Jill Pipher : Geometric discrepancy theory: directional discrepancy in 2-D

Discrepancy theory originated with some apparently simple questions about sequences of numbers. The discrepancy of an infinite sequence is a quantitative measure of how far it is from being uniformly distributed. Precisely, an infinite sequence { a1,a2, ...} is said to be uniformly distributed in [0, 1] if
lim_{n\to\infty} (1/n|{a1, a2,...an} intersect [s,t]|) = t-s.
If a sequence {ak} is uniformly distributed, then it is also the case that for all (Riemann) integrable functions f on [0, 1],
lim_{n\to\infty} (1/n\sum_{k=1}^n f(ak))=\int_0^1 f(x)dx.
Thus, uniformly distributed sequences provide good numerical schemes for approximating integrals. For example, if alpha is any irrational number in [0, 1], then the fractional part {alphak}:=ak is uniformly distributed. Classical Fourier analysis enters here, in the form of Weyl's criterion. The discrepancy of a sequence with respect to its first n entries is
D({ak},n) := sup_{s If a sequence {ak} is uniformly distributed then D({ak},n) divided by n goes to zero as n\to\infty. Van der Corput posed the following question: does there exist a sequence which is so uniformly distributed that D({ak},n) is bounded by a constant for all n? In 1945, Van Aardenne-Ehrenfest proved that the answer was: No. She proved that a lower bound existed for all sequences. Later, Roth showed that the discrepancy problem for sequences had an equivalent geometric formulation in terms of a notion of discrepancy in two dimensions. The problem in two dimensions, which is the focus of this talk, is this: Given a collection of N points in the unit square [0, 1]^2, how can we quantify the idea that it is uniformly distributed in the square? Which collections of points achieve a lowest possible discrepancy? There are many reasons to be interested in discrepancy theory, both pure and applied: sets of low discrepancy figure prominantly in numerical applications, from engineering to finance. This talk focuses primarily on theoretical issues involving measuring discrepancy in two and higher dimensions.
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• Category: Applied Math and Analysis
• Duration: 01:34:52
• Date:  October 3, 2011 at 4:25 PM
• Views: 101
• Tags:   seminar, Applied Math And Analysis Seminar