Study of Poisson geometry and its application

泊松几何是辛几何的扩展，是最近才发展起来的。然而，这一研究领域在日本并不常见，而这一研究领域在欧洲和美国都很发达。在我们的研究中，我们的使命是在日本发展这一研究领域，并与欧洲和美国在这一领域工作的研究人员合作。首先研究了变形量化的收敛性问题。对于形式变形，孔采维奇先生给出了一个很好的结果。研究变形量化的收敛性是一个重要的问题。通过我们的研究，我们发现了变形量化收敛的完全不同的特征，并提出了一种新的几何对象。我们将把对形变量化收敛性的研究作为未来的任务。在此资助下，我们于2001年和2002年举办了两次以非交换几何和d膜为主要主题的大型国际研讨会。我们会有很多来自国外和国内研究机构的参观者。我们已经出版了研究进展的论文集。Maeda被邀请参加2002年在里斯本举行的泊松国际会议，并就这个问题做了一个演讲，这对与会者来说非常有趣。我们还访问了日本国内和国外的各种会议，进行了强有力的活动。因此，我们可以有一个非交换几何的国际研究小组，它能够发展研究。

Poisson Geometry, which is the extended notion of symplectic geometry, is now developing quite recently. However, this research fields is not common in Japan, while this research field is much developing in Europe and the United State. In our Study, we have a mission to develop this research fields in Japan, and to coorporate with the researchers who are working on this fields in Europe and the Unites States. The first our development is to study the convergence problems on deformation quantizations. For the case of formal deformations, M. Kontsevich has given a pretty result. It is important problem to study the convergence for the deformation quantizations. Through our research, we found the totally different feature for the convergence of the deformation quantizations, and propose a new geometric objects on gerbes. We will extend our research on the convergence of the deformation quantization as a future task. By this grant, we have two major international symposia in 2001 and 2002, which has noncommutative geometry and D-brane as main Topics. We could have many visitors from abroad and also from domestic research institutes. We have published a proceedings for our research developments. Maeda has been invited to the Poisson 2002 international conference at Lisbon and gave a talk on this problem, which was very interesting for the participants. We have also visited various meetings in Japan and outside of Japan to make strong activities. As a results, we could have a international research groups on noncommutative geometry, which is able to develop the research.