Geometry of tensors in Nancy 2025
Journée scientifique du pôle AM2I : “Introduction à la géométrie des tenseurs” (Monday 3rd November 2025)
Location: IECL (Institut Élie Cartan de Lorraine), Université de Lorraine, rue du Jardin botanique, Villers-lès-Nancy (salle de conférences). How to get there
Date: November 3rd, 2025.
About the Workshop
This one-day workshop is an introduction to the mathematical theory of tensors, specifically from the viewpoints of algebraic geometry and commutative algebra, and its applications. This topic has indeed been attracting considerable research interest in the past few years, due to its various applications, for example in machine learning, algebraic statistics as well as signal processing.
Though it is intended to non-specialists geometers, part of the talks will be accessible to a larger audience, and everyone is welcome!
Registration
Registration is free but mandatory (the capacity is limited), by October 15th at the latest. You can register here.
Invited speakers
- Bernard Mourrain (INRIA Sophia-Antipolis)
- Pierpaola Santarsiero (Università Politecnica delle Marche, Ancona, Italy)
- Luca Sodomaco (Max-Planck-Institut, Leipzig, Germany)
- Alexander Taveira Blomenhofer (QMATH centre, University of Copenhagen, Denmark)
- Nick Vannieuwenhoven (KU Leuven, Belgium)
Schedule (to be confirmed)
| Time | Session |
|---|---|
| 10:00-10:30 | Welcome coffee |
| 10:30-11:30 | Nick Vannieuwenhoven An introduction to tensor rank decomposition (abstract) |
| 11:30-11:45 | Coffee break |
| 11:45-12:15 | Alexander Taveira Blomenhofer Additive X-rank decomposition (abstract) |
| 12:15-14:00 | Lunch break |
| 14:00-15:00 | Bernard Mourrain An algebraic-geometric view on tensor decomposition problems (abstract) |
| 15:00-15:30 | Coffee break |
| 15:30-16:00 | Pierpaola Santarsiero TBA (abstract) |
| 16:00-16:30 | Break |
| 16:30-17:00 | Luca Sodomaco Nonlinear Rayleigh quotient optimization (abstract) |
| 19:30 | Dinner in the city centre |
Nick Vannieuwenhoven
An introduction to tensor rank decomposition
Abstract: TBA
Alexander Taveira Blomenhofer
Additive X-rank decomposition
Abstract: The X-rank problem aims to write a vector as a sum of r elements of a subvariety X, where r is minimal. Common examples include tensor rank, symmetric tensor rank, Chow rank and alternating rank, where X is, respectively, the Segre variety, the Veronese variety, the Chow variety or the Grassmannian. The X-rank problem also occurs in the parameter estimation problem for Gaussian mixtures from moments. We discuss recent results on X-rank, such as identifiability and algorithms. Based on joint work with Benjamin Lovitz and Alex Casarotti.
Bernard Mourrain
An algebraic-geometric view on tensor decomposition problems
Abstract: Tensors are fundamental mathematical objects that generalize matrices to higher dimensions and play a crucial role in various domains.
Tensor decomposition has emerged as a key technique in the analysis of higher-order tensors, to reveal the underlying structure of the data, to reduce its complexity, and to facilitate more efficient computations. However, it is a challenging problem from a computational and numerical point of view. This can be explained by the complex and still unrevealed geometry hidden behind the scene.
We will explore this geometry through the prism of algebra, illustrating how Artinian algebras can be naturally associated with general tensor decompositions for multilinear, symmetric, and multisymmetric tensors.
We will review algebraic geometric approaches, which reduces tensor decomposition to direct eigenvalue and eigenvector computations when the rank is small.
When the rank is higher, the geometry is much more complex. We will see how the tensor decomposition problem can be transformed into a simultaneous diagonalization problem of extended tensors. The relationship between these varieties of extended tensors and the punctual Hilbert scheme, via families of commuting matrices will be discussed.
The various concepts will be illustrated by effective calculations on a few examples.
Pierpaola Santarsiero
TBA
Abstract: TBA
Luca Sodomaco
Nonlinear Rayleigh quotient optimization
Abstract: The nonzero critical points of the Rayleigh quotient associated with a square matrix correspond to its nonzero eigenvectors. In this work, we study the critical points of the Rayleigh quotient constrained to an algebraic variety. We relate the number of complex critical points to the Euclidean distance degree of a Veronese embedding of the variety of constraints and provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors. This presentation is about joint work with Flavio Salizzoni and Julian Weigert.
Contact Us
Organization committee:
- Clara Derand
- Anne Incerti
- Gianluca Pacienza
- Konstantin Usevich
- Institut Élie Cartan de Lorraine
- group SiMul @ CRAN.
You can email the organizers at: firstname.lastname@univ-lorraine.fr
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© 2025 Journée scientifique du pôle AM2I : “Introduction à la géométrie des tenseurs”