# 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