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Curriculum Vitae de Alessandra Bernardi

Curriculum de l’Activite Scientifique etd’Enseignement

de Alessandra Bernardi

1 Donnees Personnelles

— Nee le 27 Juin 1977 a Porretta Terme, Bologna, Italie.

— Nationalite : Italienne.

— Situation Actuelle, Decembre 2013 – Decembre 2016 : Chercheuse.

— Adresse professionnelle :

Alessandra BernardiUniversita di BolognaDipartimento di MatematicaPiazza di Porta San Donato 5I-40126 BolognaItaly

— Telephone :

— Fax (partage) :

— E-mail : .

— Page Web :

— Visiteur regulier a :— Departement de Mathematiques de l’Universite de Trento, Italie.

— Qualifiee pour des postes de Maıtre de Conference et de Professeur des Universites en Section 25(Mathematiques Pure).

2 Etudes

— Juin 1996 : Baccalaureat Scientifique, Liceo Scientifico du Polo Scolastico Maria Montessori,Porretta Terme, Bologna, Italie avec une note de 60/60.

— Le 16 Mars 2001 : Maıtrise (Laurea) de Mathematiques a l’Universite de Bologna avec unenote de 110/110 avec felicitations. Memoire de maıtrise : “Schemi 0-dimensionali e forme cano-

Turin (Italie), 17 septembre 2014 1 Alessandra Bernardi

Curriculum Vitae de Alessandra Bernardi 3 EMPLOIS

niche di polinomi omogenei”, (i.e. “Schemas 0-dimensionnels et formes canoniques de polynomeshomogenes”). Directeur : prof. Alessandro Gimigliano (Universite de Bologna).

In this thesis there are firstly presented the relations between already solved problems like“dimensions of secant varieties of Veronese varieties”, “postulations of fat points” and“Big Waring Problem”. Then the author uses Inverse Systems to compute the postulationsof n-th fat points in some cases, and the dimension of some secant varieties of varietiesparameterizing forms of the type Ld−jF where L is a linear form in 3 variables and F is aform of degree j in 3 variables.

— 2001-2005 : Doctorat en Mathematiques, Universite de Milano, Italie. Titre de la These : “Va-rieties parameterizing forms and their secant varieties”. Directeur de these : prof. AlessandroGimigliano (Universite de Bologna).

In this thesis the author studies various problems related with varieties parameterizing formsor tensors :— Secant varieties to osculating varieties to Veronese varieties (the dimensions are com-

puted in many cases) ;— Secant varieties to varieties that parameterize forms that can be written ad the product

of forms in different degrees (the dimensions are computed in some cases) ;— Varieties that parameterize forms in n + 1-variables that can be written as the product

of d linear forms (the dimension of their secant varieties are computed in many cases ;moreover it is found a contraexample to a conjecture formulated by R. Ehrenborg in 1999that states that the secant varietis to the Grassmannian G(n− 1, n+ d− 1) is the sameof the one of secant varieties of those varieties parameterizing forms in n+ 1-variablesthat can be written as the product of d linear forms) ;

— Secant varieties of Segre varieties (the author relates two different approaches for stu-dying those varieties : the one of Inverse System and a new one in RepresentationTheory).

Plusieurs parties du travail de these ont ete developpees pendant des sejours a l’etranger (voir 1.et 2. dans la Section 6 “Recherche a l’etranger” de ce Curriculum).

3 Emplois

— 2001–2005 : Doctorat en Mathematiques, Universite de Milano.

— Septembre 2005–Janvier 2006 : Research Assistant, Texas A&M University, College Station,Texas, USA.

— 28 Juin–27 Septembre 2006 : Investigador extranjero en la UCM (Universidad Complutensede Madrid, Espagne), financee par le GRUPO SANTANDER.

— Novembre 2005–Novembre 2006 : Boursiere post-doctorale (Assegnista di ricerca), Universitede Bologna, Italie.




Curriculum Vitae de Alessandra Bernardi 4 ARTICLES ET PREPRINTS

— Juillet 2008 : Teaching Assistant, MSRI - Mathematical Sciences Research Institute - (Berkeley,California - USA).

— 2 Novembre 2009– 1 Novembre 2010 : Boursiere Post-Doctoral, CIRM (Centro Internazionaleper la Ricerca Matematica) - Fondazione Bruno Kessler (Trento - Italie).

— 8 Novembre 2010–7 Novembre 2012 : Bourse Post-Doctorale Individuelle “Marie Curie” IEF(International European Fellowship) a l’INRIA (Institut National de Recherche en Informatiqueet en Automatique) - Mediterranee, Sophia Antipolis (France).

— 17 Janvier - 26 Fevrier 2011 : Visiteur, Mittag-Leffler Intitut (The Royal Swedish academy ofsciences) invitee par A. Dickenstein, S. Di Rocco, R. Piene, K. Ranestad et B. Sturmfels pour leSpring Semester 2011 “Algebraic geometry with a view towards applications”.

— Novembre, 2012–Decembre 2013 : Chercheuse, Universite de Turin (Italie).

