Integral Geometry of Algebraic Varieties

Integral geometry is the subject of averaging geometric properties of a submanifold of a homogeneous space. This is a classical topic, with its origin in the so-called Buffon's needle problem: drop a random needle on a floor made of planks and calculate the probability the needle lies across a crack. This is solved formulating accurately what “random” means, so that expectations become integrals with respect to a certain measure (on the Grassmannian of lines in this case). Generalizations of this problem lead to beautiful progress in the theory of valuations on convex bodies and stochastic processes. In this workshop we will focus on more recent developments, especially on integral geometry of (real) algebraic varieties. This allows to establish connections with intersection theory and convex geometry. At the same time, restricting to the algebraic framework, opens the possibility of effective studies of average properties, with applications to sampling methods.