American Institute of Mathematics, Palo Alto, California
William Chen, Michael Lacey, Mikhail Lifshits, and Jill Pipher
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 schedule.
A report on the workshop activities.