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

December 8 to December 12, 2008
at the

American Institute of Mathematics,
San Jose, 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

by Frank Aurzada, Fuchang Gao, Thomas Kühn, Wenbo V. Li, and Qi-Man Shao

Directional discrepancy in two dimensions

by Dmitriy Bilyk, Xiaomin Ma, Jill Pipher, and Craig Spencer

Universality of the asymptotics of the one-sided exit problem for integrated processes

by Frank Aurzada and Steffen Dereich

Path regularity of Gaussian processes via small deviations

by Frank Aurzada

A three dimensional signed small ball inequality

by Dmitriy Bilyk, Michael T. Lacey, Ioannis Parissis, and Armen Vagharshakyan

Bounds for entropy numbers of some critical operators

by M. A. Lifshits