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多様体の幾何構造変形の可視化

流形几何结构变形可视化

基本信息

批准号：
08640092
负责人：
小島定吉
金额：
$ 1.41万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至无数据
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08640092/
关键词：
多様体双曲多様体幾何構造 Dehn手術可視化

项目摘要

多様化の幾何構造の研究には,様々な数学の側面が現れる.本研究では,研究組織のメンバーの多岐にわたる専門領域と,メンバーが在籍する数理・計算科学専攻の計算機環境を拠り所に,他機関の研究者も含め相互研究交流を日常的にとりながら,幾何構造の変形の研究を進めた.具体的な変形操作として,「3次元双曲多様体の双曲デーン手術」と「サークルパッキングを用いた閉曲面の射影構造の変形」を取り上げた.前者については,コンピュータ上での仮想計算をよりどころに,錐状の特異集合を許容する3次元双曲錐多様体の大域変形の可能性について,いろいろ理論的な結果を得た.たとえば双曲結び目の分岐被覆には,1つの例外を除いて,分岐指数が3以上であれば常に双曲構造が入ることをなど.現在成果をまとまた論文を投稿中である.後者については,当初計画していた種数2以上の閉曲面の射影構造の変形論は計算に多大な困難を伴うことが分かったため,種数を1にし,サークルパッキングを使ったトーラスの複素アフィン構造の変形に対象を変えた.そして変形に従うパッキングの変化を,展開写像を通して平面上に可視化した.またこうした実験数学的なアプローチをもとに,変形が予想どおり局所的に実2次元分あることを示した.これらの実験数学的取組みや理論的考察には,分担者および海外を含む1他の機関の研究者との間の交流により得られた様々なアイデアをフルに使った.研究の性格上多くの専門家との交流が不可欠であったため,研究連絡および専門的知識の提供依頼が頻繁にあり,研究費のかなりの部分を旅費および謝金にあてた.

The study of geometric structure of diversification is contrary to the mathematical underpinnings. This study is organized in various fields, such as mathematics, computational science and computer science. Researchers from other institutions are involved in the study of geometric structures. For the specific shape operation, select "Hyperbolic operation of three-dimensional hyperbolic polyhedron" and "Projective construction of closed surface with three-dimensional hyperbolic polyhedron". In the former case, we obtain the results of the theory on the possibility of large-scale transformation of three-dimensional hyperbolic cone polyhedrons by calculating the ideal of cone shape and special set. The hyperbolic structure is divided into two parts, one part is divided into two parts, the other part is divided into two parts, the other part is divided into three parts, the other part is divided into two parts, the other part is divided into three parts, the other part is divided into two parts, the other part is divided into two parts, the other part is divided into three parts, the other part is divided into two parts, the other part is divided into two parts, the other part is divided into three parts, the other part is divided into two parts, the other part is divided into three parts, the other part is divided into two parts, the other part is divided into two parts, the other part is divided into three parts, the third part is divided into three parts, the other part is divided into three parts, the third part is divided into three parts, the fourth part is divided into four parts Now the results are in. In the latter case, it is planned that the number of projective structures of closed surfaces with more than 2 numbers will be calculated with great difficulty. The image can be visualized on a flat surface. The number of people who are interested in this topic is: This is the first time in history that the author has been involved in the study of mathematics. Research on the character of many people in the family and exchanges can not be owed, research contacts and knowledge of the door to provide frequent, research fees and part of the travel expenses.