Alessandra Bernardi

Professore associato

Dipartimento di Matematica


Via Sommarive, 14 - 38123 Povo
tel. 0461 281613
alessandra.bernardi[at]unitn [dot] it
Formazione
  • June 1996. Graduated from the high school Liceo Scientico of Polo Scolastico Maria Montessori in Porretta Terme, Bologna, Italy, with 60/60.
  • March 16, 2001 Laurea degree in Mathematics with full marks and honors (110/110 with praise) from Alma Mater Studiorum Università degli Studi di Bologna. Thesis title: "Schemi 0-dimensionali e forme canoniche di polinomi omogenei", (i.e. "0-dimensional schemes and canonical forms of homogeneous polinomials"). Advisor: Prof. A. Gimigliano (University of Bologna).
  • 2001-2005. PhD in Mathematics, Università degli Studi di Milano. Thesis title: "Varieties parameterizing forms and their secant varieties". Advisor: Prof. A. Gimigliano (University of Bologna).
Carriera accademica ed attività didattica

Employment

  • 2001-2005 PhD in Mathematics, Università degli Studi di Milano, Italy.
  • September 2005-January 2006, Research Assistant, Texas A&M University, College Station, Texas, USA.
  • June 28-September 27, 2006, Investigador extranjero en la UCM (Universidad Complutense de Madrid, Spain), financed by GRUPO SANTANDER.
  • November 2005-November 2009, Assegnista di ricerca; Alma Mater Studiorum Università degli studi Bologna, Italy.
  • July 2008, Teaching Assistant, MSRI - Mathematical Sciences Research Institute - (Berkeley, California - USA).
  • November 2, 2009 - November 1, 2010: Post Doc, CIRM (Centro Internazionale per la Ricerca Matematica) - Fondazione Bruno Kessler (Trento - Italy).
  • November 8, 2010 - November 7, 2012: Individual Marie Curie Fellow (Post Doc) IEF (International European Fellowship), INRIA (Institut National de Recherche en Informatique et en Automatique) Sophia Antipolis - Mediterranee (France).
  • January 17 - February 26, 2011: Visitor, Mittag-Leer Intitut (The Royal Swedish academy of sciences) invited by A. Dickenstein, S. Di Rocco, R. Piene, K. Ranestad and B. Sturmfels for the Spring Semester 2011 "Algebraic geometry with a view towards applications".
  • November 2012 - December 2013, Researcher RTD a), University of Torino, Department of Mathematics "Giuseppe Peano", Italy.
  • December 2013 - January 2016, Researcher RTD b), University of Bologna, Department of Mathematics, Italy.
  • November 2014 - December 2014. Long-Term Participant at Simons Institute for Theory of Computing (Berkeley, CA, USA), in the framework of the Fall Program 2014 "Algorithms and Complexity in Algebraic Geometry", Invited by P. Burgisser, JM Landsberg, K. Mulmuley, B. Sturmfels.
  • From February 2016, Associate Professor in Algebra, University of Trento, Department of Mathematics. Italy.

Teaching Activity

  • Teaching activity abroad (for PhD Students or higher)
    • ​​Research Assistant for the American PhD course in Math.: MATH 689-computational complexity geometry, I Semester A.A. 2005-2006, Texas A&M University, College Station (Texas, USA); Teaching Professor: JM Landsberg (College Station, Texas, USA).
    • Teaching Assistant for JM Landsberg (Texas A&M, Texas, USA), L.-H. Lim (UC Berkeley, California, USA) and J. Morton (UC Berkeley, California, USA) during the Graduate Workshop "Geometry and representation theory of tensors for computer science, statistics and other areas", at MSRI (Mathematical Sciences Research Institute), Berkeley (California - USA). July 2008.
    •  Main Speaker at "36th Autumn School in Algebraic Geometry: Power sum decomposition and apolarity, a geometric approach". Advanced school for PhD students and higher. Lukecin, Poland, September 1-7, 2013.
  • Teaching activity in Italy
    • Teaching Professor
      • Algebra 2, II year course of Mathematics, Univ. Torino, 2012-2013.
      • Geometria 1, I year course of Mathematics, Univ. Torino, 2012-2013.
      • Geometria, I year course of Engineering and architecture, Univ. Bologna, 2014-2015.
      • Teoria Algebrica dei Numeri, III year course of Mathematics, Univ. Trento, 2015-2018.
      • Geometria e algebra lineare, Engineering, Univ. Trento, 2016-2018.
      • Advances Commutative Algebra, Master and PhD in Mathematics, Univ. Trento 2016-2017.
      • Tensor Decomposition for Big Data Analysis, Master in Mathematics, Univ. Trento 2017-2018.
    • Tutorial Exercises 
      • Geometria e Algebra I, I year course of Mathematical and Physical Engineering, Politecnico di Milano, 2002-2003, teaching professor G. Bolondi.
      • Elementi di Analisi Matematica, Algebra e Geometria. I year course of Mechanical Engineering, Politecnico di Milano, 2003-2004, teaching professor F. Colombo.
      • Geometria e Algebra Lineare, (allievi A-K and L-Z). I year course of Management Engineering, Univ. Bologna, 2003-2004, 2006-2008 teaching professor A. Gimigliano.
      • Analisi Matematica B (allievi Ingegneri Civili). I year course of Ingegneria Civile, Milano Leonardo Politecnico (Italy), 2004-2005, teaching professor G. Verzini.
      • Geometria e Algebra (Corso integrato, moduli di Algebra e Geometria), I year course of Scienza della Formazione Primaria, Univ. Bologna, 2007-2008, teaching professor A. Gimigliano.
      • Analisi Matematica I, Geometria e Algebra Lineare (Corso integrato, moduli di Geometria e Algebra Lineare), (allievi A-K and L-Z). I year course of Management Engineering,Univ. Bologna, 2008-2010 teaching professor A. Gimigliano.
Interessi di ricerca

