Seamus Albion Ferlinc: The Selberg integral

The Selberg integral

Seamus Albion Ferlinc, IMFM, Ljubljana

Abstract: The Selberg integral is a beautiful $n$-dimensional generalisation of the Euler beta function. Originally discovered by Atle Selberg in 1941, it has, after an initial period of obscurity, found sustained applications in both mathematics and physics. I will survey the history of the Selberg integral from its first appearances in the 1940s, its rediscovery in the 1970s by Dyson, Mehta and Bombieri, to its many modern incarnations touching on topics such as random matrix theory, special functions, analytic number theory, enumerative combinatorics and conformal field theory. This will culminate in some recent joint work with Eric Rains (Caltech) and Ole Warnaar (University of Queensland) in which we derive an elliptic analogue of the Selberg integral of type $\mathrm{A}_n$.