In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials $P_N$ of degree $N$ with condition number bounded above by $N$. In this talk, we show the origin of this problem, previous knowledge until this work, its relation to other interesting mathematical problems such as Smale's 7th problem, and our main result obtained: a simple and direct answer to the Shub and Smale problem for $N=4M^2$, with $M$ a positive integer, as well as comments on its proof.

This is a joint work with Carlos Beltrán.

I will talk about the relation between optimal quadratures, Riesz (or logarithmic) energies and minimal discrepancy configurations. In particular I will discuss the use of the zeros of elliptic (or Kostlan-Shub-Smale) polynomials, among other configurations, as quadrature nodes for Sobolev spaces on the sphere. There will be many open problems throughout the talk.