Finitely generated structures are important subjects of study in various mathematical disciplines. Examples include finitely generated groups, Lie algebras and C*-algebras, etc. It is thus a fundamental question whether there exists a universal mechanism in the study of these vastly different entities. In 2009, the notion of projective spectrum for several elements (A1,…, An) in a unital Banach algebra B was defined through the multiparameter pencil A(z) = z1A1 + … + znAn; z∈Cn. This conspicuously simple definition turned out to have a surprisingly rich content. This series of 5 talks aims to give an introduction to this theory. The topics in each talk are as follows.

 

1) Definitions, examples and some general facts.

2) Maurer-Cartan form and the topology of resolvent set.

3) Hermitian metrics on resolvent set and geometric properties

4) Tuple of compact operators and kernel bundle

5) Application to group representations and a link with complex dynamics

 

The talks are self-contained and friendly to graduate students.

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