Differentiable structures on finite sets

This workshop, sponsored by AIM and the NSF, 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 in Rⁿ extends to a C^m function on the whole of Rⁿ with small norm.

For instance, suppose f : E &rarr 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 C² norm less than 1. Then f can be extended to the whole plane with C² 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 Rⁿ.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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