Objective The goal of MesuR is to deepen our knowledge of geometric and dynamical properties of a class of metric-measure spaces, called sub-Riemannian (sR) manifolds. These are generalizations of Riemannian manifolds, naturally arising in the frame of control theory and hypoelliptic operators. SR geometry is a theory in expansion, and it recently received a great impulse thanks to two ERC-StG on this topic: “GeCoMethods” (2010-2016, PI: U. Boscain), and “GeoMeG” (2017–now, PI: E. Le Donne).In this action we focus on sR manifolds endowed with intrinsic measures. These have been introduced in the frame of geometric control theory: as a key novelty, they are allowed here to have singularities, opposite to the smooth measures usually employed in the existing literature on geometry and analysis in sR manifolds. In this framework, we aim at proving:(1) sR isoperimetric inequalities for singular measures, and investigate relations with the standing Pansu’s conjecture about the shape of isoperimetric sets in the Heisenberg group;(2) Essential self-adjointness and stochastic completeness of the intrinsic sR Laplacian, amounting to prove the conjectured confinement of the heat and of quantum particles to the non-singular region;(3) Heat kernel estimates, i.e. qualitative informations on the solutions to the Heat equation for the intrinsic sR Laplacian.Our objectives will follow by proving suitable functional inequalities encoding geometric properties of the underlying space, that we call metric-measure inequalities. This will be done thanks to an original interaction between variational and control theoretic techniques, respectively typical of the backgrounds of the applicant and of the Supervisor. Through this innovative point of view, we will obtain new results in the context of sR geometry and provide new techniques to study geometry and dynamics on metric-measure spaces presenting singularities. Fields of science natural sciencesmathematicspure mathematicsgeometry Keywords sub-Riemannian geometry intrinsic measures functional and geometric inequalities isoperimetric inequalities Heat and Schrödinger equations Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Topic(s) MSCA-IF-2017 - Individual Fellowships Call for proposal H2020-MSCA-IF-2017 See other projects for this call Funding Scheme MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF) Coordinator SORBONNE UNIVERSITE Net EU contribution € 173 076,00 Address 21 rue de l'ecole de medecine 75006 Paris France See on map Region Ile-de-France Ile-de-France Paris Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00
