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Title: The Alexander phenomenon

Abstract: A famous result of H. Alexander asserts that any proper
holomorphic self-map of the unit (Euclidean) ball in higher dimensions is
an automorphism. Alexander's result has been extended to various classes
of domains including strictly pseudoconvex domains (by Pinchuk) and weakly
pseudoconvex domains with real-analytic boundary (by Bedford and Bell).
It is conjectured that any proper holomorphic self-map of a smoothly bounded
pseudoconvex domain in higher dimensions must be an automorphism.

In this talk, I shall first briefly survey some of the prominent
Alexander-type results. I shall then talk about an extension of
Alexander's Theorem to a certain class of balanced, finite type domains. I
shall also highlight how the use of dynamics in the proof offers some
insight on the aforementioned conjecture.

Jaikrishnan Janardhanan, IIT Madras