Title: Generalized Hydrogen Atoms and the Wallach Representations of the Conformal Groups

 

Abstract:

 

The hydrogen atom is the simplest atom in our universe. The generalized hydrogen atoms are hypothetical atoms for which the nucleus also carries magnetic charges. Though not yet observed in nature, these hypothetical atoms are mathematically as beautiful as the hydrogen atom, and have been recently constructed in all dimensions.

 

The classification of unitary representations of non-compact Lie groups is a fundamental unsolved problem in representation theory.  However, the unitary highest weight representations, with the Wallach representations being the very special ones,  do admit a nice classification.

 

A goal in this series of talks is to show that the (nontrivial) Wallach representations of types II and III for the conformal groups are precisely the ones that can be realized by the Hilbert spaces of bound states of the generalized hydrogen atoms. As a byproduct, there emerge both a clean geometric realization and a simple algebraic characterization for these Wallach representations.

 

Prerequisites:

 

Some basic core knowledges in modern geometry, representations, special functions, functional analysis, and quantum mechanics are very helpful.