Innovation challenge to the construction of stable maps by big data visualization techniques

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13438
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kyushu University
• Principal Investigator: SAEKI OSAMU 九州大学, マス・フォア・インダストリ研究所, 教授 (30201510)
• Co-Investigator(Kenkyū-buntansha): 小林 真人 秋田大学, 理工学研究科, 准教授 (10261645) ; 山本 卓宏 東京学芸大学, 教育学部, 准教授 (60435972)
• Co-Investigator(Renkei-kenkyūsha): TAKAHASHI Shigeo 会津大学, コンピュータ理工学部, 教授 (40292619)
• Research Collaborator: SAKURAI Daisuke Zuse Institute Berlin, 研究員
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 特異ファイバー / 安定写像 / 可視化 / ビッグデータ / 写像の構成論 / 特異点集合 / 低次元多様体 / 単純化

Outline of Final Research Achievements

An arbitrary map between differentiable manifolds can be approximated by a stable map; however, it is know that explicit construction of a stable map is extremely difficult. The main objective of this project was to develop a new systematic method to construct stable maps by applying the big data visualization techniques in computer science.
We first obtained classification results of singular fibers of stable maps in various situations, and applied them to cobordism theories. We also used them to develop a user interface for analyzing and visualizing big data. Inspired by such techniques, we succeeded in developing a new method to explicitly construct a deformation of a given map, and applied it to the study of 4-dimensional manifolds. We thus constructed various geometrically important stable maps concretely, and applied them to the study of mapping invariants. We also developed a new method to prove a simplification theorem of stable maps on manifolds.

Free Research Field

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