Differentiable structures on finite sets

August 2 to August 6, 2010

at the

American Institute of Mathematics, San Jose, California

organized by

Charles Fefferman and Nahum Zobin

Original Announcement

This workshop will focus on the recent activity in the study of Lipschitz structures on finite sets. Is there a reasonable notion of structures on a finite set involving higher degrees of smoothness?

A lot is known about whether a given function $f$ on a large finite subset $E \subset \R^n$ extends to a $C^m$ \ function on the whole of $\R^n$ with small norm.

For instance, suppose $f : E \to \R$, where $E$ is an arbitrarily large finite subset of the plane. Assume that the restriction of $f$ to any six points of $E$ can be extended to the whole plane with $C^2$ norm less than 1. Then $f$ can be extended to the whole plane with $C^2$ norm less than a universal constant.

The analogous results for Sobolev norms are at a much earlier stage. We would like to make further progress on these (and related) problems, and to explore whether there is a sensible version of these questions for finite sets not necessarily contained in $\R^n$.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets
by  Charles L. Fefferman and Garving K. Luli,  Rev. Mat. Iberoam. 30 (2014), no. 3, 875-892  MR3254995
Non-Noetherianity of Denjoy-Carleman rings of germs
by  Liat Kessler and Andreea C. Nicoara