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.

Attività di ricerca

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 C8.
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 SN1H1 ⊗ SN2H2. Tensor rank is a natural measure of the entanglement
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 ∈ SN11H1⊗ SN2H2 of the two species is a tensor product
T = T1 ⊗ T2, where Ti ∈ SNiHi The results of this paper will expand on this novel quantum effect.

Appartenenza a società e comitati scientifici
Premi e riconoscimenti
  • "Borsa di studio ENAM per studenti universitari meritevoli". Academic Years: 1996-1997, 1997-1998, 1998-1999, 1999-2000.
  • Selected among the 100 best students of the Italian high schools by University Orientation Course for at Cortona (Italy). SCUOLA NORMALE SUPERIORE DI PISA. Summer 1995.
  • Scholarship: "Best student of the Scientic Liceo, Polo Scolastico Maria Montessori" (Porretta Terme, Bologna, Italy). Financed by Lions Club of Granaglione Alto Reno, December 1996.
  • Awards for the best student of the Scientic Liceo, Polo Scolastico Maria Montessori (Porretta Terme, Bologna, Italy). Financed by City Council of Gaggio Montano (Bologna, Italy). Years: 1991-1992, 1992-1993, 1993-1994, 1994-1995, 1995-1996.
  • Award "VI Latinus Ludus". Financed by Lions Club Cattolica, Maggio 1992.
  • Award for the best student of the Scuola Media Salvo D'Acquisto (Gaggio Montano, Bologna). Year 1991-1992.
Convegni e conferenze

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). 

Altre attività

Coordination of Research Projects (Founded)

International

  • Title of the project: “Programa de visitantes distinguidos e investigadores extranjeros en la UCM”.
    Financed by: GRUPO SANTANDER.
    Duration: 3 months; period: June 28 – September 27, 2006.
    Host: Prof. E. Arrondo (Universidad Complutense de Madrid - Spain).

  • Title of the project: “Decomposition of Structured Tensors, Algorithms and Characterization”. Financed by: European Community in the framework of the program FP7-PEOPLE-2009-IEF - 252367, Individual Marie Curie Fellow.
    Duration: 2 years, period: November 2010 – November 2012.
    Supervisor: Prof. B. Mourrain (INRIA Sophia Antipolis - M ́editerran ́ee - Nice -France)

Italian

  • Title of the project: “Dimensione e ideali di variet`a delle secanti di variet`a che parametrizzano forme e/o tensori. Generazione di algoritmi per il computo del rango strutturato dei loro elementi.” Financed by: CIRM – FBK (Trento).
    Duration: 1 year; period: November 2009 – November 2010.
    Supervisor: Prof. E. Ballico (Universit`a degli Studi di Trento).

  • Title of the project: “Algebra e Geometria Algebrica”.
    Financed by: University of Torino, Department of Mathematics, in the framework of Local Founds for young researchers.
    Duration: 2 years; period: December 2013 – December 2015.
    Coordinator: Alessandra Bernardi.

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”.
In the framework of: FIRB 2013.
Principal Investigator: Alessandra Bernardi.
Mark at the national Pre-selection: 9.33/10.
Mark at the national First selection: 13/15 (admitted to the second phase only the projects with 14/15).

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: 2002-2005. Responsible: Prof. Antonio Lanteri (Universit`a degli Studi di Milano).

“Geometria sulle varietà algebriche”. Financed by: MIUR and Università degli Studi di Milano. Period: 2002-2004. 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: 2005-2006. 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: 2004-2005. 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: 2006-2007. 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: 2013-2016. Responsible: Prof. M. Mella (Univ. Ferrara).

Note

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.