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.

- Instructor: Robert Young (ryoung@cims.nyu.edu)
- Office: WWH 601
- Office hours by appointment
- Lectures: R 11--12:50, WWH 201
- Readings:
- B. Bowditch, A course on geometric group theory
- M. Bridson, The geometry of the word problem
- S. Wenger and R. Young, Constructing Hölder maps to Carnot groups
- R. Young, Notes on asymptotic cones

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 |