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 fundamentallynonlinearandgeometric. We will use tools from many different subjects, including functional analysis, Fourier analysis, PDE, and differential geometry.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 science.

**Instructor:**Prof. Jeremy Tyson**Office Location:**329 Altgeld Hall**Office Phone:**244-4132**Email:**`tyson@math.uiuc.edu`**Office Hours:**by appointment**Lecture Times:**MonWedFri 10:00-10:50**Lecture Location:**143 Altgeld Hall**Course web site:**https://math.uiuc.edu/~tyson/595s11.html.**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*, Springer, 1997. - 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.

- J. Heinonen,
**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 homework.