The goal of these talks is to discuss the history, the circumstances, and (the main aspects of) the proof of a recent result of ours, which says: a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism.

Results of this character are motivated by a remarkable theorem of H. Alexander about proper holomorphic self-maps of the Euclidean ball. The first part of the talk will focus on the meanings of the terms in the above theorem. We will give an elementary proof that the automorphism group of a bounded symmetric domain acts transitively on it. But the existence of an involutive symmetry at each point is restrictive in many other ways. For instance, it yields a complete classification of the bounded symmetric domains. This follows from the work of Elie Cartan.

Earlier results in the literature that are special cases of our result focused on different classes of domains in the Cartan classification, and have mutually irreconcilable proofs. One aspect of our work is a unified proof: a set of techniques that works across the Cartan classification. One of those techniques is the use of an algebraic gadget: the machinery of Jordan triple systems.

We will present definitions and examples related to this machinery in the first part of the talk. In the second part of the talk, we shall state a result on finite-dimensional Jordan triple systems that is essential to our work. After this, we shall discuss our proof in greater depth.

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Last updated: 15 Jul 2024