DIFFERENTIAL GEOMETRY OF HESSIAN STRUCTURES AND ITS APPLICATIONS TO INFORMATION GEOMETRY

Hessian结构的微分几何及其在信息几何中的应用

• 批准号: 13640078
• 负责人: SHIMA Hirohiko
• 金额: $ 2.56万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640078/
• 关键词: Hessian metric; Hessian structure; Codazzi structure; affine differential geometry; Kahler geometry; information geometry; family of probability distributions; アファイン微分幾何学

A pair (D,g) of a flat connection D and a Riemannian metric g is said to be a Hessian structure if g is locally expressed by Hessian with respect to D. The geometry of Hessian structures is deeply related to information geometry as well as Kahler geometry. In this project we intended to the fundamental study of Hessian structures and its application to information geometry. We published a book in Japanese titled "Hessian Geometry" including the results obtained in this project. We are now translating the book in English.

如果Hessian局部表达了Hessian相对于D。卡勒几何学。在这个项目中，我们打算对Hessian结构的基本研究及其在信息几何形状上的应用。我们在日语中出版了一本书，标题为“黑森几何”，包括该项目获得的结果。我们现在正在用英语翻译这本书。

项目成果

Shigeru Furuichi, Kenjiro Yanagi, Ken Kuriyama: "A sufficient condition on concavity of the auxiliary function appearing in quantum reliability function"INFORMATION. 6-1. 71-76 (2003)

Shigeru Furuichi、Kenjiro Yanagi、Ken Kuriyama：“量子可靠性函数中出现的辅助函数凹性的充分条件”信息。

Hiroko Shima: "Hessian Geometry"Shokabo. 268 (2001)

岛宏子：《黑森几何》Shokabo。

Yoshifumi Ando: "Fold-maps and the space of base point preserving of spheres"J. Math. Kyoto Univ.. 41. 693-737 (2002)

安藤喜文：“折叠地图和球体基点保留空间”J.

Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka J. Math.. 38. 239-249 (2001)

Katsuhiro Komiya：“等变 K 理论欧拉类和表示球面图”Osaka J. Math.. 38. 239-249 (2001)

Shigeru Furuicni, Kenjiro Yanagi, Ken Kuriyama: "A remark on concavity of the function appearing in quantum reliability function"Quantum Information Science. 67-68 (2002)

Shigeru Furuicni、Kenjiro Yanagi、Ken Kuriyama：“关于量子可靠性函数中出现的函数凹性的评论”量子信息科学。