IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Irregularity of Distributions, Harmonic Analysis, and Some Small Ball Problem

  • Dec. 6, 2007
  • 3:30 p.m.
  • LeConte 412

Abstract

It is a classical observation that no matter how cleverly one selects a number of points in the unit cube in two or higher dimensions, they must be relatively far from uniform, with respect to a variety of metrics. We will survey old and new results on this topic, with an emphasis on the so-called star-Discrepancy, associated with quasi-Monte Carlo methods. The underlying proof techniques, developed over many years by a variety of researchers, draw upon probability, and harmonic analysis. The methods are closely associated with long standing questions in Probability Theory, and Approximation Theory, which concern certain versions of the Small Ball Problem. Recent partial results in dimension three give the hope for some new insights into these questions.

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