Jordan Algebras, Finsler Geometry and Dynamics

点击翻译按钮获取中文摘要

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.