The 4th Workshop on Geometry and Topology around Young-Nam

September 22–23, 2023

Kyungpook National University, Daegu, Korea

sponsored by
National Research Foundation of Korea
Samsung Science and Technology Foundation
Poster

Commemorative Pictures

Speakers

Intensive Lectures

  • 이강혁 (경상국립대학교)
  • 정기룡 (경북대학교)

Invited Speakers

  • 김벼리 (POSTECH)
  • 박지훈 (경북대학교)
  • 송종백 (부산대학교)
  • 이경석 (POSTECH)
  • 최선미 (경북대학교)

Contributed Speakers

  • 남응모 (부산대학교)
  • 박상훈 (부산대학교)
  • 정지원 (부산대학교)

Confirmed Participants

  • 김대환 (대구대학교)
  • 김민훈 (경북대학교)
  • 김벼리 (POSTECH)
  • 김영락 (부산대학교)
  • 김유식 (부산대학교)
  • 김유정 (부산대학교)
  • 남응모 (부산대학교)
  • 박경동 (경상국립대학교)
  • 박상우 (부산대학교)
  • 박상훈 (부산대학교)
  • 박지윤 (부산대학교)
  • 박지훈 (경북대학교)
  • 서애령 (경북대학교)
  • 송종백 (부산대학교)
  • 안병희 (경북대학교)
  • 양승엽 (경북대학교)
  • 오진석 (경북대학교)
  • 우창화 (부경대학교)
  • 윤홍대 (경북대학교)
  • 이강혁 (경상국립대학교)
  • 이경석 (POSTECH)
  • 이상수 (경북대학교)
  • 이지현 (부산대학교)
  • 정기룡 (경북대학교)
  • 정지원 (부산대학교)
  • 조민서 (부산대학교)
  • 조현지 (부산대학교)
  • 주정훈 (부산대학교)
  • 최선미 (경북대학교)
  • 최영준 (부산대학교)
  • 최진주 (부산대학교)
  • 표준철 (부산대학교)
  • Thomas Hudson (DGIST)
  • Tian Xueqian (부산대학교)

Organizers

  • 김민훈 (경북대학교)
  • 안병희 (경북대학교)
  • 양승엽 (경북대학교)
  • 정기룡 (경북대학교)
  • 김유식 (부산대학교)
  • 김영락 (부산대학교)
  • 표준철 (부산대학교)
  • 우창화 (부경대학교)

Venue

Talk Schedule

9월 22일 (금)

13:20 – 13:30 Opening
Chair : 김민훈
13:30 – 14:10 정기룡 (경북대학교)
Incidence variety and Fano Plane
14:10 – 14:20 Break
14:20 – 15:00 이강혁 (경상국립대학교)
Existence of a complete holomorphic vector field and the uniformization of negatively curved Kahler manifolds I
15:00 – 15:10 Break
Chair : 우창화
15:10 – 15:40 이경석 (POSTECH)
Links and Milnor fibers of complex hypersurface singularities
15:40 – 15:50 Break
15:50 – 16:10 김벼리 (POSTECH)
On smoothing skill for topological surfaces in 4-manifolds
16:10 – 16:30 박지훈 (경북대학교)
Factor system of graphs and Combinatorial HHS
16:30 – 16:40 Break
Chair : 정기룡
16:40 – 17:00 남응모 (부산대학교)
Some classification theorems of λ-translating solitons
17:00 – 17:20 박상훈 (부산대학교)
Review of minimal generating sets of moves for surfaces immersed in the 4-space
17:20 – 17:40 정지원 (부산대학교)
An Introduction to The Complex Monge-Ampère Equation on Compact Kähler/Hermitian Manifolds
17:40 – 20:00 Dinner

9월 23일 (토)

Chair : 표준철
09:30 – 10:10 정기룡 (경북대학교)
Enumeration through moduli
10:10 – 10:20 Break
10:20 – 11:00 이강혁 (경상국립대학교)
Existence of a complete holomorphic vector field and the uniformization of negatively curved Kahler manifolds II
11:00 – 11:10 Break
Chair : 양승엽
11:10 – 11:40 송종백 (부산대학교)
The face numbers of a lattice polytopes and toric varieties
11:40 – 12:00 최선미 (경북대학교)
Effective data reduction algorithm for topological data analysis
12:00 – 12:10 Closing
12:10 – 14:00 Lunch
14:00 – 18:00 Discussion

* The schedule may vary.

