Research

My research interests have their core in Tensor Decomposition and Applications.
The view point I have on this topic comes from Algebraic Geometry and Commutative Algebra. I investigate the algebraic geometric 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.

CV_Bernardi.pdf

Publications

Papers

My published Papers can be found on both ORCID or in my publications page at the Math Dept of Trento.

Preprints

All available on arXiv.

Books

Algebra lineare e geometria analitica

A. Bernardi, A. Gimigliano.
Città studi. (first edition 2014, second edition 2018).

Quantum Physics and Geometry

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.

My Group

Post Docs

Former Post Doc

Maciej Gałązka (2024-2025). Post doc of the TensorDec Laboratory, in the framework of PRIN 2022 - Multilinear Algebraic Geometry.
Tomasz Mańdziuk (2023-2024).
Hanieh Keneshlou (2022-2024) (University of Konstanz). In the framework of Konstanzia Fellowship Program (Gender equality in science and studies).

PhD Students

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)

Alumni PhD students

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.

Master Students

Alumni

Vincenzo Antonio Isoldi (2025). Schur Apolarity and the Dual Terracini Lemma for Flag Varieties: A Geometric and Represntatio-Theoretic Approach.
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.

PEM (Excellence CV)

Federico Mazzone (2018): Title of the thesis: Il crivello quadratico.
Lorenzo Barban (2018): Title of the thesis: Sistemi inversi di Macaulay.

Selection of talks

(Last update: Feb. 2026)

Plenary speaker (conferences with parallel sessions)

15–19 Jun 2015 — MEGA: Effective Methods in Algebraic Geometry (Trento, Italy). “Tensor decomposition and homotopy continuation.”
3–7 Apr 2017 — British Mathematical Colloquium (Durham, UK). “Tensor decomposition and homotopy continuation.”
17 Apr 2023 — Inauguration of the Italian Research Day in the World (Italian Embassy, Brasilia, Brazil). Declined for other commitments.
24–26 Jun 2024 — EACA: Encuentros de Álgebra Computacional y Aplicaciones (El Escorial, Spain).

Invited speaker (conferences, workshops, programs)

