**THE MALLAT FAMILY FUND FOR RESEARCH IN
MATHEMATICS **

**invites**** you to a**

**SPECIAL LECTURE SERIES**** **

**to**** be presented by**

The lectures will be held in

Room 232

Technion - Israel Institute of Technology

Lecture
I:

**Perelman's
work on the Geometrization Conjecture – I**

An overview of Perelman's proof of the Geometrization
Conjecture for a general mathematical audience.

Lecture
II:

**Perelman's
work on the Geometrization Conjecture – II**

This will be a
continuation of the first lecture, in which several important strands of the
proof are discussed in greater detail. The lecture will be aimed at people
working in Geometry, Topology, or PDE's.

Lecture
III:

**BiLipschitz
embedding in Banach spaces**

A mapping between metric
spaces is L-biLipschitz if it stretches distances by a factor of at most L, and
compresses them by a factor no worse than 1/L. A basic problem in geometric
analysis is to determine when one metric space can be bi-Lipschitz embedded in
another, and if so, to estimate the optimal bi-Lipschitz constant. In
recent years this question has generated great interest in computer science,
since many data sets can be represented as metric spaces, and associated
algorithms can be simplified, improved, or estimated, provided one knows that
the metric space space in question can be biLipschitz embedded (with controlled
bi-Lipschitz constant) in a nice space, such as L^2 or L^1.

The lecture will
discuss several new existence and non-existence results for bi-Lipschitz
embeddings in Banach spaces. One approach to non-existence theorems is
based on generalized differentiation theorems in the spirit of Rademacher's
theorem on the almost everywhere differentiability of Lipschitz functions on
R^n. We first show that earlier differentiation based results of Pansu
and Cheeger, which proved non-existence of embeddings into R^k, generalize to
many Banach space targets, such as L^p for 1 < p < infinity. We then
focus on the case when the target is L^1, where differentiation theory is known
to fail, and the embedding questions are of particular interest in computer
science. When the domain is the Heisenberg group with its
Carnot-Caratheodory metric, we show that a
modified form of differentiation still holds for Lipschitz maps into L^1, by
exploiting a new connection with functions of bounded variation, and some very
recent advances in geometric measure theory. This leads to a proof of a
conjecture of Assaf Naor.

This is joint work with
Jeff Cheeger.