Representation theory and pseudo-differential operators. New perspectives.

Talk by Duván Cardona (Ghent University), at the Antwerp Algebra Colloquium on November 26, 2021.

The theory of pseudo-differential operators was born in the 1960s as a versatile technique to analyze several problems in partial differential equations. One of its more remarkable applications is the role that it had in the proof of the Atiyah-Singer index theorem, a celebrated result of the geometric analysis that interpolate several results of Topology: the Riemann-Roch theorem, the Hirzebruch signaturetheorem, etc. In this lecture we will give an introduction to the theory of pseudo-differential operators and we will discuss its connections with the representation theory of compact Lie groups. The interplay of both theories provides a modern perspectiveon the field that was discovered some years ago by M. Ruzhansky and V. Turunen.