# Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory (SuRiG-MeLiT)

Main funder

Funds granted by main funder (€)

571 374,00

Funding program

Academy Project, AoF (Academy of Finland)

Project timetable

Project start date: 01/09/2019

Project end date: 31/08/2023

Summary

What are the best trajectories to park a truck with several trailers? How fast can a lattice grow? These are some of the questions studied in this project because both the infinitesimal structure of controlling a truck and the asymptotic geometry of a (nilpotent) lattice are examples of metric groups: Lie groups with homogeneous distances.

The PI plans to study geometric properties of metric groups and implications to control systems and nilpotent groups. In particular, the plan is to exploit the relation between the regularity of distinguished curves, sets and maps in subRiemannian groups, volume asymptotics in nilpotent groups and embedding results. The general goal is to develop an adapted geometric measure theory.

SubRiemannian spaces and in particular Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, subelliptic PDE's and geometric group theory. The results in the project will provide more links.

The PI developed a net of high level international collaborations and obtained several results via a combination of analysis on metric spaces (differentiation of Lipschitz maps, tangents of measures and Gromov-Hausdorff limits) and the theory of locally compact groups (Lie group techniques and the solutions of the Hilbert 5th problem). The PI solved a number of open problems in the field as the analogue of Myers-Steenrod theorem on the smoothness of isometries, of Nash isometric embedding and the non-minimality of curves with corners. Some of the next aims are to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problem.

The PI plans to study geometric properties of metric groups and implications to control systems and nilpotent groups. In particular, the plan is to exploit the relation between the regularity of distinguished curves, sets and maps in subRiemannian groups, volume asymptotics in nilpotent groups and embedding results. The general goal is to develop an adapted geometric measure theory.

SubRiemannian spaces and in particular Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, subelliptic PDE's and geometric group theory. The results in the project will provide more links.

The PI developed a net of high level international collaborations and obtained several results via a combination of analysis on metric spaces (differentiation of Lipschitz maps, tangents of measures and Gromov-Hausdorff limits) and the theory of locally compact groups (Lie group techniques and the solutions of the Hilbert 5th problem). The PI solved a number of open problems in the field as the analogue of Myers-Steenrod theorem on the smoothness of isometries, of Nash isometric embedding and the non-minimality of curves with corners. Some of the next aims are to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problem.