Exploitation of new relations between differential geometry and quantum cohomology in the context of integrable systems

在可积系统的背景下利用微分几何和量子上同调之间的新关系

• 批准号: 21244004
• 负责人: GUEST Martin
• 金额: $ 16.72万
• 依托单位: Waseda University (2012)Tokyo Metropolitan University (2009-2011)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21244004/
• 关键词: 幾何学; 可積分系; 量子コホモロジー

We have made progress with some key examples, which demonstrate interesting and nontrivial phenomena. In "Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa" (M. Guest and C.-S. Lin, J. reine angew. Math., 2012, in press) the existence of a family of smooth solutions of the tt*-Toda equation was established. This was a technical breakthrough: p.d.e. methods are well suited to the noncompact case, where standard loop group methods fail. In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (M. Guest, A. Its, and C.-S. Lin, arXiv:1209.2045), a second technical breakthrough was made, by relating the global smoothness of the solutions to the monodromy data (Stokes data) of an associated linear equation. This Stokes data was computed explicitly for all globally smooth solutions of the tt*-Toda equation. We expect that these techniques will be applicable to other problems in differential geometry.

我们在一些关键示例上取得了进展，这些示例展示了有趣且重要的现象。在“Cecotti 和 Vafa 的 tt* 方程的非线性 PDE 方面”（M. Guest 和 C.-S. Lin，J. reine angew. Math.，2012 年，正在出版）中，存在一系列光滑解建立tt*-Toda方程。这是一项技术突破：p.d.e.方法非常适合标准循环组方法失败的非紧凑情况。在“Cecotti 和 Vafa I. Stokes 数据的 tt* 方程的等单性方面”（M. Guest、A.its 和 C.-S. Lin，arXiv：1209.2045）中，取得了第二个技术突破，通过将相关线性方程的单性数据（斯托克斯数据）解的全局平滑度。该斯托克斯数据是针对 tt*-Toda 方程的所有全局平滑解显式计算的。我们期望这些技术将适用于微分几何中的其他问题。

项目成果

Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces

• DOI: 10.2748/tmj/1435237041
• 发表时间: 2012-07
• 期刊: arXiv: Differential Geometry
• 作者: Hui Ma;Y. Ohnita

Geometry of Lagrangian submanifolds and isoparametric hypersurfaces

• 发表时间: 2010
• 作者: Y. Ohnita

Infrasolv-towers and smooth rigidity of geometric aspherical manifolds

几何非球面流形的次解塔和光滑刚度

• 发表时间: 2012
• 作者: S. Katayama;T. Ozawa; H. Sunagawa;佐藤文広;M.Ozawa;松本圭・塩谷亨・伊丸岡俊秀・近江政雄;長尾由希子;神島芳宣
• 通讯作者: 神島芳宣

Hyperplane arrangements, local system homology and iterated integrals

超平面排列、局部系统同源性和迭代积分

• 发表时间: 2012
• 期刊: Adv. Stud. Pure Math
• 作者: 久永 聡子;積山 薫;伊賀崎 伴彦;村山 伸樹;藤田昌久・吉川洋[編著];T. Kohno
• 通讯作者: T. Kohno

Chaotic inflation in F (R) supergravity

F(R)超重力中的混沌暴胀

• 发表时间: 2010
• 期刊: Physics Letters
• 作者: Matsui M;Takeuchi A;Katagiri M;Matsuda Y;Suzuki M;二木昭人;澁谷覚(単著);山口真美;Arturo Kohatsu-Higa;K. Ono;小澤正直;伊吹山知義;S.V.Ketov
• 通讯作者: S.V.Ketov