My research interests are in the field of Algebraic Geometry, Algebra and their applications. In particular:

- Secant varieties, their dimensions and ideals;
- 0-dimensional schemes and their postulations;
- Varieties parameterizing polynomials and/or tensors both in the complex case and in the real case (Veronese variety, Segre variety, Grassmannians, Flag varieties, Homogeneous varieties);
- Rank of symmetric tensors and structured tensors;
- Uniqueness of the decomposition of a tensor;
- algebraic and numerical algorithms for the tensor decomposition both in the complex and real case;
- Applications to telecommunications, complexity theory, data analysis, phylogenetics and physics, in paticular I am recently interested in application to Quantum Information.

The main objective of my present research is the kick-off of an independent research line under my own responsibility on the topics on which I have accumulated international experience in the first stages of my scientific career, namely the DECOMPOSITION OF STRUCTURED TENSORS and the COMPUTATION OF THEIR STRUCTURED RANK.

The Tensor Decomposition (TD) problem from linear and multilinear algebra point of view consists of writing a structured tensor as a minimal linear combination of r indecomposable tensors of the same structure, r being the rank. In geometrical terms, dealing with rank 1 structured tensors, corresponds to studying subvarieties of Segre varieties that parametrize rank 1 tensors of a certain structure.

One of the central problems in this field is the determination of ALGORITHMS to compute the structured rank of a given tensor. Up to now, the only available ones are the classical Sylvester algorithm for complex symmetric tensors in 2 variables and its modern generalization to partially symmetric tensors, developed during my stay at INRIA with Brachat, Comon and Mourrain. My scientific project is to proceed further investigating other cases, starting from skew-symmetric ones.

The main geometric tool to tackle these problems are SECANT VARIETIES that allow to naturally study a closely related concept of rank, the so called border rank. The most direct strategy to know the border rank of a structured tensor would be to test it in the equations of certain secant varieties. Despite the great interest that the mathematical community has dedicated to this area for decades, the determination of the equations of secant varieties is among the most significant open problems even from a pure algebraic geometric point of view. So far, all available techniques to compute IDEALS OF SECANT VARIETIES of varieties parameterizing tensors (VPT) combine algebraic geometry and representation theory in group theory. As varieties of this kind are homogeneous varieties for the action of some group, their ideal can be described in terms of irreducible Schur modules invariant for the action of the same group. I have learned those techniques during my visits at the Texas A&M University with Prof. JM Landsberg who has been one of the firsts that brought them into the field of TD. Moreover I have already had the opportunity of helping PhD students, both mathematicians and applied ones, in tackling TD open problems during my TA for the PhD course at MSRI -- Berkeley (2008).

Another project that I have is to further exploit these techniques to obtain new results on specific secant varieties.

Another problem that has stimulated important advances in this field is the one of the dimensions of secant varieties, which has led to the introduction of concepts such as APOLARITY and 0-DIMENSIONAL HILBERT SCHEMES to this context. I plan to extend the concept of apolarity to more general classes of structured tensors starting from skew-symmetric ones. This will serve to extend the Alexander-Hirschowitz theorem and determine the dimension of secant varieties of various VPT, as well as to write algorithms for computing the rank of the corresponding structured tensors. Apolarity and 0-dimensional Hilbert schemes naturally appear in generalized singular value decompositions based on Henkel matrices, which are the key tool of all existing TD algorithms.

The algorithmic part of my projects started firstly with pure algebraic methods for the decomposition of symmetric and partially symmetric tensors. Now, thanks to my international network, I am becoming interested also in the numerical side of this problem, in particular I am developing a numerical algorithm that will allow to find the solution for TD with the software Bertini. I have built up this project together with one of the developers of the software Bertini (in particular J. Hauenstein) and with B. Mourrain. This is an ambitious but very realistic project and we will involve PhD students and/or Post Doc's that would be interested in it.

The invitation to participate at the workshop in Palo Alto during the summer 2008 gave me a more insite interest in the APPLICATIONS (multilinear techniques for data analysis in signal processing for telecommunication; algebraic statistics; geometric approaches for the P?=NP problem; hidden variables problems in phylogenetics and medical engineering; entanglement in quantum information theory).

