There are many examples in mathematics, both pure and applied, in which
problems with symmetric formulations have non-symmetric solutions.
Sometimes this symmetry breaking is total, as in the example of
turbulence, but often the symmetry breaking is only partial. One technique
that can sometimes be used to constrain the symmetry breaking is
reflection positivity. It is a simple and useful concept that will be
explained in the talk, together with some examples. One of these concerns
the minimum eigenvalues of the Laplace operator on a distorted hexagonal
lattice. Another example that we will discuss is a functional inequality
due to Onofri.
The talk is based on joint work with E. Lieb.

In single-molecule microscopy it is necessary to reconstruct a signal that consists of positive point sources from noisy observations of the spectrum of the signal in the low-frequency band [−fc,fc]. It is shown that the problem can be solved using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the support of the signal, i.e., on how many point sources can occur within an interval of length 1.87/fc. The stability estimate is complimented by a converse result: the performance of convex algorithm is nearly optimal. The results are generalized to multi-dimension signals. Applications in microscopy are briefly discussed.

Discussion about the paper "Bivariate Polynomials Modulo Composites and their Applications" by Dan Boneh and Henry Corrigan-Gibbs