Topological study of real singularities and manifolds using fibring structures
| Project/Area Number |
16K05140
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Keio University (2018)
Tohoku University (2016-2017) |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2016-04-01 – 2019-03-31
|
| Project Status |
Completed (Fiscal Year 2018)
|
| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
|
|---|
| Keywords | 特異点 / 安定写像 / トーリック型コンパクト化 / ３次元多様体論 / 接触構造 / 結び目理論 / 低次元トポロジー / 多面体 / 特異点論 / 幾何学 |
|---|
| Outline of Final Research Achievements |
We studied the information of manifolds and singularities using singular fibers of fiber bundles given by polynomial mappings and stable maps, and further described their global information. In the study of singularities at infinity of polynomial mappings, we proved that, in two-variable real polynomial case, the atypical values of singularities at infinity can be determined by using toric compactifications and toric resolutions. For complex plane curve singularities, we gave a way to describe the embeddings of the Milnor fibers into the Milnor ball by polyhedrons called shadows. This result was obtained by using the real morsification and A'Campo's divides. Concerning the study of 3-manifolds appearing on the boundary of fibrations of 4-manifolds, we studied a certain correspondence between flow-spines and contact structures.
|
| Academic Significance and Societal Importance of the Research Achievements |
多様体間の写像は数学のみならず自然科学全般において重要な道具である。多様体間の滑らかな写像は、ほとんどの場合において特異点をもつ。滑らかな点における現象は把握し易いが、特異点における現象を理解するためには、その特異点の性質を深く知る必要がある。本研究では、多項式写像の無限遠の特異点や複素平面曲線特異点などの位相的性質について、これまでに知られている結果よりも更に深い理解を得ることに成功した。これらの結果は、今後の３次元多様体論と４次元多様体論を結び付ける研究や、多変数多項式写像の特異点および高次元多様体のファイバー束構造の研究のための基礎として、重要な役割を果たすものである。
|
Report
(4 results)
Research Products
(21 results)