This interest has been its first realization in the writing and winning of my Marie Curie project at INRIA in collaboration with B. Mourrain and P. Comon in the telecommunications field (the knowledge of the TD of a tensor allows to solve problems of Blind Identification and of Tensor polyadic decomposition for Antenna Array Processing).

I will pursue this direction by a long visit at Grenoble CNRS in the equipe directed by P. Comon.

Another applied side that I intend to work on will be the one on the effective decomposition of noisy tensors (namely tensors coming from concrete data analysis). I will work on this together with L. De Lathauwer (KU Leuven, Belgio) who has already written a package in MATLAB for the TD of noisy tensors. I will also want to tackle the problem of identifiability of a noisy tensor together with N. Vannieuwenhoven (PhD student at KU Leuven, Belgio) who is already working on this from a more computational point of view and who has recently contacted me for an interesting collaboration.

## Meetings and Conferences

#### Conferences and lectures

**Talks in Italian and international conferences**

“Osculating varieties of Veronesean and their higher secant varieties”, December 10, 2004 - CMS 2004 Winter Meeting, Montreal (Quebec, Canada).

“Variet`a delle secanti a variet`a che parametrizzano forme ottenute come prodotto di forme lineari”, May 29, 2006, Giornate di Geometria Algebrica e argomenti correlati VIII, Univ. Trieste.

**[Invited]** “Secant Varieties and Ideals of varieties parameterizing certain symmetric tensors”, July 17, 2008, MSRI (Mathematical Sciences Research Institute) (Berkeley, California, USA).

“Sylvester’s Algorithm”, June 10, 2009, Workshop on tensors and interpolation, Nice, France.

“From the Waring problem to tensor rank through secant varieties”, March 18, 2010, SAGA Winter School, Auron, Nice, France.

“Decomposition of Homogeneous Polynomials”, September 15, 2010, Workshop on Tensor Decompositions and Applications (TDA 2010). September 13–17 2010. Monopoli, Bari, Italy.

“Applicazioni recenti di risultati classici su variet`a delle secanti a variet`a che parametriz- zano tensori. Dal problema di Waring al rango di tensori”, November 22, 2010, Progressi Recenti in Geometria Reale e Complessa, October 17–22, 2010, Levico Terme (Trento, Italy).

“Secant varieties and Rank of tensors”, February 1, 2011, Mittag-Leffler Institute, Spring Semester: “Algebraic Geometry with a view towards applications” 17 January – 15 June 2011.

**[Invited]** “Ranks of Tensors, related varieties and applications”, November 18, 2011, Genova- Torino-Milano Seminar: some topics in Commutative Algebra and Algebraic Geometry, November 17–18, 2011, Milano (Italy).

**[Invited]** “Algebraic Geometry in Signal processing, Phylogenetic and Quantum Physics”, Collo- quium Politecnico di Torino, May 30, 2013, Politecnico di Torino, Italy.

**[Invited]** “Tensor Ranks”, 2013 SIAM Conference on Applied Algebraic Geometry. August 1, 2013, Fort Collins (Colorado, USA).

**[Invited] Main Speaker** at the 36th Autumn School in Algebraic Geometry on “Power sum decomposition and apolarity, a geometric approach”. September 1-7, 2013, Lukecin, Poland.

**[Invited]**, [Declined for family reasons (maternity leave)] Invited Speaker to the “Tensors and Optimization” Conference, May 19–22, 2014 for the SIAM Optimization Meeting, San Diego (CA, USA).

**[Invited]**, [Declined for family reasons (maternity leave)] Invited Speaker for Computational Nonlinear Algebra, for the ICERM Conference, June 2–6, 2014, Brown University (Providence, USA).

**[Invited]** “Cactus Varieties of Cubic Forms: Apolar Local Artinian Gorenstein Rings”, November 13, 2014, inside the Workshop Tensors in Computer Science and Geometry in the framework of the fall program 2014 ”Algorithms and Complexity in Algebraic Geometry“, Invited by P. Bu ̈rgisser, JM Landsberg, K. Mulmuley, B. Sturmfels.

