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

Original Announcement

This workshop 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.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:
Small deviations for a family of smooth Gaussian processes
Directional discrepancy in two dimensions
Universality of the asymptotics of the one-sided exit problem for integrated processes
Path regularity of Gaussian processes via small deviations
A three dimensional signed small ball inequality
Bounds for entropy numbers of some critical operators