Luca Rizzi's seminars at SISSA

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENT
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**Tuesday 27.10.2015, 14:00  Luca Rizzi, Ecole Polytechnique, Paris 
(SISSA, room 133)*

Title: A sub-Riemannian Santaló formula with applications to 
isoperimetric inequalities and Dirichlet spectral gap of hypoelliptic 
operators

Abstract: We prove a sub-Riemannian version of the classical Santaló 
formula: a result in integral geometry that describes the intrinsic 
Liouville measure on the unit cotangent bundle in terms of the geodesic 
flow.

As an application, we derive (p-)Hardy-type and isoperimetric-type 
inequalities for a compact domain with Lipschitz boundary and negligible 
characteristic set. Moreover, we prove a universal (i.e. curvature 
independent) lower bound for the first Dirichlet eigenvalue of the 
intrinsic sub-Laplacian, All our results are sharp for the 
sub-Riemannian structures on the hemispheres of the complex and 
quaternionic Hopf fibrations.

If time allows, we discuss an interesting bound on the first Dirichlet 
eigenvalue on fundamental domain of the Heisenberg nilmanifold (the 
quotient of the 3D Heisenberg group by a cocompact lattice).

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**Wednesday 28.10.2015, 16:00  Luca Rizzi, Ecole Polytechnique, Paris 
(SISSA, room 133)*

Title: Intrinsic random walks in Riemannian and sub-Riemannian geometry 
via volume sampling

Abstract: We relate some basic constructions of stochastic analysis to 
differential geometry, via random walk approximations. The motivation is 
largely to explore how one can pass from geodesics to diffusion (and 
hence their infinitesimal generators) on sub-Riemannian manifolds, which 
is interesting in light of the fact that geodesics are relatively well 
understood, while there is no completely canonical notion of 
sub-Laplacian on a general sub-Riemannian manifold. However, even in the 
Riemannian case, this random walk approach illuminates the geometric 
significance of Ito and Stratonovich stochastic differential equations.