Positive definite functions and their applications

Abstract:  

For a measurable set A in the euclidean space  (or, more generally, in any LCA group), the autocorrelation function 1_A * 1_{-A}  is positive definite, i.e. its Fourier transform is nonnegative. This simple fact has far reaching applications in various branches of mathematics. I will review some of the most striking applications in recent years, and explain the connections to my own work on Fuglede's spectral set conjecture and the density of planar sets avoiding unit distances.