Topics in geometry: Nilpotent groups and subriemannian geometry
Overview
This course is an introduction to subriemannian geometry and the geometry of nilpotent groups. Nilpotent groups are the simplest noncommutative groups. Their simplicity means that they appear in many areas of mathematics, especially geometry and geometric group theory. Their noncommutativity leads to distinctive and unusual geometry that make them a productive source of examples and a useful tool in a mathematician's toolbox.
In this course, we will study the geometry and analysis of nilpotent groups, possibly including topics such as:
- subriemannian metrics and manifolds
- lattices, large-scale geometry and asymptotic cones
- embeddings and metric geometry
- geodesics, surfaces, and geometric measure theory
Basics
- Instructor: Robert Young (ryoung@cims.nyu.edu)
- Office: WWH 601
- Office hours: by appointment
- Lectures: WWH 517, Tuesdays, 9:00-10:50
Outline
Lecture 1: Geometric group theory and large-scale geometry
Further reading:
- Bowditch, A course on geometric group theory
Lecture 2: Nilpotent groups: algebra and geometry
Further reading:
- Le Donne, A Primer on Carnot Groups
- Lukyanenko, Fillings in Nilpotent Groups
Lecture 3: Scaling limits and subriemannian geometry
Further reading:
Lecture 4: Carnot groups and asymptotic cones
Further reading:
- Notes from lecture 4
- Agrachev, Barilari, Boscain, A comprehensive introduction to sub-Riemannian geometry
- Fréderic Jean, Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning
Lecture 5: Scaling limits and maps between groups and spaces
Further reading:
Lecture 6: Pansu's theorem and quasi-isometries of nilpotent groups
Further reading:
- Enrico Le Donne, Lecture notes on sub-Riemannian geometry from the Lie group viewpoint (Section 9.1: Rademacher Theorem)
Lecture 7: Amenability and Shalom's theorem
Further reading: