We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process we propose a point processs in odddimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by Bondarenko, Radchenko and Viazovska in [BRV13]. Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously.

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In a recent article [AZ15], Alishahi and Zamani discuss the spherical ensemble, a rotationally invariant determinantal point process on S 2 . In this paper we extend this process in a natural way to the 2ddimensional sphere S 2d . We prove that the expected value of the Riesz s-energy associated to this determinantal point process has a reasonably low value compared to the known asymptotic expansion of the minimal Riesz s-energy.Date: June 28, 2018. 2010 Mathematics Subject Classification. 31C12 (primary) and 31C20, 52A40 (secondary). Definition 2.1. A DPP of N points on S d has isotropic associated reproducing kernel K(p, q) if there exists a function f : [0, 2] −→ R such that |K(p, q)| = f (||p − q||) for all p, q ∈ S d .When the kernel is isotropic, we say that the DPP is rotationally invariant. A weaker property will be that of homogeneus intensity.