MesuR · Metric-measure inequalities in sub-Riemannian manifolds
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2019-09-01 → 2021-08-31
- EU contribution
- €173,076
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Metric-measure inequalities in sub-Riemannian manifolds
MesuR has been devoted to deepen our knowledge of geometric and dynamical properties of a class of metric-measure spaces, called sub-Riemannian (sR) manifolds. From a scientific point of view, MesuR allowed us to obtain new isoperimetric inequalities in a class of singular structures and to identify a purely sR behavior in a class of sR inequalities, called Hardy inequalities, and in the heat/quantum confinement of sR manifolds. We also treated heat kernel estimates in sR manifolds when a magnetic field influences the dynamic: this study was not previewed in the DoA, but it naturally arises as an open problem connected to MesuR's Objectives. Finally, I had the chance to open my research field to a new theme and collaborate on the applications of sR model of the visual cortex to human contrast perception. From a wider perspective MesuR offered me the opportunity to to disseminate the results of the project to a wide (scientific and non-scientific) public, to deepen and enlarge my competences, produce high-level research and compete for positions in academia. This is supported by the fact that, during the first year of MesuR, I won a position as Researcher (RTD-A) at the University of Padova, that started in September 2020.
Data: CORDIS, © European Union
Project 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.
Original text from CORDIS.
Participants
- SORBONNE UNIVERSITE · ParisCoordinatorFrance
Links
Data: CORDIS, © European Union