10 Dec 2004 — CMS 2004 Winter Meeting (Montreal (Quebec), Canada). “Osculating varieties of Veronesean and their higher secant varieties.”
17 Jul 2008 — MSRI (Mathematical Sciences Research Institute) (Berkeley, CA, USA). “Secant Varieties and Ideals of varieties parameterizing certain symmetric tensors.”
13–17 Sep 2010 — Workshop on Tensor Decompositions and Applications (TDA 2010) (Monopoli, Bari, Italy). “Decomposition of Homogeneous Polynomials.”
22 Nov 2010 — Progressi Recenti in Geometria Reale e Complessa (Levico Terme, Trento, Italy). “Applicazioni recenti di risultati classici su varietà delle secanti a varietà che parametrizzano tensori. Dal problema di Waring al rango di tensori.”
18 Nov 2011 — Genova–Torino–Milano Seminar (Milano, Italy). “Ranks of Tensors, related varieties and applications.”
1 Feb 2011 — Mittag-Leffler Institute, Spring Semester (17 Jan–15 Jun 2011). “Algebraic Geometry with a view towards applications.” (Sweden). “Secant varieties and Rank of tensors.”
30 May 2013 — Colloquium, Politecnico di Torino (Torino, Italy). “Algebraic Geometry in Signal processing, Phylogenetic and Quantum Physics.”
1 Aug 2013 — SIAM Conference on Applied Algebraic Geometry (Fort Collins, Colorado, USA). “Tensor Ranks.”
19–22 May 2014 — SIAM Optimization Meeting (San Diego, CA, USA). “Tensors and Optimization.” Declined for maternity leave.
2–6 Jun 2014 — ICERM Conference (Brown University, Providence, USA). “Computational Nonlinear Algebra.” Declined for maternity leave.
13 Nov 2014 — Workshop “Tensors in Computer Science and Geometry” (Simons Institute for Theory of Computing, Berkeley, USA). “Cactus Varieties of Cubic Forms: Apolar Local Artinian Gorenstein Rings.”
2–4 Dec 2015 — MAG2015 (Barcelona, Spain). “Singularities of plane rational curves via projections.”
10–13 Jun 2015 — AMS–EMS–SPM Joint Meeting (Porto, Portugal). “On the cactus variety of cubic forms.” (Invited by T. Iarrobino; session #12 Commutative Artinian Algebras and Their Deformations.)
7–12 Sep 2015 — Convegno UMI (Siena, Italy). “A geometric view of the splitting type for plane rational curves.” (Invited; section S18 Geometria algebrica.)
18–22 Jan 2016 — Workshop on Tensor Decompositions and Applications (TDA 2016) (Leuven, Belgium). “Tensor Decomposition and Homotopy Continuation.”
12–14 Dec 2016 — Tensors: Algebra meets Numerics (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany). “Tensor Decomposition and Homotopy Continuation.”
31 Jul – 4 Aug 2017 — SIAM 2017 (Georgia Institute of Technology, Atlanta GA, USA). “On the identifiability of skew-symmetric tensors.” Invited in the section "Applied Algebraic Geometry AG17".
24–29 Sep 2018 — Varieties and Group Actions (IMPAN, Warsaw, Poland). “Skew-symmetric tensor decomposition.” (Program: Varieties: Arithmetic and Transformations.)
12–16 Nov 2018 — Computational Nonlinear Algebra (ICERM Conference, Brown University, Providence, USA). “Algorithms for rank, tangential rank and cactus rank of a homogeneous polynomial.”
18 May 2021 — Workshop IV: Efficient Tensor Representations for Learning and Computational Complexity (IPAM, UCLA, USA). “On the Dimension of Tensor Network Varieties.”
2–3 Dec 2021 — A New Paradigm for Sustainable Development in Agriculture: Mathematics & AI get into the Field (Milano, Italy). “High order singular value decomposition for plant diversity estimation.”
18–22 Jul 2022 — AMS–SMF–EMS Joint International Meeting (Grenoble, France). “On the regularity of certain apolar schemes.”
22–26 Aug 2022 — AEWM General Meeting 2022 (Finland, Aalto University). “On the dimension of Tensor Network Varieties.”
20 Sep 2022 — AGATES Kickoff workshop (IMPAN, Warsaw, Poland). “On the regularity of certain apolar schemes.”
4–7 Oct 2022 — AGATES Workshop “Tensors from the physics viewpoint” (IMPAN, Warsaw, Poland). “On the dimension of Tensor Network Varieties.”
9–12 May 2023 — Secant vs Cactus Workshop (Toulouse, France). “Algorithms for rank and cactus decomposition of polynomials.”
22–24 May 2023 — Commutative Algebra Towards Applications (Torino, Italy). “Cactus rank behind complexity theory and quantum physics.”
24–26 Jul 2023 — Third Meeting of Women in Mathematics (University of Coimbra, Portugal). (Representative of UMI at the round-table discussion among Presidents of EU Math societies)
10–13 Jul 2023 — Homemade Algebraic Geometry (Arrondo 60) (Alcalá de Henares, Madrid, Spain). “Homemade recipe for apolar natural schemes.”
14 Dec 2023 — Tensor Methods Reunion Conference 2 (TMRC2) (Lake Arrowhead, Los Angeles, USA). “Generalized Additive Decomposition.”
8 Feb 2024 — Tensor Networks (TNK2024) (IPAM, Los Angeles, USA). “Defective Tensor Network varieties.”
6–10 May 2024 — The Geometry of Hilbert Schemes of Points (Levico Terme (TN), Italy). “Schemes evinced by generalized additive decompositions and their regularity.”
13–17 May 2024 — JM Invariant at 60: A Conference on Tensor Invariants in Geometry and Complexity Theory (Auburn University, USA). “On the dimension of certain Cactus varieties.”
3 Aug 2024 — BM60: A Workshop for Bernard Mourrain's 60th Birthday (Leipzig, Germany). “Overview of Bernard's Algorithm for Polynomial Decomposition: Advances and Developments.”
6–7 Feb 2025 — Women in Mathematics 2025 (Palermo, Italy). “Geometric and Algebraic Tools in Interdisciplinary Research.”
10–12 Feb 2025 — Gianfranco Casnati's legacy (Torino, Italy). “Generalized additive decompositions and the regularity of schemes.”
24 Jun 2025 — Jordan Types of Artinian Algebras and Geometry of Punctual Hilbert Schemes (Iarrobino 80) (Nice, France). “A refinement of the local cactus algorithm.”
10 Jul 2025 — SIAM AG meeting (session “Tensors in Algebraic Geometry”) (Madison, USA). “On Tensor Network varieties.”
11 Dec 2025 — Semester Program: Linear Algebra and Complexity Theory (Simons Institute, Berkeley, California, USA). “From Apolar Schemes to Generalized Affine Decompositions.”

