MAG - Multilinear Algebraic Geometry

Abstract:

Multilinear maps are ubiquitous in mathematics, especially in their symmetric version representing both homogeneous polynomials and symmetric tensors. The vastness of interests that they open penetrates the depths of algebraic, geometric and numerical research which by their nature take roots in classical algebraic geometry and find an outlet in highly relevant open problems such as complexity theory.

In this project we aim to study the geometry of these spaces describing multilinear phenomena, with a focus on secant varieties and the connected concept of rank harmonizing the whole project around one of the most important unsolved problems in complexity theory like the complexity of Matrix Multiplication (MaMu).

The interest in these multilinear phenomena is increasing due to their interdisciplinary nature, which is spread over several fields of Mathematics (Multilinear Algebra, Commutative Algebra, Algebraic Geometry, Combinatorics, Numerical Analysis). Tensors are nowadays widely used in several emerging technologies: Signal processing, Machine learning, Algebraic statistics, Game Theory, Complexity.

We list here the four main topics:

1. Complexity of Matrix Multiplication
2. Identifiability
3. Zero-dimensional schemes and cactus varieties
4. Critical points of distance function and low rank optimization