2015 Fiscal Year Final Research Report
New developments of singularity theory of mappings
| Project/Area Number |
23244008
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyushu University |
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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)
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| Co-Investigator(Kenkyū-buntansha) |
OHMOTO Toru 北海道大学, 大学院理学研究院, 教授 (20264400)
YOKURA Shoji 鹿児島大学, 学術研究院理工学域理学系, 教授 (60182680)
IWASE Norio 九州大学, 大学院数理学研究院, 教授 (60213287)
KAMADA Seiichi 大阪市立大学, 理学研究科, 教授 (60254380)
SAKUMA Kazuhiro 近畿大学, 理工学部, 教授 (80270362)
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| Co-Investigator(Renkei-kenkyūsha) |
ISHIKAWA Masaharu 東北大学, 大学院理学研究科, 准教授 (10361784)
FUKUI Toshizumi 埼玉大学, 理工学研究科, 教授 (90218892)
ISHIKAWA Goo 北海道大学, 大学院理学研究院, 教授 (50176161)
YAMAMOTO Minoru 弘前大学, 教育学部, 准教授 (40435475)
TAKASE Masamichi 成蹊大学, 理工学部, 准教授 (30447718)
ASHIKAGA Tadashi 東北学院大学, 工学部, 教授 (90125203)
KATANAGA Atsuko 信州大学, 学術研究院, 准教授 (20373128)
KOBAYASHI Mahito 秋田大学, 大学院理工学研究科, 准教授 (10261645)
YAMAMOTO Takahiro 九州産業大学, 工学部, 准教授 (60435972)
TAKEUCHI Kiyoshi 筑波大学, 数理物質系, 教授 (70281160)
TAKATA Toshie 九州大学, 大学院数理学研究院, 准教授 (40253398)
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| Project Period (FY) |
2011-04-01 – 2016-03-31
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| Keywords | 特異点 / 多項式写像芽 / 特異点消去 / 埋め込み / はめ込み / 特異ファイバー / 同境群 / データ可視化 |
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| Outline of Final Research Achievements |
The main purpose of this project was to indicate new directions of the singularity theory of differentiable mappings in our era of modern Mathematics. First, we completely solved affirmatively the long-standing classical problem of Milnor about the existence of nontrivial polynomial map germs of dimension 6 to 3, by utilizing the topology of configuration spaces. We also completely determined the conditions for a special generic map to be desingularized from a viewpoint of immersions and embeddings. We classified singular fibers of stable maps on 3-manifolds with boundary, established the notion of cobordism group of Morse functions on surfaces with boundary, and showed that it is a cyclic group of order two. Finally, we applied such a theory of singular fibers to the data visualization and developed a new user interface. Summarizing, we could broadly show the new directions for the development of the singularity theory of differentiable mappings.
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| Free Research Field |
位相幾何学
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