“On the cactus variety of cubic forms”. AMS-EMS-SPM Joint meeting (Porto), June 10-13, 2015.

**[Invited] Invited Speaker** at MEGA Effective Methods in Algebraic Geometry, “Tensor decom- position and homotopy continuation”. Trento, 15–19 June, 2015.

“A geometric view of the splitting type for plane rational curves”. September 8, 2015, Convegno UMI, September 7–12, 2015, Siena, Italy.

**[Invited] Invited Speaker** at MAG2015, December 2–4, 2015, Barcellona, Spain.

"Tensor Decomposition and Homotopy Continuation", Workshop on Tensor Decompositions and Applications (TDA 2016) January 18-22, 2016, Leuven, Belgium.

**[Invited]** "Fat points schemes", Research Station on Commutative Algebra June 13-17, 2016, Seoul, Corea.

**[Invited]** "Tensor Decomposition and Homotopy Continuation", Tensors: Algebra meets Numerics, Max Planck Institute for Mathematics in the Sciences, December 12-14, 2016, Leipzig, Germany.

**[Invited]** "Tensor decomposition and homotopy continuation", British Mathematical Colloquium, April 3-7, 2017, Durham, UK.

"On the identifiability of skew-symmetric tensors", SIAM 2017, Georgia Institute of Technology, Atlanta GA, USA; July 31-August 4, 2017.

**Invited Talks in Italian and foreign Universities**

“Secant varieties to osculating varieties of Veronesean” , February 18, 2005 - Departamento de A ́lgebra, Universidad Complutense de Madrid. (Madrid, Spain).

“Secant varieties and Big Waring Problem”, October 7, 2005, Mathematical Department, Texas A&M University (College Station,Texas, USA).

“Secant varieties to osculating varieties of Veronese Varieties”, September 4, 2008 - Departamento de A ́lgebra, Universidad Complutense de Madrid, (Madrid, Spain).

“Rappresentazione di varietà algebriche”, October 28, 2008, Univ. Bologna (Italy).

“Variet`a che parametrizzano polinomi completamente decomponibili”, March 13, 2009, Univ. Firenze (Italy).

“Sylvester’s Algorithm”, June 10, 2009 - Workshop on tensors and interpolation, Nice (France).

“Dal problema di Waring alle telecomunicazioni”, December 10, 2009, Univ. Trento (Italy).

“Dal problema di Waring alle telecomunicazioni”, April 20, 2010, Univ. Ancona (Italy).

“Un assaggio di scienza nell’iconografia russa”, June 17, 2010, Univ.Trento (Italy).

“Varietà delle secanti a variet`a che parametrizzano tensori: attualità del problema di Waring, aspetti geometrici correlati ed applicazioni”, October 7, 2010. Univ. Trieste (Italy).

“Polynomial and Tensor Decompositions”, March 22, 2011, GALAAD–INRIA, Sophia Antipolis M ́editerran ́ee, France.

“Decomposition of Structured Tensors, Algorithms and Characterization”, May 9, 2011, Multimedia Geometry Seminars, Univ. Trento (Italy).

“Varietà delle secanti: dimensioni, ideali e rango di tensori”, May 23, 2011, Poltiecnico di Torino (Italy).

“Tensor decompositions: achievements and developments”, October 26, 2011, Univ. Trento (Italy).

“Ranghi di Tensori”, November 16, 2011, Univ. Torino (Italy).

“Decomposition of partially symmetric tensors”, December 2, 2011, Univ. Firenze (Italy).

“Tensor Decomposition: a link between Algebraic Geometry and Applications”, April 4, 2012, Univ. Bologna (Italy).

“Various approaches for polynomial decomposition”, October 23, 2012, Univ. Pau (France).

“A generalization of Sylvester Algorithm”, December 4, 2012, Universidad Complutense de Madrid (Spain).

“On the local cactus rank of generic cubic forms”, December 4, 2014, Simons Institute for Theory of Computing (Berkeley, CA, USA), seminar in the framework of the fall program 2014 “Algorithms and Complexity in Algebraic Geometry”.

