Hadamard ranks of algebraic varieties

Speakers: Alessandro Oneto (Università di Genova)\n\nThe additive notion of rank with respect of an embedded algebraic variety is a classical notion that employs the language of secant varieties to give a general framework for matrix rank, tensor rank, and other similar notions. In the last decade, motivated by the study of particular algebraic statistical models called Restricted Boltzmann Machines, it has been introduced the notion of Hadamard product of two algebraic varieties, namely, the Zariski closure of the coordinate-wise product of all possible pairs of points in the Cartesian product. This definition can be used to construct Hadamard powers of algebraic varieties and define Hadamard ranks, which can be regarded as a multiplicative version of secant varieties and the classical notion of rank. The multiplicative nature of the construction allows methods from tropical geometry to be successfully applied to these problems. In this talk, I will introduce these notions and present recent results and open problems. This is based on joint works with Dario Antolini, Edoardo Ballico, Guido Montufar and Nick Vannieuwenhoven.\n\nhttps://indico.math.cnrs.fr/event/17003/