부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title: A pants decomposition of a rose 

Abstract: A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. For example, by forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. Through the fact that a rose is homotopy equivalent to a compact surface with boundary, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.

Title: Deformation rigidity of some quasi-homogeneous varieties with Picard number one 

Abstract: We investigate the global deformation rigidity results of rational homogeneous manifolds of Picard number one which were developed by Hwang, Mok and others. In particular, we focus on the role of varieties of minimal rational tangents. Starting with similar ideas, we introduce some recent global deformation rigidity results of some quasi-homogeneous varieties, symmetric varieties and horospherical varieties, with Picard number one. This talk will be a very introductory presentation compiling known results.

Title: Geometric realizations of birational maps

Abstract: In this talk we aim to describe the rich relation between C*-actions on projective varieties and birational geometry. In the first part, we show how any C*-action induces naturally a birational map among the geometric quotients of the variety parametrizing general orbits. We then try to answer the opposite question: given a birational map, can we construct a variety, with a C*-action, such that the birational map among the quotients coincide with the starting one? If so, we call this variety a geometric realization of the birational maps. Geometric realization are algebraic analogs of cobordism for Morse theory. We survey which birational maps admit a geometric realization, and provide explicit examples in the toric setting. This talk is based on joint works with G. Occhetta, E. A. Romano, L. Solá Conde and S. Urbinati.

Title: Some open problems in syzygies and related topics

Abstract: In this talk, we introduce some interesting problems in syzygies and related fields. Although we are focusing on syzygies, we also introduce some problems in other areas of projective algebraic geometry. The open problems we introduce include Eisenbud-Goto regularity conjecture, Green’s conjecture, and Hartshorne's conjecture. The purpose of this talk is to present the problems easily so that people whose research area is not algebraic geometry can also catch the idea of the problems.

Title: Optimality of Gerver's Sofa 

Abstract: We resolve the moving sofa problem, posed by Moser in 1966, which asks for the maximum area of a connected planar shape that can move around the right-angled corner of a L-shaped hallway with unit width. We confirm the conjecture made by Gerver in 1994 that his construction, known as Gerver's sofa, with 18 curve sections attains the maximum area 2.2195…. (학부생의 참석도 환영합니다)

Title. Non-extendablity of Shelukhin's quasimorphism and non-triviality of Reznikov's class

Abstract. Shelukhin constructed a quasimorphism on the universal covering of the group of Hamiltonian diffeomorphisms for a general closed symplectic manifold. We prove the non-extendability of that quasimorphism for certain symplectic manifolds, such as a blow-up of torus and the product of a surface of genus at least two and a closed symplectic manifold. As its application, we prove the non-vanishing of Reznikov's characteristic class for the above symplectic manifolds. This is a joint work with Mitsuaki Kimura (Osaka Dental University), Shuhei Maruyama (Kanazawa University), Takahiro Matsushita (Shinshu University), Masato Mimura (Tohoku University). 

Title: Equivariant Lagrangian correspondence and Seidel representation 

Abstract:  In this talk, we first review a joint work (with Siu-Cheong Lau and Conan Leung) on Floer theory for equivariant Lagrangian correspondences, applying to resolve of a conjecture of Teleman on the mirror constructions of Hamiltonian G-spaces Y and their symplectic quotients X using moment level correspondences. 

Time permitting, we discuss an ongoing joint work (with the same authors) on another application to Seidel representation, which is obtained from counting sections of Seidel spaces E associated to Hamiltonian S^1 spaces X, by realising X as symplectic quotients of E. In particular, we generalise a result of Chan-Lau-Leung-Tseng on the images of Seidel elements under closed-string mirror symmetry of toric semi-Fano manifolds X.

Title: Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

Abstract: The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about groups of 'area'-preserving diffeomorphisms on non-orientable manifolds. In this talk, I will discuss the recent study about groups of density-preserving diffeomorphisms on non-orientable manifolds, especially on the Möbius band. Here, the density is a natural concept that generalizes volume without concerning orientability. This talk is based on the joint work with S.Maruyama.

Title: Vietoris-Rips complex, Persistent Homology, and Gromov-Hausdorff distance

Abstract: The Vietoris–Rips complex is a type of simplicial complex defined by the metric structure of an underlying space. More precisely, given a metric space 

X and a positive real number r>0, the Vietoris–Rips complex VR(X;r) is the simplicial complex whose vertex set is X, and where a finite subset {x_0,...,x_p} of X forms a p-simplex whenever diam({x_0,...,x_p})<r

Recently, Vietoris–Rips complexes have emerged as a central object of study in two active research areas:

(1) Topological Data Analysis (TDA) and (2) Global Metric Geometry.

In TDA, a key pipeline for extracting topological features from data involves building a filtration of Vietoris–Rips complexes across scales and analyzing the associated persistent homology. To fully understand the power and limitations of persistent homology as a tool for data analysis, it is essential to investigate the homotopy types of Vietoris–Rips complexes, especially when the underlying space is a manifold.

In Global Metric Geometry, inspired by TDA and shape matching problems, computing the Gromov–Hausdorff distance between manifolds has become increasingly important. Remarkably, we have recently obtained sharp bounds on the Gromov–Hausdorff distance between spheres of different dimensions by employing thickenings via Vietoris–Rips complexes. Furthermore, we have shown that these bounds are tight.

In this presentation, I will explore these developments and highlight the role of Vietoris–Rips complexes in both TDA and Global Metric Geometry.

Title: Lefschetz pencils on a complex projective plane from a topological viewpoint 

Abstract: Any projective surface admits a Lefschetz pencil, and more generally every symplectic $4$-manifold admits a Lefschetz pencil and vice versa. Moreover, a Lefschetz pencil can be characterized topologically by means of its monodromy factorization in the mapping class group of the fiber surface. In this talk, we present a topological construction of symplectic Lefschetz pencils analogous to the holomorphic Lefschetz pencils of degree $d$ curves in $\mathbb{C}P^2$ for arbitrary $d \geq 4$. Moreover, we prove that the genus $3$ holomorphic Lefschetz pencil on $\mathbb{C}P^2$ is isomorphic to a topologically constructed Lefschetz pencil for $d=4$.