"Sul rango tipico delle forme ternarie", May, 13 2016, Bologna, in the framework of the work group ``Geometria Algebrica Reale e Tensori a.a. 2015-2016."

"ABC dell'Entanglement", December 19, 2016, Bologna, in the framework of the work group ``Geometria Algebrica e Tensori 2016-2017".

**Posters**

"Tensor decomposition and homotopy continuation", February 13, 2017, METT VII, Univ. Pisa (Italy).

**Other presentations **

“Sulle funzioni convesse”, February 27, 2002, Univ. Trieste (Italy).

“Dimostrazione del teorema di Darboux”, September 27, 2002, Univ. Trieste (Italy).

“Programma di Sarkisov in dimensione 2 per la classificazione degli Spazi Fibrati di Mori secondo la Teoria di Mori”, July 18, 2002, Univ. Milano (Italy).

“Esposizione dell’articolo di G. Canuto Curve associate e Formule di Pluker nelle Grassmaniane, apparso su “Inventiones Mathematicae”, 53, 77-90 (1979)”, January 15, 2003, Univ. Milano (Italy).

“How one’s can calculate all the differential invariants of Seg(Pn × Pn) ∩ H, where H is a generic hyperplane. Understand this as a homogeneous variety of Sln+1C”, February 13, 2003, Univ. Trieste (Italy).

“Un’introduzione al problema dello studio della Postulazione dei Punti Grassi”, March 19, 2003, Univ. Milano (Italy).

“Un’introduzione al problema dello studio della Postulazione dei Punti Grassi e recenti appli- cazioni”, May 23, 2003, Univ. Pavia (Italy).

“Waring type problems and Auxiliary varieties Associated to Veronese varieties”, October 6, 2004, Queen’s University (Kingston, Ontario, Canada).

“Secant varieties to the Osculating varieties of the Veronesean”, October 13, 2004, Queen’s Univer- sity (Kingston, Ontario, Canada).

“Varietà delle secanti alle Veronese e applicazioni algebriche”, January 26, 2005, Departamento de A ́lgebra Universidad Complutense de Madrid (Madrid, Spain).

“Varietà delle secanti alle varietà tangenziali ed osculanti a varietà di Veronese”, February 2, 2005, Departamento de Algebra,Universidad Complutense de Madrid (Madrid,Spain).

“Construction of Cominuscule Varieties”, October 6, 2005, Texas A&M University (College Station, Texas, USA).

“An introduction to Representation Theory”, November 2, 2005, Texas A&M University (College Station, Texas, USA).

“An introduction to de Rham Cohomology I, II, III”, November 17, 18, 22, 2005, Texas A&M University (College Station, Texas, USA).

“On Alexander-Hirshowitz theorem via Lemma d’Horace”, December 1, 2005, Texas A&M Univer- sity (College Station, Texas, USA).

“Lemma d’Horace differentielle”, December 5, 2005, Texas A&M University (College Station, Texas, USA).

“Dall’Algebra Lineare a questioni irrisolte”, May 15, 2008, Talk inside the “Corso di Laurea Algebra Superiore”, Dipartimento di Matematica, Univ. Bologna (Italy).

“Ideale delle variet`a di Segre-Veronese”, June 12, 2008, Univ. Genova (Italy).

“Rango e rango simmetrico di tensori simmetrici.”, March 3, 2009 , Univ. Bologna (Italy).

“Algorithms for computing the rank of a tensor”, February 11, 2011, Mittag-Leffler Institute, Spring Semester: “Algebraic Geometry with a view towards applications” 17 Janaury – 15 June 2011 (Sweeden).

“Tenseurs”, March 8, 2011, GALAAD–INRIA, Sophia Antipolis Mediterranee (France).

"Geometry of tensor decomposition starting from spin squeezed states" October, 28 2016, Trento (Italy).

"Un po' di matematica per i Tensor Network con breve introduzione fisica." December, 1, 2017, Trento (Italy).

