Topological toric theory and combinatorics

这些年来，我一直致力于从拓扑学的角度发展代数几何中的环变差理论。与此同时，英国和俄罗斯也从相同的角度独立开展了一项研究，现在这门新兴学科被称为环拓扑学。在这项研究中，我主要与塔拉斯·帕诺夫、陆志等人共同研究拓扑学与组合学的关系，为这门新学科的发展做出了努力。特别是，从2006年5月底开始，我在大阪市立大学组织了为期一周的环状拓扑学国际会议。除了我之外，组织者是Megumi Harad(多伦多)、Yael Karshon(多伦多)和Taras Panov(莫斯科)。本次会议主要由21COE计划《关注结点的广角数学基础的构成》支持，其他资源有松本茂、帕诺夫、桥本吉和这是我的研究基金。幸运的是，我们有140名参与者，其中一半是…更多的是外国人。我们意识到很多人对这个新兴领域感兴趣。至于我自己的研究，我得到了以下结果：(1)Gullmin-Zara研究了具有环面作用的流形的拓扑与图论之间的关系。这是一项非常有趣的研究，因为它将拓扑学的思想引入到图的研究中。在他们的研究的启发下，我们引入了环面图的概念，并证明了它的等变上同调与交换代数中已经引入的单纯偏序集的面环一致。这项研究的论文已经发表在《高级数学》杂志上。212(2007)，458-483。(2)等变上同调包含了环面流形的丰富几何，非常适合于环面流形的研究。在这项研究中，我证明了等变上同调完全区分环面流形。(3)Bott塔是一个迭代CP^1丛，它提供了一族很好的环面流形。我与塔拉斯·帕诺夫一起研究了这些Bott塔的拓扑分类(论文已提交)。此外，我还与董友普一起研究了高位塔的拓扑分类，这是对高位塔的高维推广。较少

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