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Dmytro Bilyk : Discrepancy Theory and Analysis

In this talk, we shall look at discrepancy theory through the prism of harmonic and functional analysis. Discrepancy theory deals with finding optimal approximations of continuous objects by discrete sets of points and quantifying the inevitably arising errors (irregularities of distribution). This field lies at the interface of several areas of mathematics: approximation, probability, discrete geometry, number theory. Historically, methods of analysis (Fourier techniques, Riesz product, wavelet expansions etc) played a pivotal role in the development of the subject.

A number of exciting new connections of discrepancy theory to other fields were discovered recently and are not yet fully understood. These include approximation theory (metric entropy of spaces with mixed smoothness, hyperbolic approximations), probability (small deviations of Gaussian processes, empirical processes), harmonic analysis (small ball inequality, Sidon theorem), compressed sensing etc.

We shall describe some of the recent results in the field, the main ideas and methods, and numerous relations to other areas of mathematics.

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