimal model program for 3-folds. In this talk, I will present another approach to the conjecture using the deformation theory of Pinkham and the anti-MMP. This is a joint work with Dongsoo Shin.
  
  

• 정승조 _ 전북대학교
  
  Hypersurface singularities via $D$-modules
  
  For hypersurface singularities, there are many interesting invariants, e.g. Milnor numbers, Tjurina numbers, log canonical thresholds, Berstein--Sato polynomials, Brian\c{c}on--Skoda exponents, etc. It turns out that the theory of $D$-modules plays a big role in studying these invariants. This talk gives a gentle introduction to the theory and we discuss further possible applications of the theory.
  
  

• 좌동욱 _ 고등과학원
  
  Hamiltonian and Lagrangian Floer theory of isolated singularities
  
  This talk will start with a review of several topological invariants of isolated singularities such as Milnor numbers, Lefschetz numbers, and multiplicities. As hypersurface singularities have a symplectic nature, those topological numbers admit symplectic enhancement. After briefly introducing Floer theory, I will explain how such numbers can be obtained from Hamiltonian and Lagrangian Floer theory of singularities. I will explain how this enhancement leads us to the solution to the classical problem such as Zariski conjecture. Then, I will introduce the recent joint work with Cheol-Hyun Cho, Hanwool Bae, and Wonbo Jeong which categorifies other classical operations such as variation homomorphism and Seifert parings.

주관

• 박희상 _ 건국대학교

• 신동수 _ 충남대학교

후원

• 고등과학원

• 한국연구재단

문의

• 신동수: dsshin@cnu.ac.kr