# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Tractable approximations to nonnegative polynomials with applications to the joint spectral radius

The problem of recognizing nonnegativity of a multivariate polynomial has a celebrated history, tracing back to Hilbert’s 17th problem. In recent years, there has been much renewed interest in the topic because of a multitude of applications in applied and computational mathematics and the observation that one can optimize over an interesting subset of nonnegative polynomials using “sum of squares (SOS) optimization”. In this talk, we give a brief overview of the developments in this field and then focus on two recent results. In part (i), we show that the joint spectral radius of a finite set of matrices is less than one if and only if there exists a polynomial norm (i.e., a norm which is the d-th root of a degree-d homogeneous polynomial) that decreases under the application of all matrices.

##### Some solved and unsolved problems in single-particle cryo-EM

The determination of 3D protein structures by electron cryomicroscopy has been astonishingly successful in some cases, but quite unsuccessful in others. I will first discuss the status of our work on 3D structure determination of a membrane protein. Then, from the perspective of an experimentalist, I will describe some remaining problems with single-particle reconstruction that I hope might yield to mathematical solutions.