My research interests have their core in Tensor Decomposition and Applications.
The view point I have on this topic comes from Commutative Algebra. I investigate the algebraic properties of varieties and schemes associated to classes of tensors and the computational aspects of generation of algorithms. I am also interested in various application effects of tensor decomposition, for example in Quantum Physics and Data Science.
My published Papers can be found on both ORCID or in my publications page at the Math Dept of Trento.
All available on arXiv.
A. Bernardi, A. Gimigliano.
Città studi. (first edition 2014, second edition 2018).
Editors: E. Ballico, A. Bernardi, I. Carusotto, S. Mazzucchi, V. Moretti.
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 25). (2019). ISBN: 978-3-030-06122-7.
Hanieh Keneshlou (2022-) (University of Konstanz). In the framework of Konstanzia Fellowship Program (Gender equality in science and studies).
Maciej Gałązka (2024-2025). Post doc of the TensorDec Laboratory, in the framework of PRIN 2022 - Multilinear Algebraic Geometry.
Tomasz Mańdziuk (2023-2024).
Oriol Reig Fité (ongoing XL Cycle) (MSC-Doctoral Network, In the frame of Horizon 2020- TENORS fellow)
Enrica Barrilli (ongoing XL Cycle) (MSC-Doctoral Network, In the frame of Horizon 2020- TENORS fellow)
Asma Momal (ongoing – XXXIX Cycle) (In the framework of PNRR DM 117: Industrial PhD in collaboration with BlueTensor).
Vincenzo Galgano (Excellence Program for PhD students of the Mathematical PhD school of Trento). Defended in December 2023. Title of the thesis: Secant varieties of Spinor varieties and of other generalized Grassmannians.
Papers that are coming out from the thesis:
- Identifiability and singular locus of secant varieties to spinor varieties, Galgano, V., arXiv:2302.05295.
- Equivariant Euler characteristics on permutohedral varieties, Galgano, V., Keneshlou, H., Michalek, M., arXiv:2302.04598.
- Identifiability and singular locus of secant varieties to Grassmannians, Galgano, V., Staffolani, R., arXiv:2212.05811.
- Graph States and the Variety of Principal Minors, Galgano, V., Holweck, F. AMPA, 2023.
Followup: (2024-) postdoctoral researcher Max-Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG) in Dresden, Germany.
Valentina Amitrano (Quantum@TN PhD program). Defended in 2023, Co-advised with F. Pederiva. Title of the thesis: Digital quantum computing for many-body simulation.
Followup: (2023-) Pst-doc INFN Ferrara.
Claudia De Lazzari (Excellence Program for PhD students of the Mathematical PhD school of Trento, Quantum@TN PhD program). Defended in December 2022. Title of the thesis: Algebraic, geometric and numerical methods for Tensor Network Varieties.
Papers that came out from the thesis:
- The Linear Span of Uniform Matrix Product States, De Lazzari, C., J Motwani, H., Seynnaeve, T., SIGMA 18 (2022), 099, 18 pages.
- Dimension of tensor network varieties, Bernardi, A., De Lazzari, C., Gesmundo, F., Communications in Contemporary Mathematics, 2022, 2250059.
Followup: (2022- ) CDL is the Quantum Cryptographer - QTI SRL
Pierpaola Santarsiero. Defended in April 2022. Title of the thesis: Identifiability of small rank tensors and related problems.
Papers that came out from the thesis:
- An algorithm for the non-identifiability of rank-3 tensors, Santarsiero, P. BUMI, 2023, 16(3), pp. 595–624.
- Terracini locus for three points on a Segre variety, Ballico, E., Bernardi, A., Santarsiero, P., arXiv:2012.00574.
- Identifiability of Rank-3 Tensors, Ballico, E., Bernardi, A., Santarsiero, P., Mediterranean Journal of Mathematics, 2021, 18(4), 174.
Followup: (2022) PS was Postdoctoral researcher, MPI MiS. (2023) Post-doc University of Osnabrück, Germany. (2024) Post-doc in Bologna.
Reynaldo Staffonali. Defended on February 2022. Title of the thesis: Schur apolarity and how to use it.
Papers that came out from the thesis:
- Schur apolarity, Staffolani, R., JSC, 2023, 114, pp. 37–73.
- A note on the maximal rank, Bernardi, A., Staffolani, R. European Journal of Mathematics, 2022, 8, pp. 94–100.
Vincenzo Antonio Isoldi.
Jose Enrique Manrique Santillana (2024). Internship at Cancer Center Amsterdam. Perineural Invasion in Pancreatic Cancer: Clinical significance using Multiple and SVD-Based Imputation.
Simone Patscheider (2023), Geometric Measure of Entanglement via Tensor Decomposition. Master in Mathematics. Internship at Equal1 (Dublin).
Elena Zordan (2021), Data is the new oil only for the right people: the silent problem of data gap that fosters inequalities (Co-advised. Main advisor: Luigi Amedeo Bianchi). Master in Data Science.
Martina Iannacito (2019), HOSVD for multispectral images. A numerical approach to plant biodiversity estimation.
Papers that came out form this thesis:
- High order singular value decomposition for plant diversity estimation, Bernardi, A., Iannacito, M., Rocchini, D., Bolletino dell Unione Matematica Italiana, 2021, 14(4), pp. 557–591.
- rasterdiv—An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back, Rocchini, D. at al, Methods in Ecology and Evolution, 2021, 12(6), pp. 1093–1102.
Daniele Taufer (2017): Symmetric Tensor Decomposition (Thesis for the final defense of Galileian School of Higher Education - University of Padova).
Paper that came out of this thesis:
- Waring, tangential and cactus decompositions, Bernardi, A., Taufer, D., Journal des Mathematiques Pures et Appliquees, 2020, 143, pp. 1–30.
Federico Mazzone (2018): Title of the thesis: Il crivello quadratico.
Lorenzo Barban (2018): Title of the thesis: Sistemi inversi di Macaulay.