Conference Agenda

Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven.

For a point p and a variety X in a projective space, the X -rank of p is the minimal n such that p lies on an n-secant (n-1)-space to X. I consider the variety V(p,X) of such n-secants, will recall the cases when X is a rational or elliptic curve and explain some surface cases. A very nice result of Iliev and Manivel for tensors C^3 x C^3 have applications on these issues.

This talk is concerned with the discriminant locus of a vector bundle (that is, the locus of singular sections of the vector bundle).

The discriminant locus of a vector bundle is generally expected to be an irreducible hypersurface. The purpose of this talk is three-fold. The first is to discuss when the discriminant locus of the vector bundle has such expected properties, the second is to show that if the discriminant locus of the vector bundle is an irreducible hypersurface, then its degree is expressible as a function of the vector bundle, and the third is to talk about an application of the discriminant locus of a vector bundle to eigenvectors of tensors.

Tensor decomposition problems appear in many areas such as Signal Processing, Quantum Information Theory, Algebraic Statistics, Biology, Complexity Analysis, etc as a way to recover hidden structures from data. The decomposition is a representation of the tensor as a weighted sum of a minimal number of indecomposable terms. This problem can be seen as a sparse recovery problem from sequences of moments. We will develop this analogy and present an algebraic approach to address the decomposition problem, via duality and Hankel operators. We will analyze the varieties of moments associated to low rank decompositions, investigate their defining equations and some of their properties that can be exploited in the decomposition problem. Links with the Hilbert scheme of points will be presented. Examples exploiting these properties will illustrate the approach.

Let X be a Segre-Veronese product of projective spaces and denote with X* its dual variety. In this talk, we outline the main properties of the ``Euclidean Distance polynomial'' (ED polynomial) of X*, as a remarkable example of a more general theory on ED polynomials developed in a recent work with Ottaviani. Given a tensor T, the roots of the ED polynomial of X* at T correspond to the singular values of T. Moreover, we describe the variety of tensors that fail to have the expected number of singular vector tuples, counted with multiplicity. This variety is, in general, a non-reduced hypersurface and its equation is, up to scalars, the leading coefficient of the ED polynomial of X*.