Conference Dining

Conference Dinner

9월 22일(금) 17:40 ~ 20:00

라미아에밀리

대구 중구 달구벌대로 2061-2 (대구 중구 계산동2가 171), 지도

Conference Lunch

9월 23일(토) 12:10 ~ 14:00

산

대구 중구 남성로 53-1 산 한정식 (대구 중구 동성로3가 81-2 산 한정식), 지도

Title & Abstract

Abstracts

Intensive Lectures

Speaker
이강혁 (Gyeongsang National University)
Title
Existence of a complete holomorphic vector field and the uniformization of negatively curved Kähler manifolds
Abstract
The uniformization theorem of the Riemann surfaces says that a simply connected Riemann surface is conformally equivalent to Riemann sphere $\mathbb{C}P^1$ or the complex plane $\mathbb{C}$ or the unit disc $\Delta$. This implies that every Riemann surface is covered by these three model spaces. Especially most Riemann surfaces (e.g. compact Riemann surface of $g\ge2$) is covered by the unit disc by classical theories such as the Schwarz lemma and the Gauss- Bonnet theorem. This crucial feature is based on the curvature negativity of the Poincaré metric of $\Delta$. In Several Complex Variables, theories on the Poincaré metric have been developed under name of the geometric function theory. However, there are still many problems to be solved for the multidimensional uniformization. A fundamental problem is the characterization of hermitian symmetric spaces of noncompat type as a covering of a negatively curved compact Kähler manifold. A key approach in this study is to show the existence of a complete holomorphic vector field. In this talk, I will introduce classical results and problems on the uniformization first. Then the the potential rescaling method for the existence of complete holomorphic vector fields will be discussed. This is a collaboration work with Y.-J. Choi and S. Seo.
Speaker
정기룡 (Kyungpook National University)
Title
Incidence variety and Fano Plane
Abstract
In this presentation, we will introduce the concept of incidence variety and provide several illustrative examples. Additionally, we will explore the connection between the Fano plane and a simple Diophantine equation. The content of the talk is tailored for graduate students.
Title
Enumeration through moduli
Abstract
In this presentation, we will begin by explaining the concept of moduli, followed by a discussion on various enumerative problems and their corresponding solutions. The content of the talk is specifically designed for graduate students.

Invited Talks

Speaker
김벼리 (POSTECH)
Title
On smoothing skill for topological surfaces in 4-manifolds
Abstract
In this talk, I will talk about a new smoothing skill for topologically embedded surfaces and disks in smooth 4-manifolds that provides topological isotopies to smooth surfaces and disks. This result is motivated from recent David Gabai’s Light bulb theorem. As an application, we can get some result which leading us to “topological=smooth” in dimension 4 for isotopy classifications of certain disks and spheres.
Speaker
박지훈 (Kyungpook National University)
Title
Factor system of graphs and Combinatorial HHS
Abstract
A Hierarchically Hyperbolic Space (HHS) is an unified framework for studying the Mapping class groups of surfaces and the CAT(0) cube complex introduced by Behrstock-Hagen-Sisto in 2017. By its deep geometric, structural consequences, finding an HHS structure for a given space or group is an interesting question, but is usually challenging because of the complicated definition of HHS. To remedy such difficulty, Behrstock- Hagen-Martin-Sisto introduced a Combinatorial HHS as a simpler criterion for verifying HHS structure via the curve graph machinary for the Mapping class groups. The CHHS criterion is, however, not the optimal one as it cannot cover the case of the curve graph of CAT(0) cube complex, although its HHS structure combinatorially encoded in the curve graph. In this talk, we will briefly review the (combinatorial) HHS theory and modify the CHHS criterion to cover the cubical curve graph and the intersection graph of the system of subspaces. If time permits, as an application we extends the factor system machinary of the CAT(0) cube complex to the CAT(0) prism complex and explain related topics.
Speaker
송종백 (Pusan National University)
Title
The face numbers of a lattice polytopes and toric varieties
Abstract
Billera—Lee '79 and Stanley '80 proved McMullen's g-conjecture, which provides us a complete combinatorial characterization of a simplicial (or simple) polytope in terms of its face numbers. Stanley's proof for the necessity part of this conjecture employed toric geometry, where he used the relationship between the face numbers of lattice polytopes and the Betti numbers of rationally smooth toric varieties corresponding to a simple polytope. In this talk, we will discuss a topological background of this relationship and see how it might be extended to singular toric varieties.
Speaker
이경석 (POSTECH)
Title
Links andMilnor fibers of complex hypersurface singularities
Abstract
Links of complex hypersurface singularities provide many interesting examples in the low-dimensional topology. It is a natural and interesting task to understand how the topology of the links is related to the real/complex geometry of the hypersurface singularities. In the first part of this talk, I will discuss a brief history of the subject. Then I will discuss how certain topological invariants of links of hypersurface singularities are related to their Milnor fibers. The last part of this talk is based on joint works (some in progress) with Anatoly Libgober and Nikolai Saveliev.
Speaker
최선미 (Kyungpook National University)
Title
Effective data reduction algorithm for topological data analysis
Abstract
Topological data analysis (TDA) has recently emerged as one of the most interesting tools in the data science toolbox. TDA provides a methodology for analyzing the shape of a dataset using tools and prospects from algebraic topology. However, the computational complexity makes it quickly infeasible to process large datasets, especially those with high dimensions. In this talk, we introduce a preprocessing strategy called the Characteristic Lattice Algorithm (CLA), which allows users to reduce the size of a given dataset as desired while maintaining geometric and topological features in order to make the computation of TDA feasible or to shorten its computation time. In addition, we derive a stability theorem and an upper bound of the barcode errors for CLA based on the bottleneck distance. This is joint work with Jinseok Oh, Jeong Rye Park, Seung Yeop Yang, and Hongdae Yun.

