S EMINARIO DE A N ÁLISIS Departamento de Matemáticas. Universidad Autónoma de Madrid Viernes, 3 de abril de 2009 11:00 h. en el aula C-XV-520 Armen Vagharshakyan Georgia Institute of Technology Discrepancy and the Small Ball Inequality Resumen: The Small Ball Inequality refers to the lower bound for the L∞ norm of linear combinations of multidimensional Haar functions adapted to rectangles of a fixed volume. We obtain the first non-trivial improvement over the average case bound in dimensions four and higher. The relevant results are known in dimension 2, a result due to W. Schmidt and M. Talagrand, with important contributions from Halasz and Temlyakov. J. Beck established a prior result in dimension 3, whose argument we extend. The relation of these results to the Discrepancy theory is also outlined. This is joint work with Dmitriy Bilyk and Michael Lacey.