Finite metric spaces and their embeddings

Manor Mendel
CMI, Caltech

Properties of finite metric spaces and in particular their
bi-Lipschitz embeddability has become an active research area in
theoretical computer science in the last decade.  Since finite metric
spaces are common objects in the study algorithmic problems, a deeper
understanding of their geometrical structure often leads to better
algorithmic tools.

In the first talk (Nov/19) we will review basic results in the field:
Bourgain's embedding, Johnson-Lindenstrauss' dimension reduction
lemma, and Karp-Bartal's embedding into probabilistic trees -
illustrating their algorithmic use with some examples.  Time
permitting we will also sketch proofs of these theorems.

In the second talk (Dec/3) we will discuss the following Ramsey-type
question in finite metric spaces: "Given n, what is the largest m such
that any n-point metric contains a subset of size m which resembles a
Euclidean point set?"  This question is motivated both from pure-math
perspective as a non-linear counterpart to Dvoretzky's theorem and
from Computer-Science perspective as a tool in proving lower bounds
for some optimization problems defined on metric spaces.

Both talks will be self contained.