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.