Jordan Algebras, Finsler Geometry and Dynamics

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Pascual Jordan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，但结果证明它与数学的不同领域有许多联系，包括李代数，微分几何和数学分析。有限维Jordan代数是由Jordan、von Neumann和Wigner在他们著名的1934年论文中进行代数分类的。我们项目的核心在于用Koecher和Vinberg独立发现的锥几何来描述有限维Jordan代数，这是一个美丽而深远的特征。它们的特征与实际流形的黎曼几何有着惊人的联系。对于无限维实约当代数，没有这样的特征是已知的。然而，PI， Co-PI和Walsh最近的研究表明，在无限维中，根据锥体的Finsler几何及其相关的有序结构，存在真实Jordan代数的替代特征。这个项目的第一个主要目标是建立任意维的实约旦代数的这种特征。这些新颖的特征将为数学分析的应用开辟新的途径，就像有限维的特征一样，并极大地促进了我们对几何和约当代数之间根深蒂固的相互作用的理解。对称锥及其相关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杂域上全纯映射的动力学。在这种情况下，存在着著名的Denjoy-Wolff定理，它完整地描述了复平面上开单位盘的不动点自由全纯自映射的动力学。近年来出现了一系列的活动来建立Denjoy-Wolff定理在其他环境中的类似，包括可能无限维空间中的复杂域和各种真实的Finsler度量空间。真正的芬斯勒度量空间中特别有趣的一类是希尔伯特度量空间，它是克莱因的双曲空间模型的自然推广，还有汤普森的锥度规。我们的第二个主要目标是建立Denjoy-Wolff型定理在对称锥上，它可以是无限维的，并在相应的复杂管域上，通过利用真实和复杂设置之间的新联系，相关的约旦代数结构，以及潜在的Finsler几何。PI（锥体上的度量和芬斯勒几何，以及在实际动力系统中的应用）和Co-PI（几何和分析中的乔丹结构，以及它们在复杂动力系统中的应用）的互补研究专长将是项目成功结果的关键。

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.