Multiplication kernels for differential equations (Ilia Gaiur - IHES)

Multiplication kernels are integral operators that produce tensor powers of
differential equations. Restricted to the diagonal, they become similar to
lifts in the Langlands program. One needs to develop tools for their explicit
construction. I will discuss two approaches to this problem: a formal-analytic
approach of M. Kontsevich and A. Odesskii, and a Betti approach with rigid
local systems introduced by V. Golyshev, A. Mellit, V. Rubtsov, and D. van
Straten. I will illustrate both through explicit examples. To demonstrate the
first approach, I will present joint results with V. Rubtsov and D. van
Straten on the higher Bessel functions and explain the geometric origin of
their kernels. For the second approach, I will discuss ongoing work with V.
Golyshev on multiplication kernels for Heun local systems. I will also show
how higher-order analogues of the Heun equations, introduced by P. Boalch, N.
Katz, and C. Simpson, appear naturally in the framework of multiplication
kernels.