# Geometry of subRiemannian groups: regularity of finite-perimeter sets, geodesics, spheres, and isometries with applications and generalizations to biLipschitz homogenous spaces (research costs)

Main funder

Funder's project number: 319205

Funds granted by main funder (€)

- 140 000,00

Funding program

Project timetable

Project start date: 01/09/2018

Project end date: 31/08/2020

Summary

The PI plans to build a research group that studies geometric properties of subRiemannian spaces and their implications to metric spaces

and nilpotent groups.

In particular, the plan is to exploit the relation between the regularity of distinguished sets and maps in subRiemannian groups, volume

asymptotics in nilpotent groups, and embedding results.

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 this project will provide more links between such areas.

The general goal for understanding subRiemannian geometries is to develop an adapted Geometric Measure Theory.

In the last years the PI has developed a net of high level international collaborations and, together with his collaborators, has obtained

several results via a mixed 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 Hilbert fifth problem).

This has allowed the PI to solve a number of open problems in the field, e.g., the analogue of Myers & Steenrod Theorem on the

differentiability of isometries

and the analogue of Nash Theorem on path isometric embeddings.

One of the next aims is to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets.

Many more related questions are still open and the goal of this project is to tackle them.

The PI received his first degree at Scuola Normale Superiore di Pisa (with Marco Abate as advisor) and his PhD from Yale University

(with Bruce Kleiner as advisor).

and nilpotent groups.

In particular, the plan is to exploit the relation between the regularity of distinguished sets and maps in subRiemannian groups, volume

asymptotics in nilpotent groups, and embedding results.

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 this project will provide more links between such areas.

The general goal for understanding subRiemannian geometries is to develop an adapted Geometric Measure Theory.

In the last years the PI has developed a net of high level international collaborations and, together with his collaborators, has obtained

several results via a mixed 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 Hilbert fifth problem).

This has allowed the PI to solve a number of open problems in the field, e.g., the analogue of Myers & Steenrod Theorem on the

differentiability of isometries

and the analogue of Nash Theorem on path isometric embeddings.

One of the next aims is to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets.

Many more related questions are still open and the goal of this project is to tackle them.

The PI received his first degree at Scuola Normale Superiore di Pisa (with Marco Abate as advisor) and his PhD from Yale University

(with Bruce Kleiner as advisor).

Principal Investigator

Other persons related to this project (JYU)

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