Math 595 Section MTM (Mapping Theory in Metric
Spaces) Spring 2011
The goal of the course is to introduce modern techniques and results
in the study of mappings between metric spaces. A major theme will be
the influence of the geometry of both source and target on the
prevalence and structure of various classes of mappings. In contrast
with the real- or vector-valued case, the theory of metric
space-valued mappings is fundamentally nonlinear
and geometric. We will use tools from many different subjects,
including functional analysis, Fourier analysis, PDE, and differential
A significant part of the course will be devoted to extension and
embedding theorems for finite metric spaces. This subject is
characterized by an intricate interplay betweeen geometry,
combinatorics and functional analysis. In the latter part of the
course we will consider metric spaces with other specific structure
arising from synthetic differential geometry or sub-Riemannian
geometry. One of the main goals of the course is to discuss a
significant recent theorem of Cheeger and Kleiner on the bi-Lipschitz
nonembeddability of the sub-Riemannian Heisenberg group into certain
Banach spaces, and the application of that theorem to questions from
the geometry of finite metric spaces motivated by algorithmic computer
- Instructor: Prof. Jeremy Tyson
- Office Location: 329 Altgeld Hall
- Office Phone: 244-4132
- Office Hours: by appointment
- Lecture Times: MonWedFri 10:00-10:50
- Lecture Location: 143 Altgeld Hall
- Course web site:
- Prerequisites: Real analysis (Math 540) is absolutely
required; functional analysis (Math 541) is encouraged (students may
consider enrolling simultaneously in Math 541). Some fluency with
Riemannian geometry may be helpful in the latter part of the course,
but we will give a self-contained introduction to the sub-Riemannian
geometry of the Heisenberg group.
- Texts: There is no required textbook. I will hand out
drafts of a book in preparation (Z.M. Balogh, K. Fassler and J.T.
Tyson, Mapping Theory in Metric Spaces).
Here are a few other references:
- J. Heinonen, Lectures on Analysis in Metric Spaces,
Universitext, Springer, 2001.
- M. M. Deza and M. Laurent, Geometry of Cuts and Metrics,
- N. Linial, Finite metric spaces: combinatorics, geometry,
algorithms, ICM Proceedings, 2002.
- P. Indyk and J. Matousek, `Low distortion embeddings of finite
metric spaces', Chapter 8 in Handbook of Discrete and
Computational Geometry, second edition, 2004.
- Y. Benjamini and J. Lindenstrauss, Geometric Nonlinear
Functional Analysis, vol. 1, AMS Colloquium Publications, 2000.
See especially Chapter 8.
- Grading: There will be occasional homework
assignments. The draft textbook which I am writing contains a number
of problems and exercises. A few of these will be assigned. Students
are encouraged to try other problems; I welcome any feedback from
the class concerning the text and the problems.
I will ask each student to give a lecture or mini-lecture near the end
of the semester, on some topic related to the course material.
Grades will be determined based on class participation and