Scientific International Events

Over the last 60 years, multilinear algebra made its way into the applied sciences. Just as matrices play a central role in linear algebra, tensors are the main actors of multilinear algebra. From mathematics to computer science, from engineering to physics, the reason for their success is that they provide a natural and compact way to store information. A standard way to extract information from tensors is to consider structured decompositions, with respect to some notion of rank which typically has a natural geometric interpretation. Algebraic geometry provides the right tools to study these structured decompositions. Geometry plays a major role in two areas of the study of tensors: the theory of secant varieties and the spectral theory of tensors. The first one is an object of a long line of research, starting from classical work from the XIX and early XX century to state-of-the-art pure and applied mathematics. The second one is a young subject, which has strong connections to optimization and witnessed promising developments in the last few years. This minisymposium brings together experts in the field as well as young researchers with the purpose of providing an overview on recent developments in the role of algebraic geometry in the study of tensor decomposition. In this way, the minisymposium offers a theoretical perspective which is complementary to the minisymposium “Tensors in Applications”.

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. These topics raise challenging computational problems, but also the theory behind them is far from fully understood. Algebraic geometry has already played an important role in the study of tensors. It has shed light on: the ill-posedness of tensor approximation problems, the generic number of decompositions of a rank-r tensor, the number and structure of tensor eigen- and singular tuples, the number and structure of the critical points of tensor approximation problems, and on the sensitivity of tensor decompositions among many others. This minisymposium focuses on recent developments on the geometry of tensors and their decompositions, their applications, and mathematical tools for studying them, and is a sister minisymposium to "The algebra and geometry of tensors 2: structured tensors" organized by A. Bik, J. Draisma and E. Ventura.

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.