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.
