Speaker: Dmitry Faifman (U Toronto)

Title: The integral geometry of indefinite signature.

Abstract. Integral geometry goes back to Buffon's needle problem and the Cauchy-Crofton formula. Slightly less ancient is the theory of convex valuations, which goes back to Dehn's solution of Hilbert's third problem. In Euclidean space, the two central results of the theory are Hadwiger's characterization of the intrinsic volumes as the unique invariant continuous valuations, and the kinematic formulas (Blaschke, Santalo, Chern, Federer). Following the rapid progress in the theory due to Alesker's foundational work on McMullen's conjecture, the integral geometry of other spaces came within grasp, most notably Hermitian geometry (Alesker, Bernig, Fu, Solanes). In this talk, we will survey the relevant parts of the general theory of valuations, and then discuss the integral geometry of an indefinite quadratic form. In particular, we will present the analogue of Hadwiger's theorem, and discuss the existence of invariant Crofton formulas, which can be seen as a first step towards kinematic formulas. An important ingredient is the evaluation of a new Selberg-type integral. Some of the results are based on joint works with S. Alesker and A. Bernig.

Title: The integral geometry of indefinite signature.

Abstract. Integral geometry goes back to Buffon's needle problem and the Cauchy-Crofton formula. Slightly less ancient is the theory of convex valuations, which goes back to Dehn's solution of Hilbert's third problem. In Euclidean space, the two central results of the theory are Hadwiger's characterization of the intrinsic volumes as the unique invariant continuous valuations, and the kinematic formulas (Blaschke, Santalo, Chern, Federer). Following the rapid progress in the theory due to Alesker's foundational work on McMullen's conjecture, the integral geometry of other spaces came within grasp, most notably Hermitian geometry (Alesker, Bernig, Fu, Solanes). In this talk, we will survey the relevant parts of the general theory of valuations, and then discuss the integral geometry of an indefinite quadratic form. In particular, we will present the analogue of Hadwiger's theorem, and discuss the existence of invariant Crofton formulas, which can be seen as a first step towards kinematic formulas. An important ingredient is the evaluation of a new Selberg-type integral. Some of the results are based on joint works with S. Alesker and A. Bernig.

## Date:

Mon, 29/05/2017 - 11:00 to 13:00

## Location:

Rothberg B220 (CS bldg)