This course will cover topics in the quantitative geometry of manifolds, groups, and spaces. Quantitative geometry uses tools from geometry and analysis to study the asymptotics of a space: how curves and surfaces in the space behave at different scales, for instance, or how geometric invariants such as systoles affect the shape of a space. In this course, we will develop tools to measure how the geometry of a space changes as its scale or complexity increases and use these tools to describe spaces arising from topology and geometric group theory.
Topics to be covered may include: filling inequalities; systolic geometry; asymptotic cones; embedding problems; and uniform rectifiability and its applications to geometry.
| 9/6 | Overview, Cayley graphs and quasi-isometries | Lecture notes |
| 9/13 | Dehn functions and filling functions | |
| 9/20 | Equivalence of Dehn functions | |
| 9/27 | Equivalence of Dehn functions/Federer-Fleming | Lecture notes |
| 10/4 | Higher-order filling functions/Heisenberg group | |
| 10/11 | Hölder maps to the Heisenberg group | Figures Lecture notes |
| 10/18 | Filling inequalities for the Heisenberg group |
Lecture notes |
| 10/25 | Gromov-Hausdorff convergence and ultralimits |
Lecture notes |
| 11/1 | Asymptotic cones | |
| 11/8 | Asymptotic cones and filling inequalities |
Lecture notes Notes on asymptotic cones (2008) |
| 11/15 | Filling inequalities in nilpotent and Carnot groups |
Lecture notes |
| 11/29 | Systolic geometry in two dimensions |
Lecture notes |