Abstracts

Abstract: The Heisenberg group is a meeting ground for several different subjects: quantum mechanics, signal analysis, representation theory, partial differential equations, several complex variables, and differential geometry, among others. In these lectures we will examine the role of the Heisenberg group in these subjects and the ways in which its study leads into further developments in them.

Abstract: Let {x} be the one-peridic "sawtooth" function

{x} = xE(x) – ½

(where E(x) is the integer part of x). Davenport series are of the form

f(x) = Σ an {nx}.

Functions of this type were initially introduced by Hecke in the 1920s, and, under their general form by Davenport in the 1940s. They now lie at a crossroad between many areas related with harmonic analysis: When the series Σ an is not absolutely convergent, their pointwise convergence properties has been the occasion to introduce new summation methods. When the series Σ an is absolutely convergent, these functions have discontinuities at the rational numbers. This fact has many implications: these functions are then multifractal; i.e. the sets of points where they have a given regularity are fractal sets; these sets can be characterized by Diophantine approximation properties; furthermore they have paradoxical properties: indeed, they fall in the class of "sets with large intersection" introduced by Falconer.

Therefore the study of the properties of these series will give us an occasion to explore some unexpected connexions between harmonic analysis, arithmetic functions, and geometric measure theory.

Abstract: It is often easier to show that a typical object of type A will have a property B than to actually find an object of type A with property B. I will give several examples which occur in the study of measures and their Fourier transforms.