Contributed Talks

Speaker
남응모 (Pusan National University)
Title
Some classification theorems of $\lambda$-translating solitons
Abstract
$\lambda$-translating solitons are natural generalizations of translating solitons which are special solutions of the mean curvature flow. Since translating solitons often occur as some type of singularities, it is important to study ($\lambda$-)translating solitons. Moreover, in terms of variational viewpoint, $\lambda$-translating solitons are related to isoperimetric problems. In this talk, I will give some classification results on these solitons. More specifically, under the constancy of the mean curvature, a $\lambda$-translating soliton must be a hyperplane or split into constant mean curvature hypersurface and Euclidean line. Furthermore, under the constancy of norm of the second fundamental form, a graphical $\lambda$-translating soliton must be a hyperplane. These results hold both on the Euclidean space and Minkowski space.
Speaker
박상훈 (Pusan National University)
Title
Review of minimal generating sets of moves for surfaces immersed in the 4-space
Abstract
Cup move, cap move, band slide, band swim makes a minimal generating set of spacial moves between diagrams of equivalent surfacelinks embedded in $\mathbb{R}^4$. M. Hughes, S. Kim, and M.Miller added three moves: intersection/band slide, intersection/band pass, and intersection/band swim to construct a generating set of moves relating equivalent immersed surface-links. For planar moves, K. Yoshikawa introduced eleven moves for surface-links embedded in $\mathbb{R}^4$. With three extra moves, S. Kamada, A. Kawauchi, J. Kim, and S.Y. Lee extended Yoshikawa’s moves to a set of moves relating equivalent immersed surface-links. Michal Jablonowski derives a minimal generating set of spatial moves for diagrams of surfaces immersed in the 4-space, which translates into a generating set of planar moves. In this talk, based on his paper, I intend to review how he built a minimal generating set.
Speaker
정지원 (Pusan National University)
Title
An Introduction to The Complex Monge-Ampère Equation on Compact Kähler/Hermitian Manifolds
Abstract
After Calabi proposed his conjecture, only few mathematicians believed it to be true. In fact, most people thought it was "too good to be true". However, it was eventually proven by Shing-Tung Yau in 1978. And in 2010, Tosatti and Weinkove more generalized the results of Yau on Hermitian Manifolds. In this talk, I will introduce the conjecture proposed by Eugenio Calabi in 1954 and its solution by Yau and how the solution can be used to construct Kähler-Einstein metrics. Furthermore, I will briefly introduce the results of Tosatti andWeinkove.