Riemann-Hilbert problems and large determinants; the example of the Kontsevich integral

02/09/2016 - 15:30
Marco Bertola, Concordia University et CRM
CRM, UdeM, Pavillon André-Aisenstadt, 2920, ch. de la Tour, salle 4336

 Riemann-Hilbert problems are particular boundary value problems in the complex plane. When they depend on parameters one can associate, under certain conditions, a "tau function" whose zeroes indicate the non-solvability of the problem, and hence play the same role as a determinant does for linear systems.

There are cases in which this tau function is literally a determinant (finite or infinite dimensional); in these cases certain asymptotic questions can addressed using analysis of the Riemann-Hilbert problem. I will focus on one example that is related to the Kontsevich matrix integral; this integral gives (in a formal sense) the generating function for certain "intersection numbers" on the moduli space of Riemann surfaces in the limit where the size of the matrix being integrated tends to infinity. The approach via Riemann Hilbert method allows a non-formal treatment of this limit. Based on a joint project with Mattia Cafasso, Angers.

Last edited by on Thu, 02/04/2016 - 14:54