Discrepancy theory and related questions (Dmitriy Bilyk, University of Minnesota)

The talk will concentrate on open questions related to the optimal bounds for the discrepancy of an $N$-point set in the $d$-dimensional unit cube. The so-called star-discrepancy measures the difference between the actual and expected number of points in axis-parallel rectangles, and thus measures the equidistribution of the set. This notion has been explored by H. Weyl, K. Roth, and many others, however many questions still remain open, especially in higher dimensions.  We shall discuss the two main conjectures on the order of star-discrepancy and present evidence in support of each one, as well as their connections to various areas of mathematics. In addition, we shall talk about discrepancy in other geometrical settings (rotated rectangles, balls, points on the sphere etc).