Invited talks in Italian and foreign universities (short selection)

18 Feb 2005 — Departamento de Álgebra, Universidad Complutense de Madrid (Madrid, Spain). “Secant varieties to osculating varieties of Veronesean.”
7 Oct 2005 — Mathematical Department, Texas A&M University (College Station, Texas, USA). “Secant varieties and Big Waring Problem.”
29 May 2006 — Giornate di Geometria Algebrica e argomenti correlati VIII (Univ. Trieste, Italy). “Varietà delle secanti a varietà che parametrizzano forme ottenute come prodotto di forme lineari.”
4 Sep 2008 — Departamento de Álgebra, Universidad Complutense de Madrid (Madrid, Spain). “Secant varieties to osculating varieties of Veronese Varieties.”
28 Oct 2008 — Univ. Bologna (Italy). “Rappresentazione di varietà algebriche.”
13 Mar 2009 — Univ. Firenze (Italy). “Varietà che parametrizzano polinomi completamente decomponibili.”
10 Jun 2009 — Workshop on tensors and interpolation (Nice, France). “Sylvester's Algorithm.”
10 Dec 2009 — Univ. Trento (Italy). “Dal problema di Waring alle telecomunicazioni.”
18 Mar 2010 — SAGA Winter School (Auron, Nice, France). “From the Waring problem to tensor rank through secant varieties.”
20 Apr 2010 — Univ. Ancona (Italy). “Dal problema di Waring alle telecomunicazioni.”
17 Jun 2010 — Univ. Trento (Italy). “Un assaggio di scienza nell'iconografia russa.”
7 Oct 2010 — Univ. Trieste (Italy). “Varietà delle secanti a varietà che parametrizzano tensori: attualità del problema di Waring, aspetti geometrici correlati ed applicazioni.”
22 Mar 2011 — GALAAD–INRIA, Sophia Antipolis Méditerranée (France). “Polynomial and Tensor Decompositions.”
9 May 2011 — Multimedia Geometry Seminars, Univ. Trento (Italy). “Decomposition of Structured Tensors, Algorithms and Characterization.”
23 May 2011 — Politecnico di Torino (Italy). “Varietà delle secanti: dimensioni, ideali e rango di tensori.”
26 Oct 2011 — Univ. Trento (Italy). “Tensor decompositions: achievements and developments.”
16 Nov 2011 — Univ. Torino (Italy). “Ranghi di Tensori.”
2 Dec 2011 — Univ. Firenze (Italy). “Decomposition of partially symmetric tensors.”
4 Apr 2012 — Univ. Bologna (Italy). “Tensor Decomposition: a link between Algebraic Geometry and Applications.”
23 Oct 2012 — Univ. Pau (France). “Various approaches for polynomial decomposition.”
4 Dec 2012 — Universidad Complutense de Madrid (Spain). “A generalization of Sylvester Algorithm.”
5 Sep 2024 — University of Bern (Switzerland). “On Generalized decompositions of a Polynomial.”
3 Dec 2025 — San Francisco State University (USA). “Generalized decompositions of a Polynomial.”

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