**Additional Conferences and Schools Attended **

“Summer School Perugia”, Perugia (Italy), July 29 – September 1, 2001.

“Pragmatic 2003”, Catania (Italy), June 9 – 28, 2003.

“Interpolation problem and Projective embeddings”, Torino (Italy), September 15 – 20, 2003;

“Workshop on Algebraic curves, monodromy and related topics”, Milano (Italy), April 1-2, 2004.

“International school on Projective Geometry”, Anacapri (Italy), June 1–5, 2004.

“Projective Varieties with unexpected geometric properties”, Siena (Italy), June 8–12, 2004.

“School/WorkshoponComputationalAlgebraforAlgebraicGeometryandstatistics”,Torino(Italy), September 6 – 11, 2004.

“Rt. 81 conference in honor of Graham Evans and Workshop on Resolution (for young researchers)”, Cornell University of Ithaca, New York, USA, October 1–3, 2004.

“AGaFE, Geometry of Algebraic Varieties”, Ferrara (Italy), June 22–25, 2005.

“Texas Geometry and Topology conference”, Austin, Texas (USA), September 30 – October 2, 2005.

“Geometric and Probabilistic Methods in group theory and dynamical systems”, November 4–6, 2005, Texas A&M University, College Station, Texas (USA).

“Harvey/Polking conference, Singularities in Analysis, Geometry and Topology”, November 11–13, 2005, Rice University, Houstin, Texas (USA).

“INDAM workshop: Geometry of projective varieties” (Roma), September 30 – October 4, 2008.

“Conference on Classical and recent aspects in the study of projective varieties. In honor of Lucian Badescu on the occasion of his 65th birthday”, Genova (Italy), January 21–22, 2010.

“INdAM Conference “Complex Geometry””, Levico Terme, Trento (Italy), May 31 – June 4, 2010.

“Summer school: Geometry of tensors and applications”, Sophus Lie Conference Center, Nord- fjordeid (Norway), June 14 – 18, 2010.

“School (and Workshop) on The Minimal Model Program and Shukurov’s ACC Conjecture”, Povo (Trento), June 5 – 10, 2010.

“International Conference on Perspectives on Algebraic Varieties”, Levico Terme, Trento (Italy), September 5–10, 2010.

“Algebraic Geometry in the sciences”, Oslo (Norway), January 10–14, 2011.

“CIAM workshop: An afternoon of biology and mathematics”, KTH, Stockholm (Sweden), February 4, 2011.

“Solving polynomial equations”, KTH, Stockholm (Sweden), February 21–25, 2011.

“MEGA 2011: Effective Methods in Algebraic Geometry” , Stockholm University (Stockholm, Sweden), May 30–June 3, 2011.

“Genova, Torino, Milano Seminar: Some Topics in Commutative Algebra and Algebraic Geometry”, June 28–29, 2012, Torino (Italy).

“Groebner Bases, Curves, Codes and Cryptography”, July 30 – August 10, 2012, Trento (Italy).

“School (and Workshop) on Invariant Theory and Projective Geometry”, September 17 – 22, 2012, Trento (Italy).

“3rd SAGA Workshop”, October, 9–11, 2012, Trento (Italy).

“Genova-Torino-Milano Seminar: Some Topics in Commutative Algebra and Algebraic Geometry”, January 28–29, 2014 (Politecnico di Milano, Italy).

“Vector Bundles Days II, Pau-Trieste Workshop on Vector Bundles and Related Topics. On the occasion of Emilia Mezzetti’s 60th birthday”, January 29–31, 2014 (Trieste, Italy).

## Memberships

#### Memberships in societies and scientific committees

- From January 1st, 2017. Member of the editorial board of the SIAM Journal on Matrix Analysis and Applications (SIMAX).
- From June 2017, Member of the Advisory Board of MEGA (Effective Methods in Algebraic Geometry).
- From April 2015 to April 2019. Segretario aggiunto of the UMI (Unione Matematica Italiana).
- Form September 2015. Member of the UMI Working Group: "Gruppo Risorse Umane".
- From September 2015. Member of the editorial committee of the electronic journal: "UMI Newsletter" and NUMI “Notiziario dell’Unione Matematica Italiana”
- President of the commission for Tricerri Prize - UMI (2017).
- Member of the commission for Enriques Prize - UMI (2017).
- President of the commission for Cotoneschi Prize - UMI (2017).
- President of the commission for Baldassarri Prize - UMI (2018).

