ABSTRACT DMS-9705775 Kolountzakis Kolountzakis will study properties of translational tilings of Euclidean Space using Harmonic Analysis. A function f(x) is said to "tile" the Euclidean space with the discrete "tile set" A if, when translated at the locations of A, it adds up to a constant w (the "weight" of the tiling), almost everywhere. One obtains an equivalent condition for this to happen: the Fourier transform of the distribution with one point mass at each point of A must be supported at the zeros of the FT of the function f, plus the point 0. Among the problems Kolountzakis proposes to study are: (a) the structure of tilings of the line with tiles of non-compact support, (b) the periodic tiling conjecture in higher dimension, (c) results related to the uniqueness of the Poisson Summation formula, (d) further estimates on the Steinhaus problem, (e) the problem of simultaneous fundamental domains in general abelian groups and (f) non-translational tilings of space via methods of non-abelian harmonic analysis. The study of tilings of space using a certain building block called the "tile" has significant connections with crystallography. Interest in this has been renewed after the, relatively recent, discovery that non-periodic tilings (i.e., pavings of space which are not "determined" by how they look in an arbitrarily large but finite part of the space) do exist in nature. The question of which are the possible ways that space can be tiled is the paramount one in this field. The proposed research will help to understand the situation in higher dimensions. Another domain of application of the proposed methods concern the theory of Wavelets
