Alessandra Bernardi
Education 



Academic career and teaching activities 

Employment
Teaching Activity


Research interests 

My research interests are in the field of Algebraic Geometry, Algebra and their applications. In particular: Secant varieties, their dimensions and ideals; 0dimensional 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 kickoff 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 skewsymmetric 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 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 0DIMENSIONAL HILBERT SCHEMES to this context. I plan to extend the concept of apolarity to more general classes of structured tensors starting from skewsymmetric 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 0dimensional 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). 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 

Research work 

My research interests are in the field of Algebraic Geometry, Algebra and their applications. In particular: Secant varieties, their dimensions and ideals; 0dimensional 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 kickoff 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 skewsymmetric 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 0DIMENSIONAL HILBERT SCHEMES to this context. I plan to extend the concept of apolarity to more general classes of structured tensors starting from skewsymmetric ones. This will serve to extend the AlexanderHirschowitz 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 0dimensional 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. 

Memberships in societies and scientific committees 



Awards and honours 



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, MittagLeffler 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 TorinoMilano 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 17, 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”. AMSEMSSPM Joint meeting (Porto), June 1013, 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 1822, 2016, Leuven, Belgium. [Invited] "Fat points schemes", Research Station on Commutative Algebra June 1317, 2016, Seoul, Corea. [Invited] "Tensor decomposition and homotopy continuation", British Mathematical Colloquium, April 37, 2017, Durham, UK. "On the identifiability of skewsymmetric tensors", SIAM 2017, Georgia Institute of Technology, Atlanta GA, USA; July 31August 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. 20152016." "ABC dell'Entanglement", December 19, 2016, Bologna, in the framework of the work group ``Geometria Algebrica e Tensori 20162017". 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, 7790 (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 AlexanderHirshowitz 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 SegreVeronese”, 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, MittagLeffler 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 12, 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). “GenovaTorinoMilano Seminar: Some Topics in Commutative Algebra and Algebraic Geometry”, January 28–29, 2014 (Politecnico di Milano, Italy). “Vector Bundles Days II, PauTrieste Workshop on Vector Bundles and Related Topics. On the occasion of Emilia Mezzetti’s 60th birthday”, January 29–31, 2014 (Trieste, Italy). 

Other work 

Coordination of Research Projects (Founded) International
Italian
Research projects that passed the first steps of the evaluation process Title of the project: “Geometry of varieties parameterizing tensors and applications to their decomposition”. Participation to other Research Projects “Questioni di Geometria, Topologia e Algebra”. Financed by: Universit`a degli Studi di Milano, Dipartimento di Matematica “Federigo Enriques”. Period: 20022005. Responsible: Prof. Antonio Lanteri (Universit`a degli Studi di Milano). “Geometria sulle varietà algebriche”. Financed by: MIUR and Università degli Studi di Milano. Period: 20022004. Responsible: Prof. Antonio Lanteri (Universit`a degli studi di Milano), Prof. Alessandro Verra (Universit`a di Roma III). “Geometria sulle varietà algebriche”. Financed by: MIUR e Universitò degli Studi di Milano. Period: 20052006. Responsible: Prof. Lambertus Van Geemen (Univerisit`a degli Studi di Milano), Prof. Alessandro Verra (Universit`a di Roma III). “Project PRIN 2004”. Financed by: National government funds. Period: 20042005. Responsible: Prof. Angelo Vistoli (Universit`a degli Studi di Bologna). “RFO 2006 funds”. Financed by: Università degli studi di Bologna. Period: 2006. Responsible: Prof. Mirella Manaresi (Universit`a degli Studi di Bologna). “Project PRIN 2006”. Financed by: National government funds. Period: 20062007. Responsible: Prof. Mirella Manaresi (Universit`a degli Studi di Bologna). “RFO 2007 funds”. Financed by: Università degli studi di Bologna. Period: 2007. Responsible: Prof. Mirella Manaresi (Universit`a degli Studi di Bologna). “RFO 2013 funds”. Financed by: Università di di Bologna. Period: 2013. Responsible: Prof. Mirella Manaresi (Univ. Bologna). “Project PRIN 2013”. Financed by: National government funds. Period: 20132016. Responsible: Prof. M. Mella (Univ. Ferrara). 

Notes 

Abilitazione to Professore Ordinario (Full Professor) in Mathematics in the Italian universities. Valid from 28/03/2017 to 28/03/2023. Qualification to Professeur in the section 25 (Pure Mathematics) in the French universities. Obtained in February 2012. Qualification to Maître de Conference in the section 25 (Pure Mathematics) in the French universities. Obtained in February 2012. Leaves: From February 13 to July 13, 2014. Maternity Leave. 