Periodic Reporting for period 1 - TENORS (Tensor modEliNg, geOmetRy and optimiSation)
Berichtszeitraum: 2024-01-01 bis 2025-12-31
For the first 24 months of the project, the project has been implemented in line with the overall objectives. In terms of theoretical advances, we have had new results on the study of tensor spaces (objective 1), including generalized additive decomposition, Segre–Veronese varieties, and q-symmetric tensors, on high performance tensor computation (objective 2) such as the development of new tools for cactus rank computation, for polynomial optimization, exploiting tensor invariants, low-rank approximations for PDEs, and low rank structures in semidefinite programming. The advances on new applications of tensors methods (objective 3) include the applications of network methods for quantum systems, improved techniques for quantum separability analysis, and scalable approaches to semidefinite problems in physics and finance. Regarding the training and collaboration (objective 4), we have had successful completion of all planned events, including training activities and industrial secondments. The technological, economical and societal impact of the project (objective 5) is in progress but without quantitative measurements yet. Overall, the project performance in the first 24 months have obtained strong scientific output with activities fully on schedule and aligned with project objectives.
Activities performed and main achievements are:
- Significant theoretical advancements on new methods for Generalized Additive Decomposition, on the non-defectivity of secant varieties to Segre-Veronese varieties, and the geometric characterization of Lissajous varieties with their applications to dynamical systems. There has been tangible outputs, including the development of open-source software (Julia packages), the release of preprints, and submissions to international conferences such as MEGA 2026. The strong level of interaction between the partners exemplified by joint supervision and cross-node research activities (e.g. between Inria, UKON, MPI, and UniFI)—validates the network’s collaborative approach.
- New contributions on the theoretical foundations and the algorithmic development for high-performance tensor computations, ranging from symbolic and numerical tensor decomposition algorithms to tropical methods, symmetry-aware models, low-rank discretizations for PDEs, and optimization techniques for tensor-based models.
- Significant advances on multiple fronts: from new semi-analytical and numerical frameworks for tensor-network methods in quantum condensed matter physics, to strengthened hierarchies and certification techniques for quantum entanglement detection, to scalable approaches for quantum optimal control and tensor-based modelling of quantum information networks, for symmetry-exploiting optimisation.
- Consolidation of a multidisciplinary research environment across the TENORS network. Doctoral Candidates have produced significant scientific results, on tensor decomposition problems, polynomial optimization for quantic computing and high performance tensor computing and isogeometric analysis. Several of these contributions have already resulted in preprints or journal submissions, demonstrating scientific maturity and ensuring wide dissemination within the international community.
- The interactions among universities, research institutes, and industrial partners have strengthened the methodological coherence of the work package, ensuring that theoretical advances translate into computational tools and that algorithmic developments respond to concrete needs in modelling, simulation, and optimisation. Industrial secondments have further enhanced this transfer of knowledge, providing DCs with valuable experience in applied environments and contributing to the long-term impact of the network.
- Training and community-building actions, seminars, masterclasses, internal workshops, learning week(s) and joint supervision across institutions, have played a crucial role in supporting the DCs and in reinforcing the scientific ecosystem of the project. These activities have also contributed to establishing a shared research language across fields such as algebraic geometry, numerical analysis, optimisation, and data science, which is essential for the success of a project of the scope of TENORS.
- Significant theoretical advancements on new methods for Generalized Additive Decomposition, on the non-defectivity of secant varieties to Segre-Veronese varieties, and the geometric characterization of Lissajous varieties with their applications to dynamical systems. There has been tangible outputs, including the development of open-source software (Julia packages), the release of preprints, and submissions to international conferences such as MEGA 2026. The strong level of interaction between the partners exemplified by joint supervision and cross-node research activities (e.g. between Inria, UKON, MPI, and UniFI)—validates the network’s collaborative approach.
- New contributions on the theoretical foundations and the algorithmic development for high-performance tensor computations, ranging from symbolic and numerical tensor decomposition algorithms to tropical methods, symmetry-aware models, low-rank discretizations for PDEs, and optimization techniques for tensor-based models.
- Significant advances on multiple fronts: from new semi-analytical and numerical frameworks for tensor-network methods in quantum condensed matter physics, to strengthened hierarchies and certification techniques for quantum entanglement detection, to scalable approaches for quantum optimal control and tensor-based modelling of quantum information networks, for symmetry-exploiting optimisation.
- Consolidation of a multidisciplinary research environment across the TENORS network. Doctoral Candidates have produced significant scientific results, on tensor decomposition problems, polynomial optimization for quantic computing and high performance tensor computing and isogeometric analysis. Several of these contributions have already resulted in preprints or journal submissions, demonstrating scientific maturity and ensuring wide dissemination within the international community.
- The interactions among universities, research institutes, and industrial partners have strengthened the methodological coherence of the work package, ensuring that theoretical advances translate into computational tools and that algorithmic developments respond to concrete needs in modelling, simulation, and optimisation. Industrial secondments have further enhanced this transfer of knowledge, providing DCs with valuable experience in applied environments and contributing to the long-term impact of the network.
- Training and community-building actions, seminars, masterclasses, internal workshops, learning week(s) and joint supervision across institutions, have played a crucial role in supporting the DCs and in reinforcing the scientific ecosystem of the project. These activities have also contributed to establishing a shared research language across fields such as algebraic geometry, numerical analysis, optimisation, and data science, which is essential for the success of a project of the scope of TENORS.
The impact of TENORS’ activities within the scientific community is becoming increasingly visible thanks to the publication of articles in journals, which has already begun and is expected to grow further in the near future, as well as the growing number of invitations to DCs and network members to present their work at international conferences
- new method for Generalized Additive Decomposition; new results on the non-defectivity of Segre-Veronese varieties; on varieties parametrized by trigonometric polynomials ; and on the decomposition of q-symmetric tensors.
- new method for computing the cactus rank; new results for the computation of invariants of even-order tensors for the orthogonal group; new efficient methods for low-rank approximation of tensors appearing in Galerkin methods for the solution of Partial Differential Equations;
- New advances on the theory and applications of tensor network geometry to Bogoliubov-de Gennes models in condensed matter physics; improvement of Doherty-Parillo-Spedaglieri hierarchies for the analysis of separable states in quantum computing; new methods for producing rigorous computer-assisted proofs of semidefinite bounds in quantum information science; new tensor network based methods for large quantum systems under time-changing controls for applications in financial mathematics; new efficient method for solving large-scale semidefinite problems with symmetries, that appear in financial services.
- new method for Generalized Additive Decomposition; new results on the non-defectivity of Segre-Veronese varieties; on varieties parametrized by trigonometric polynomials ; and on the decomposition of q-symmetric tensors.
- new method for computing the cactus rank; new results for the computation of invariants of even-order tensors for the orthogonal group; new efficient methods for low-rank approximation of tensors appearing in Galerkin methods for the solution of Partial Differential Equations;
- New advances on the theory and applications of tensor network geometry to Bogoliubov-de Gennes models in condensed matter physics; improvement of Doherty-Parillo-Spedaglieri hierarchies for the analysis of separable states in quantum computing; new methods for producing rigorous computer-assisted proofs of semidefinite bounds in quantum information science; new tensor network based methods for large quantum systems under time-changing controls for applications in financial mathematics; new efficient method for solving large-scale semidefinite problems with symmetries, that appear in financial services.