## Research interests

#### Research interests

My research interests are in the field of Algebraic Geometry, Algebra and their applications. In particular:

*Secant varieties, their dimensions and ideals; 0-dimensional schemes and their postulations; Varieties parameterizing polynomials and/or tensors both in the complex case and in the real case (Veronese variety, Segre variety, Grassmannians, Flag varieties, Homogeneous varieties); Rank of symmetric tensors and structured tensors; Uniqueness of the decomposition of a tensor; algebraic and numerical algorithms for the tensor decomposition both in the complex and real case; Applications to quantum information, telecommunications, complexity theory, data analysis, phylogenetics and physics. *

The main objective of my present research is the kick-off of an independent research line under my own responsibility on the topics on which I have accumulated international experience in the first stages of my scientific career, namely the DECOMPOSITION OF STRUCTURED TENSORS and the COMPUTATION OF THEIR STRUCTURED RANK.

The Tensor Decomposition (TD) problem from linear and multilinear algebra point of view consists of writing a structured tensor as a minimal linear combination of *r* indecomposable tensors of the same structure, *r* being the rank. In geometrical terms, dealing with rank 1 structured tensors, corresponds to studying subvarieties of Segre varieties that parametrize rank 1 tensors of a certain structure.

One of the central problems in this field is the determination of ALGORITHMS to compute the structured rank of a given tensor. Up to now, the only available ones are the classical Sylvester algorithm for complex symmetric tensors in 2 variables and its modern generalization to partially symmetric tensors, developed during my stay at INRIA with Brachat, Comon and Mourrain. My scientific project is to proceed further investigating other cases, starting from skew-symmetric ones.

The main geometric tool to tackle these problems are SECANT VARIETIES that allow to naturally study a closely related concept of rank, the so called border rank. The most direct strategy to know the border rank of a structured tensor would be to test it in the equations of certain secant varieties. Despite the great interest that the mathematical community has dedicated to this area for decades, the determination of the equations of secant varieties is among the most significant open problems even from a pure algebraic geometric point of view. So far, all available techniques to compute IDEALS OF SECANT VARIETIES of varieties parameterizing tensors (VPT) combine algebraic geometry and representation theory in group theory. As varieties of this kind are homogeneous varieties for the action of some group, their ideal can be described in terms of irreducible Schur modules invariant for the action of the same group. I have learned those techniques during my visits at the Texas A&M University with Prof. JM Landsberg who has been one of the firsts that brought them into the field of TD. Moreover I have already had the opportunity of helping PhD students, both mathematicians and applied ones, in tackling TD open problems during my TA for the PhD course at MSRI – Berkeley (2008).

Another project that I have is to further exploit these techniques to obtain new results on specific secant varieties.

Another problem that has stimulated important advances in this field is the one of the dimensions of se- cant varieties, which has led to the introduction of concepts such as APOLARITY and 0-DIMENSIONAL HILBERT SCHEMES to this context. I plan to extend the concept of apolarity to more general classes of structured tensors starting from skew-symmetric ones. This will serve to extend the Alexander- Hirschowitz theorem and determine the dimension of secant varieties of various VPT, as well as to write algorithms for computing the rank of the corresponding structured tensors. Apolarity and 0-dimensional Hilbert schemes naturally appear in generalized singular value decompositions based on Henkel matrices, which are the key tool of all existing TD algorithms.

The algorithmic part of my projects started firstly with pure algebraic methods for the decomposition of symmetric and partially symmetric tensors. Thanks to my international network, I become interested also in the numerical side of this problem, in particular I developed a numerical algorithm that allows to find the solution for TD with the software Bertini. I have built up this project together with one of the developers of the software Bertini (J. Hauenstein), with B. Mourrain and N. Daleo.

