Small ball inequalities in analysis, probability, and irregularities of distribution

December 8 to December 12, 2008

at the

American Institute of Mathematics, Palo Alto, California

organized by

William Chen, Michael Lacey, Mikhail Lifshits, and Jill Pipher

This workshop, sponsored by AIM and the NSF, will be devoted to a theme common to Irregularity of Distributions, Approximation Theory, Probability Theory and Harmonic Analysis.

In each of these subjects, there are outstanding conjectures in dimensions three and higher that stipulate that functions which satisfy certain conditions on its mixed derivative are necessarily large in sup norm. One of these conjectures, possibly the most well known, concern uniform lower bounds on the star discrepancy of the classical discrepancy problem. Recently there has been progress, in that new non-trivial bounds for the star Discrepancy has been found in all dimensions, extending prior work of Wolfgang Schmidt in two dimensions, and Jozsef Beck in three dimensions.

The related questions in Probability Theory concern Small Ball inequalities for the Brownian Sheet, and other processes. In Approximation Theory, one seeks estimates of the Kolmogorov Entropy of Mixed Derivative Sobolev spaces. An important tool in these questions is the study of hyperbolic Haar series in sup norm.

This workshop will survey these different conjectures, seeking both commonalities and differences between these conjectures, describe recent advances, and discuss proof techniques and strategies.

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.

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