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Order Structures, Jordan Algebras and Geometry
Description and aim
The concept of a Jordan algebra has a long and rich history in mathematics. It was originally introduced by Pascual Jordan 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 theory, geometry, and mathematical analysis. The finite dimensional Euclidean Jordan algebras were characterized by Koecher and Vinberg in terms of symmetric cones. Their characterization provides a striking link with Riemannian geometry of real manifolds. For infinite dimensional real Jordan algebras no such characterization is known. Recent findings, however, indicate that in infinite dimensions there may exist alternative characterizations of real Jordan algebras in terms of the geometry of cones and their associated order structure.
The main objective of this workshop is to explore the possibilities of establishing such alternative characterizations of Jordan algebras, to discuss the challenges that come with it, and to gain a deeper understanding of the geometry that is encoded in a Jordan algebra.
Core topics of the workshop include:
1. Finsler geometries on cones in order unit spaces.
2. Geometric aspects of cones in JB-algebras.
3. Cones as Banach-Finsler manifolds.
4. Order-antimorphisms on cones and Jordan structures.
5. Cones as symmetric spaces.