The invitation to participate at the workshop in Palo Alto during the summer 2008 gave me a more insite interest in the APPLICATIONS (multilinear techniques for data analysis in signal processing for telecommunication; quantum information; algebraic statistics; geometric approaches for the P?=NP problem; hidden variables problems in phylogenetics and medical engineering; entanglement in quantum information theory).

This interest has been its first realization in the writing and winning of my Marie Curie project at INRIA in collaboration with B. Mourrain and P. Comon in the telecommunications field (the knowledge of the TD of a tensor allows to solve problems of Blind Identification and of Tensor polyadic decomposition for Antenna Array Processing).

Another application that I am interested in is QUANTUM INFORMATION. I firstly started studying this topic together with I. Carusotto with whom we focused on the possibility of using tensor rank as a measure of entanglement. I pursued my interest in quantum information and I began organize a joint seminar together with CNR “Quantum Information, Algebra and Geometry Workgroup” which is a cycle

of interdisciplinary meetings in which we try to lay the foundations for a common language between Geometry, Algebra, Analysis and Physics starting from Quantum Information: https://me.unitn.it/alessandra-bernardi/research/quantum-information-algebra-and-geometry-workgroup. I wrote a project for “Programma professori visitatori” that was entirely financed by INDAM where I organized a PhD Course in “Quantum Information Theory and Geometry” with teaching Professor: J.M. Landsberg at the Univ. Trento in June-July 2017. I co-organized together with E. Ballico, I. Carusotto, S. Mazzucchi, V. Moretti an International workshop on Quantum Physics and Geometry, in Levico Terme (Trento, Italy). July 4-6, 2017. With the same team we are editing a volume of for the Lecture Notes of the Unione Matematica Italiana entitled Quantum Information and Geometry where we are collecting various contributes for an interdisciplinary introduction to quantum information for mathematicians and physics (Luca Chiantini, Frédéric Holweck, J.M. Landsberg, FM Ciaglia, Alberto Ibort, Giuseppe Marmo, Davide Pastorello, Bassano Vacchini).

My approach to the problem of measuring the entanglement via the classification of tensor rank is well visible in a paper that I wrote together with D. Vanzo “A new class of non-identifiable skew-symmetric tensors” (recently accepted by Annali di Matematica https://doi.org/10.1007/s10231-018-0734-z) where we started studying the uniqueness of skew-symmetric tensor decomposition of certain skew sym- metric tensors, one of the techniques we used was the classification of the orbit closures of SL(8) in P( 3 C8), which has a very nice physical interpretation: it gives precisely the stratification of the entanglement measure of a 3 fermions in* C^{8}*.

Nowadays I am working together with E. Ballico, M. Christandl and F. Gesmundo on W-states and ten- sor ranks submultiplicativity: In case we have two diiffrent species of indistinguishable bosonic particles, the relevant Hilbert space is

*S*. Tensor rank is a natural measure of the entanglement

^{N1}H_{1}⊗ S^{N2}H_{2}of the corresponding quantum state and strict submultiplicativity of partially symmetric rank reflects the unexpected fact that entanglement does not simply “add up” in the composite system formed by multiple bosonic systems, even if the states

*T ∈ S*of the two species is a tensor product

^{N1}_{1}H_{1}⊗ S^{N2}H_{2}*T = T*, where

_{1}⊗ T_{2}*T*The results of this paper will expand on this novel quantum effect.

_{i}∈ S^{Ni}H_{i}## Research activities

#### Research work

My research interests are in the field of Algebraic Geometry, Algebra and their applications. In particular:

*Secant varieties, their dimensions and ideals; 0-dimensional schemes and their postulations; Varieties parameterizing polynomials and/or tensors both in the complex case and in the real case (Veronese variety, Segre variety, Grassmannians, Flag varieties, Homogeneous varieties); Rank of symmetric tensors and structured tensors; Uniqueness of the decomposition of a tensor; algebraic and numerical algorithms for the tensor decomposition both in the complex and real case; Applications to telecommunications, complexity theory, data analysis, phylogenetics and physics.*

The main objective of my present research is the kick-off of an independent research line under my own responsibility on the topics on which I have accumulated international experience in the first stages of my scientic career, namely the DECOMPOSITION OF STRUCTURED TENSORS and the COMPUTATION OF THEIR STRUCTURED RANK.

