Yuri Berest - Lacunae for hyperbolic differential operators with variable coefficients

Lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, M. Atiyah, R. Bott and L. Gårding (1970-73) created a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. The purpose of the present talk is to report on new developments in this direction. We start with a review of classical results on lacunas (related mostly to the famous Hadamard's conjecture for second order operators). We present new examples and give a solution of (a restricted version of) Hadamard's problem for wave-type operators on Minkowski spaces. In the second part of the talk we explain how these results can be extended to the case of higher order hyperbolic operators. Our goal is to give a generalization of the Petrovsky-Atiyah-Bott-Gårding theory to certain classes of partial differential operators with singular coefficients. The final part of our talk will be devoted to the discussion of various connections with algebraic geometry (structure of rings of differential operators on singular varieties), representation theory (finite reflection groups) and integrable systems.