Study of Poisson geometry and its application

泊松几何是辛几何的推广，是近年来发展起来的一个新概念。然而，这一研究领域在日本并不普遍，而这一研究领域在欧洲和美国正在大力发展。在我们的研究中，我们有一个使命，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进行交流。我们的第一个发展是研究形变量子化的收敛问题。对于形式变形的情形，M.孔采维奇给出了一个很好的结果。研究形变量子化的收敛性是形变量子化的一个重要问题。通过我们的研究，我们发现了变形量化的收敛性的完全不同的特点，并提出了一种新的几何对象gerbes。我们将把形变量子化的收敛性作为一个未来的任务来扩展我们的研究。在此资助下，我们于2001年和2002年举办了两次以非对易几何和D-膜为主要专题的国际学术讨论会。我们会有很多来自国外和国内研究机构的参观者。我们已经出版了一份研究进展的论文集。前田已被邀请到泊松2002年国际会议在里斯本，并就这个问题，这是非常有趣的与会者发表了讲话。我们还参加了日本国内外的各种会议，开展了强有力的活动。因此，我们可以有一个国际性的非对易几何研究小组，这是能够发展的研究。

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.