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Jordan Algebras, Finsler Geometry and Dynamics

乔丹代数、芬斯勒几何和动力学

基本信息

批准号：
EP/R044228/1
负责人：
Bas Lemmens
金额：
$ 39.6万
依托单位：
University of Kent
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR044228%2F1
关键词：
Jordan Algebras Finsler Geometry Dynamics

项目摘要

The concept of a Jordan algebra has a long and rich history in mathematics. It was originally introduced by Pascual Jordan in the nineteen-thirties as a way of finding alternative settings for quantum mechanics, but it turned out to have numerous connections with distinct areas of mathematics including, Lie algebras, differential geometry, and mathematical analysis. The finite dimensional Jordan algebras were classified algebraically by Jordan, von Neumann and Wigner in their famous 1934 paper. At the heart of our project lies a beautiful, and far-reaching, characterisation of the finite dimensional Jordan algebras in terms of the geometry of cones discovered independently by Koecher and Vinberg. Their characterisation provides a striking link with the Riemannian geometry of real manifolds. For infinite dimensional real Jordan algebras no such characterisation is known. Recent findings in works by the PI, Co-PI and Walsh, however, indicate that in infinite dimensions there exist alternative characterisations of real Jordan algebras in terms of the Finsler geometry of cones and their associated order structure. The first main objective of this project is to establish such charactersations of real Jordan algebras in arbitrary dimensions. These novel characterisations will open up new pathways to applications in mathematical analysis, as did the finite dimensional one, and enormously advance our understanding of the deep seated interplay between geometry and Jordan algebras. Symmetric cones and their associated tube domains are important settings for analysis and dynamics, both in finite and infinite dimensional spaces. In recent decades, complex dynamics has been a rapidly developing field. A central theme is to understand the dynamics of holomorphic maps on complex domains. In that context there exists the famous Denjoy-Wolff theorem which completely describes the dynamics of fixed-point free holomorphic self-maps of the open unit disc in the complex plane. Recent years has seen a flurry of activity to establish analogous of the Denjoy-Wolff theorem in other settings including, complex domains in possibly infinite dimensional spaces and a variety of real Finsler metric spaces. Particularly interesting classes of real Finsler metric spaces are Hilbert's metric spaces, which are natural generalisations of Klein's model of real hyperbolic space, and Thompson's metric on cones. Our second main objective is to establish Denjoy-Wolff type theorems on symmetric cones, which can be infinite dimensional, and on the corresponding complex tube domains, by exploiting novel connections between the real and complex settings, the associated Jordan algebra structures, and the underlying Finsler geometry.The complementary research expertise of the PI (metric and Finsler geometry on cones, and applications in real dynamical systems) and the Co-PI (Jordan structures in geometry and analysis, and their applications in complex dynamical systems) will be key to the successful outcome of the project.

约当代数的概念在数学中有着悠久而丰富的历史。它最初是由帕斯夸尔·乔丹在20世纪30年代引入的，作为寻找量子力学替代设置的一种方式，但事实证明，它与不同的数学领域有许多联系，包括李代数，微分几何和数学分析。有限维的约旦代数分类代数约旦，冯诺依曼和维格纳在其著名的1934年文件。在我们的项目的核心在于一个美丽的，和深远的，表征有限维约旦代数的几何锥独立发现的Koecher和Vinberg。他们的特征提供了一个引人注目的联系与黎曼几何的真实的流形。对于无限维真实的Jordan代数，没有这样的特征是已知的。然而，PI，Co-PI和沃尔什的著作中最近的发现表明，在无限维中，根据锥的Finsler几何及其相关的序结构，存在真实的Jordan代数的其他特征。本项目的第一个主要目标是建立任意维数的真实的Jordan代数的特征。这些新的特征将开辟新的途径，在数学分析中的应用，因为有限维的一个，并极大地推进我们的理解之间的深层次的相互作用几何和约旦代数。对称锥及其相关的管域是有限维和无限维空间中分析和动力学的重要设置。近几十年来，复杂动力学是一个迅速发展的领域。一个中心主题是理解复域上全纯映射的动力学。在这种情况下，存在著名的Denjoy-Wolff定理，它完全描述了复平面上开单位圆盘的不动点自由全纯自映射的动力学。近年来，人们开始在其他环境中建立类似的Denjoy-Wolff定理，包括可能无限维空间中的复域和各种真实的Finsler度量空间。特别有趣的类真实的芬斯勒度量空间是希尔伯特的度量空间，这是自然的概括克莱因模型的真实的双曲空间，和汤普森的度量锥。我们的第二个主要目标是通过利用真实的和复杂的设置之间的新联系，相关的Jordan代数结构和潜在的Finsler几何，在对称锥上建立Denjoy-Wolff型定理，这可以是无限维的，并在相应的复杂管域上。（锥上的度量和芬斯勒几何，以及在真实的动力系统中的应用）和Co-PI（几何和分析中的约旦结构，以及它们在复杂动力系统中的应用）将是该项目成功的关键。