Poisson Geometry, Contact Geometry and Quantization Problems

点击翻译按钮获取中文摘要

This research project aims several problems in Poisson geometry and contact geometry, and quantization problems. Specially, our project focus on the treatments for the quantization problems geometrical point of view. The noncommutative differential geometry is one of our target of our research project. In this project, we had several results on convergent deformation quantization problem. Grebes appears naturally from the construction of the star exponential functions of quadratic forms. Namely, we consider the set of quadratic forms in the complex Weil algebra which forms a Lie algebra isomorphic to sp(n, C). When we consider the esponential functions for these objects, we might expect the complex version of the metaplectic Lie group. Since this is simply connected, we could not handle it. We invested this object by using the explicit computations and have it is in the category of grebes with multiplications. However, it can be described more geometry in terms of the connections. The second problem is to study the invariant deformation quantization problems and construct a convergent star product for ax+b group case. We obtain the universal star product formula. The third result is to study the closed star product. We describe a general settings for obtain how to get the Hochschild cocycle via Stokes formula. This results is still working on and we expect it should be related to the deformation quantization problems for infinite dimensional case. To obtain these results, we have lot of workshops by inviting overseas and domestic researchers together with the research partners. As conclusions, we have fruitful research results which have high evaluation internationally, and also establish the international research network for this area by this grant. This project is still working and will continue for the next project.