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

Title. Polyhedral products and its applications to the persistent module

Abstract. In the standard pipeline of TDA, one constructs certain filtrations of simplicial complexes from a data set X, such as Vietoris--Rips complexes or Cech complexes. By applying the homology functor,  we obtain the standard persistent homology of X. In fact, in algebraic topology, taking the cohomology functor, hence the persistent cohomology, yields a richer algebraic structure than the usual persistent homology. Going further, one can consider certain resolutions of the face rings of simplicial complexes in the filtration to apply the Tor-functor, thereby defining the persistent Tor-algebra. It has much richer algebraic structures than the usual persistent cohomology. In this talk, we introduce the notion of a moment-angle complex (a particular type of the polyhedral product) which serves as a topological counterpart to the Tor-algebra of the simplicial complex. We then discuss the relevant persistent modules and their stabilities. It is based on the joint work with A. Bahri, I. Limonchenko, T. Panov and D. Stanley.

Title. Legendrian Floer cohomology and its spectral invariants via contact instantons

Abstract: Floer theory plays a crucial role in symplectic geometry. In contact geometry, Floer theory is typically studied using symplectization. However, with the concept of contact instantons introduced by Hofer and further developed by Oh, it is possible to formulate Floer theory directly on a contact manifold without relying on symplectization. In this talk, I will first give a brief overview of Lagrangian Floer cohomology on cotangent bundles, and then I will construct Legendrian Floer cohomology using contact instantons in the one-jet bundle. As an application, I will explain how to construct spectral invariants using Floer cohomology. This is joint work with Yong-Geun Oh.

Title: Symplectic fillings of unit cotangent bundles of spheres 

Abstract: The classification of symplectic fillings of contact manifolds is a central topic in symplectic and contact topology. In particular, there have been significant uniqueness results regarding symplectic fillings of certain contact manifolds, including contact spheres and unit cotangent bundles. In this talk, we present a uniqueness theorem for unit cotangent bundles of spheres and discuss its implications for symplectic cobordisms. This talk is based on joint work with Takahiro Oba.

Title: Homogeneous dynamics in Diophantine approximation and the Oppenheim conjecture. 

Abstract: Homogeneous dynamics explores the dynamical properties of (smooth) group actions on homogeneous spaces-specifically, the quotient spaces of Lie groups. In this talk, we will investigate the application of homogeneous dynamics to number theory through two significant examples: Diophantine approximation which examines the approximation of irrational numbers by rational numbers, and the Oppenheim conjecture which focuses on the distribution of images of integral vectors under the certain irrational indefinite quadratic form. These topics highlight the influential contribution of Dani and Margulis, respectively. Furthermore, we will discuss the role of hyperbolic geometry in these critical examples. This talk involves joint work with Seonhee Lim and Keivan Mallahi-Karai and with Prasuna Bandi and Anish Ghosh.

Title. The Mapper Algorithm in Topological Data Analysis 

Abstract. Topological Data Analysis (TDA) uses techniques from topology to extract valuable insights from data. Topology, which studies the shape and structure of spaces, enables TDA to uncover the underlying shape of a dataset. The Mapper algorithm, a network-based visualization technique in TDA, has shown applications in various fields. However, the Mapper algorithm requires tuning several parameters to generate a ``nice" Mapper graph. Focusing on a parameter called 'cover' in the Mapper algorithm, we present an algorithm that optimizes the cover of a Mapper graph by repeatedly splitting it based on a statistical test for normality.

Title: Legendre transformation related to toric plurisubharmonic functions and its application

Abstract: For arbitrary toric plurisubharmonic functions, we will present a criterion for admitting a decreasing equisingular approximation with analytic singularities. Our results are motivated by a recent result of Guan for toric plurisubharmonic functions of the diagonal type. In this study, the Legendre transformation of a toric plurisubharmonic function plays a central role. Recently, the significance of Legendre transformations on complex geometry and algebraic geometry has been raised. We will review the notion of Legendre transformations and introduce their application to find equisingular approximations.