Jordan Algebras, Finsler Geometry and Dynamics

乔丹代数的概念在数学上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年代提出的，作为寻找量子力学的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。Jordan，von Neumann和Wigner在他们著名的1934年的论文中对有限维Jordan代数进行了代数分类。我们项目的核心是用Koecher和Vinberg独立发现的锥几何来刻画有限维Jordan代数的美丽和深远的特征。它们的刻画提供了与实流形的黎曼几何的惊人联系。对于无限维实Jordan代数，这种刻画是未知的。然而，PI、Co-Pi和Walsh最近的工作结果表明，在无限维实Jordan代数存在关于锥的Finsler几何及其相关的序结构的另一种刻画。这个项目的第一个主要目标是建立任意维实Jordan代数的这种刻画。这些新颖的刻画将为数学分析的应用开辟新的途径，就像有限维的刻画一样，并极大地促进我们对几何和Jordan代数之间深层相互作用的理解。无论是在有限维空间还是无限维空间中，对称圆锥及其关联的管域都是分析和动力学的重要设置。近几十年来，复杂动力学一直是一个发展迅速的领域。一个中心主题是理解复数域上全纯映射的动力学。在这一背景下，存在着著名的Denjoy-Wolff定理，它完整地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，在可能无限维空间中的复域和各种实Finsler度量空间中，已经有一系列的活动在其他环境中建立类似的Denjoy-Wolff定理。特别有趣的实芬斯勒度量空间类是Hilbert度量空间，它是Klein的实双曲空间模型的自然推广，以及锥上的Thompson度量。我们的第二个主要目标是通过利用真实和复杂环境、相关的Jordan代数结构和基本Finsler几何之间的新联系，在对称锥(可以是无限维的)和相应的复管域上建立Denjoy-Wolff类型的定理。PI(锥上的度量和Finsler几何，以及在真实动力系统中的应用)和Co-PI(几何和分析中的Jordan结构，以及它们在复杂动力系统中的应用)的互补研究专长将是该项目成功的关键。

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.