Université McMaster, 5 - 8 décembre 2014
The theory of Newton--Okounkov bodies relates algebra and geometry outside the framework of toric geometry. This relationship is useful in many directions. For algebraic geometry it provides elementary proofs of intersection-theoretic analogues of the geometric Alexandrov--Fenchel inequalities and of their local version for the intersection multiplicity of primary ideals in a local ring. In invariant theory it gives analogues of the Bernstein--Kushnirenko theorem for horospherical varieties and some other manifolds with an action of a reductive group. In abstract algebra it allows one to introduce a broad class of graded algebras whose Hilbert functions are not necessarily polynomial at large values of the argument but have polynomial asymptotics. In geometry it provides a transparent proof of the known Alexandrov--Fenchel inequality and of its new analog for coconvex bodies.