Speaker
Ujué Etayo, CUNEF Universidad
Abstract
The construction of well-distributed finite point configurations is a classical theme in real and complex algebraic geometry—through Vandermonde-type extremal problems and Fekete points—and it has become increasingly relevant in metric questions motivated by applications, where one needs quantitative discretizations of algebraic varieties. Two complementary ways to measure uniformity are the minimization of interaction energies (logarithmic/Riesz-type and related kernels) and the control of discrepancy, understood as integration error for suitable test sets or test functions.
In this talk, I will discuss determinantal point processes (DPPs) as canonical probabilistic models of repulsive point configurations tailored to algebraic-geometric data. In the complex setting, DPPs defined from spaces of holomorphic sections (equivalently, from Bergman kernels associated with high tensor powers of an ample line bundle) can be viewed as a natural “randomization” of Fekete configurations, with joint densities governed by Vandermonde-type determinants. I will explain how the determinantal correlation structure leads simultaneously to low energy and low discrepancy/transport behavior, and how, in the algebro-geometric framework, the empirical measures converge toward canonical equilibrium (Monge–Ampère) measures determined by the underlying metric. Concrete illustrations include projective spaces, spheres and tori, where one can compare with optimal asymptotic rates.
The aim is to highlight DPPs as a unifying bridge between metric energies, discrepancy notions, and algebraic structures (Bergman kernels and Vandermonde determinants) in metric algebraic geometry.