MATH-GA.2400-002, Fall 2018: Topics in Geometry: Quantitative Geometry

Basic information

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.

Course outline

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