## Columbia Mathematics Department Colloquium

Ultrametric skeletons

by

# Assaf Naor

## Courant Institute, NYU

Abstract:

Let (X,d) be a compact metric space, and let mu be a Borel

probability measure on X. We will show that any such metric measure space

(X,d,mu) admits an "ultrametric skeleton": a compact subset S of X on which

the metric inherited from X is approximately an ultrametric, equipped with

a probability measure nu supported on S such that the metric measure space

(S,d,nu) mimics useful geometric properties of the initial space (X,d,mu).

We will make this geometric picture precise, and explain a variety of

applications of ultrametric skeletons in analysis, geometry, computer

science, and probability theory.

Joint work with Manor Mendel.

probability measure on X. We will show that any such metric measure space

(X,d,mu) admits an "ultrametric skeleton": a compact subset S of X on which

the metric inherited from X is approximately an ultrametric, equipped with

a probability measure nu supported on S such that the metric measure space

(S,d,nu) mimics useful geometric properties of the initial space (X,d,mu).

We will make this geometric picture precise, and explain a variety of

applications of ultrametric skeletons in analysis, geometry, computer

science, and probability theory.

Joint work with Manor Mendel.