The Tensor Decomposition (TD) problem from linear and multilinear algebra point of view consists of writing a structured tensor as a minimal linear combination of r indecomposable tensors of the same structure, r being the rank. In geometrical terms, dealing with rank 1 structured tensors, corresponds to studying subvarieties of Segre varieties that parametrize rank 1 tensors of a certain structure.

One of the central problems in this eld is the determination of ALGORITHMS to compute the structured rank of a given tensor. Up to now, the only available ones are the classical Sylvester algorithm for complex symmetric tensors in 2 variables and its modern generalization to partially symmetric tensors, developed during my stay at INRIA with Brachat, Comon and Mourrain. My scientic project is to proceed further investigating other cases, starting from skew-symmetric ones.

The main geometric tool to tackle these problems are SECANT VARIETIES that allow to naturally study a closely related concept of rank, the so called border rank. The most direct strategy to know the border rank of a structured tensor would be to test it in the equations of certain secant varieties. Despite the great interest that the mathematical community has dedicated to this area for decades, the determination of the equations of secant varieties is among the most signicant open problems even from a pure algebraic geometric point of view. So far, all available techniques to compute IDEALS OF SECANT VARIETIES of varieties parameterizing tensors (VPT) combine algebraic geometry and representation theory in group theory. As varieties of this kind are homogeneous varieties for the action of some group, their ideal can be described in terms of irreducible Schur modules invariant for the action of the same group. I have learned those techniques during my visits at the Texas A&M University with Prof. JM Landsberg who has been one of the rsts that brought them into the field of TD. Moreover I have already had the opportunity of helping PhD students, both mathematicians and applied ones, in tackling TD open problems during my TA for the PhD course at MSRI - Berkeley (2008). Another project that I have is to further exploit these techniques to obtain new results on specic secant varieties.

Another problem that has stimulated important advances in this eld is the one of the dimensions of secant varieties, which has led to the introduction of concepts such as APOLARITY and 0-DIMENSIONAL HILBERT SCHEMES to this context. I plan to extend the concept of apolarity to more general classes of structured tensors starting from skew-symmetric ones. This will serve to extend the Alexander-Hirschowitz theorem and determine the dimension of secant varieties of various VPT, as well as to write algorithms for computing the rank of the corresponding structured tensors. Apolarity and 0-dimensional Hilbert schemes naturally appear in generalized singular value decompositions based on Henkel matrices, which are the key tool of all existing TD algorithms.

The algorithmic part of my projects started rstly with pure algebraic methods for the decomposition of symmetric and partially symmetric tensors. Now, thanks to my international network, I am becoming interested also in the numerical side of this problem, in particular I am developing a numerical algorithm that will allow to nd the solution for TD with the software Bertini. I have built up this project together with one of the developers of the software Bertini (in particular J. Hauenstein) and with B. Mourrain. This is an ambitious but very realistic project and we will involve PhD students and/or Post Doc's that would be interested in it.

The invitation to participate at the workshop in Palo Alto during the summer 2008 gave me a more insite interest in the APPLICATIONS (multilinear techniques for data analysis in signal processing for telecommunication; algebraic statistics; geometric approaches for the P?=NP problem; hidden variables problems in phylogenetics and medical engineering; entanglement in quantum information theory). This interest has been its first realization in the writing and winning of my Marie Curie project at INRIA in collaboration with B. Mourrain and P. Comon in the telecommunications field (the knowledge of the TD of a tensor allows to solve problems of Blind Identication and of Tensor polyadic decomposition for Antenna Array Processing). I will pursue this direction toghether with the equipe directed by P. Comon. Another applied side that I intend to work on will be the one on the effective decomposition of noisy tensors (namely tensors coming from concrete data analysis). I will work on this together with L. De Lathauwer (KU Leuven, Belgio) who has already written a package in MATLAB for the TD of noisy tensors.