Department colloquium - Gilles Pisier

Title: The saga of Sidon sets

Speaker: Gilles Pisier (Professor at Universite Paris Sorbonne and Distinguished Professor, A.G. and M.E. Owen Chair of Mathematics at Texas A&M)

Gilles Pisier

Abstract:  Sidon sequences of integers {n_k} are a generalization of the sequences that are lacunary "a la Hadamard", as for example the sequence {2^k}. They have a long history, starting with Zygmund, Kahane, Rudin, Salem, Malliavin to name a few, with a golden age in the 1960's and 1970's when many researchers working on Fourier analysis were interested in "thin sets", of which Sidon sets are perhaps the simplest and most basic example. Sidon sequences are closely connected with "independent" sequences, in several distinct ways. The sequence of characters {e^{in_kt}} behaves analytically like a stochastically independent sequence, but also Sidon sequences can be characterized arithmetically by the fact that in some proportional sense they do not satisfy any non trivial linear relation over {-1,0,1}, just like for linear independence in a vector space. We will present the main theorems about Sidon sets in Abelian discrete groups such as Z, as well as (if time permits) some of the non-commutative generalizations. In the latter case {n_k} can be replaced either by a sequence of unitary representations on a compact non-Abelian group, which connects the subject with random matrices, or by a sequence in a discrete non-amenable group such as a free group.

The department invites all to coffee, tea and cakes at 14:45 in the 4th-floor lunchroom.