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Topological toric theory and combinatorics

拓扑环面理论和组合数学

基本信息

批准号：
17540092
负责人：
MASUDA Mikiya
金额：
$ 2.05万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540092/
关键词：
Toric manifold fan convex polytope equivariant cohomology Bott tower GKM graph topology combinatorics トーリック多様体論 face ring グラフ理論

项目摘要

I have beein developing the theory of toric varieties in algebraic geometry from a topological point of view for these years. At the same time, a research from the same viewpoint has been independently developed in England and Russia, and now this emerging subject is called Toric Topology. In this research, I mainly studied the relation between topology and Combinatorics jointly with Taras Panov, Zhi Lu etc, and have made effort for developing this new subject. In particular, I organized an international conference on Toric Topology for a week from the end of May 2006 at Osaka City University. The organizers are Megumi Harad (Toronto), Yael Karshon(Toronto) and Taras Panov(Moscow) except me. This conference was supported mainly by 21COE program "Constitution of wide-angle mathematical basis focused on knots", and other resources are research grants of Shigenori-Matsumoto, Taras Panov, Yoshitake Hashimoto and this my research grant. Fortunately, we had 140 participants and half of them … More are foreigners. We realized that many people are interested in this emerging area.As for my own research, I got the following outcome.(1) Gullemin-Zara have studied the relation between the topology of manifolds with torus actions and graph theory. This is a very interesting study because it introduces ideas of topology to the study of graphs. Motivated by their research, we have introduced the notion of torus graph and showed that its equivariant cohomology agrees with the face ring of a simplicial poset which was already introduced in commutative algebra. The paper of this research has appeared in Adv. Math. 212 (2007), 458-483.(2) It is known that equivariant cohomology contains a rich geometrical of a toric manifold and fits very well to the study of toric manifolds. In this research I have proved that equivariant cohomology completely distinguishes toric manifolds.(3) A Bott tower is an iterated CP^1 bundle and provides a good family of toric manifolds. I worked with Taras Panov on the topological classification of those Bott towers (the paper is submitted). Also, I worked with Dong Youp Suh on the topological classification of higher Bott towers which are higher dimensional generalization of Bott towers. Less

这些年来，我一直在从拓扑学的角度发展代数几何中的环变理论。与此同时，英国和俄罗斯也独立开展了一项基于相同观点的研究，现在这一新兴学科被称为托利拓扑。在本研究中，我主要与Taras Panov， Zhi Lu等人共同研究了拓扑与组合学的关系，为这一新兴学科的发展做出了努力。特别是，从2006年5月底开始，我在大阪城市大学组织了为期一周的环面拓扑国际会议。组织者是Megumi Harad（多伦多），Yael Karshon（多伦多）和Taras Panov（莫斯科），除了我。本次会议主要由21COE项目“构建聚焦于节的广角数学基础”支持，其他资源为Shigenori-Matsumoto、Taras Panov、Yoshitake Hashimoto的研究经费以及本人的研究经费。幸运的是，我们有140名参与者，其中一半是外国人。我们意识到很多人对这个新兴领域感兴趣。对于我自己的研究，我得到了如下的结果。(1) Gullemin-Zara研究了具有环面作用的流形拓扑与图论之间的关系。这是一个非常有趣的研究，因为它将拓扑学的思想引入了图的研究。在他们研究的启发下，我们引入了环面图的概念，并证明了环面图的等变上同调与交换代数中已经引入的简单偏序集的面环一致。论文发表在《数学学报》2007年第212期，第458-483页。(2)已知等变上同调包含了丰富的环流形几何，非常适合于环流形的研究。在本研究中，我证明了等变上同调完全区分环流形。(3)伯特塔是一个迭代的CP^1束，它提供了一个很好的环形流形族。我和塔拉斯·帕诺夫（Taras Panov）一起研究了那些博特塔的拓扑分类（论文已提交）。另外，我和Dong Youp Suh一起研究了高博特塔的拓扑分类这是博特塔的高维泛化。少

项目成果

期刊论文数量（14）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Equivariant cohomology determines (quasi) toric manifolds

等变上同调确定（拟）复曲面流形

DOI：
发表时间：
2006
期刊：
数理解析研究所講究録 1517
影响因子：
0
作者：
T. Nagano;T. Aikou;K. Miyajima;T. Aikou;T. Aikou;K. Miyajima;愛甲正(分担);愛甲正;Mikiya Masda;Mikiya Masuda;Mikiya Masuda;Mikiya Masuda;Mikiya Masuda;Mikiya Masuda
通讯作者：
Mikiya Masuda

Torus graphs and simplicial posets