— Decembre, 2013–Decembre 2016 : Chercheuse, Universite de Bologna (Italie).

— [Pevu] November 2014–December 2014 : Long-Term Participant a Simons Institute for Theoryof Computing (Berkeley, CA, USA), Fall Program 2014 “Algorithms and Complexity in AlgebraicGeometry”, Invitee par P. Burgisser, JM Landsberg, K. Mulmuley, B. Sturmfels.

4 Articles et Preprints

4.1 Livres

Algebra lineare e geometria analitica. A. Bernardi, A. Gimigliano. Citta studi. 2014.

4.2 These de Doctorat

0. “Varieties parameterizing forms and their secant varieties”(Settore Scientifico Disciplinare MAT/03).Alessandra BernardiDirecteur de These : prof. Alessandro Gimigliano (Univ. de Bologna).These Effectuee aupres du Departement de Mathematiques “Federigo Enriques” de l’Universite deMilanoThese soutenue le 13 Fevrier 2006.

Le RESUME de la these est dans la Section 2 de ce Curriculum.

4.3 Articles dans des revues

1. “Varieta che parametrizzano forme e loro varieta delle secanti”A. Bernardi.Bollettino U.M.I. La Matematica nella Societa nella Cultura. Serie VIII, Vol. X–A, Agosto 2007,191–194.

Le RESUME est dans la Section 2 de ce Curriculum.

2. “On generalized Kummer of rank-3 vector bundles over a genus 2 curve”A. Bernardi, D. Fulghesu.



Le Matematiche (Catania) Vol. LVIII (2003) - Fasc. II pp. 237–255 (2005). MR2216133 (2007b :14075).DOI : non assigne

Let X be a smooth projective complex curve and let UX(r, d) be the moduli space of semi-stable vector bundles of rank r and degree d on X. It contains an open Zariski subsetUX(r, d)s which is the coarse moduli space of stable bundles, i.e. vector bundles satisfyinginequality dF

rF< dE

rE. The complement UX(r, d) \ UX(r, d)s parametrizes certain equivalence

classes of strictly semi-stable vector bundles which satisfy the equality dF

rF= dE

rE. Each

equivalence class contains a unique representative isomorphic to the direct sum of stablebundles. Furthermore one considers subvarieties SUX(r, L) ⊂ UX(r, d) of vector bundle ofrank r with determinant isomorphic to a fixed line bundle L of degree d. In this work westudy the variety of strictly semi-stable bundles in SUX(3,OX), where X is a genus 2 curve.We call this variety the generalized Kummer variety of X and denote it by Kum3(X). Recallthat the classical Kummer variety of X is defined as the quotient of the Jacobian varietyJac(X) = UX(1, 0) by the involution L 7→ L−1. It turns out that our Kum3(X) has a similardescription as a quotient of Jac(X)× Jac(X) which justifies the name. We will see that thefirst definition allows one to define a natural embedding of Kum3(X) in a projective space.The second approach is useful in order to give local description of Kum3(X).

4.4 Articles dans des revues internationales a comite de lecture

3. “Some defective secant varieties to osculating varieties of Veronese surfaces”A. Bernardi, M. V. Catalisano.Collect. Math. 57 (2006), no. 1, pp. 43–68. MR2206180 (2007d :14096).DOI : non assigne

We consider the k-osculating varieties Ok,d to the Veronese d−uple embeddings of P2. Bystudying the Hilbert function of certain zero-dimensional schemes Y ⊂ P2, we find thedimension of Os

k,d, the (s − 1)thsecant varieties of Ok,d, for 3 ≤ s ≤ 6 and s = 9, and wedetermine whether those secant varieties are defective or not.

4. “Osculating varieties of Veronese Varieties and their higher secant varieties”A. Bernardi, M.V. Catalisano, A. Gimigliano e M. Ida.Canad. J. Math. Vol. 59 (3), 2007 pp. 488–502. MR2319156 (2008g :14095).DOI : non assigne

We consider the k-osculating varieties Ok,n.d to the (Veronese) d−uple embeddings of Pn.We study the dimension of their higher secant varieties via inverse systems (apolarity).By associating certain 0-dimensional schemes Y ⊂ Pn to Os

k,n,d and by studying theirHilbert function we are able, in several cases, to determine whether those secant varietiesare defective or not.

5. “Ideals of varieties parameterized by certain symmetric tensors”.A. Bernardi.J. Pure Appl. Algebra 212 (6), 2008 pp. 1542–1559. MR2391665 (2009c :14106).DOI : 10.1016/j.jpaa.2007.10.022

The ideal of a Segre variety Pn1 × · · · × Pnt ↪→ P(n1+1)···(nt+1)−1 is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of



Segre-Veronese varieties. The main tool is the concept of “weak generic hypermatrix” whichallows us to treat also the case of projection of Veronese surfaces from a set of general pointsand of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.

6. “Secant varieties to osculating varieties of Veronese embeddings of Pn.”A. Bernardi, M.V. Catalisano, A. Gimigliano e M. Ida.J. Algebra 321 (2009) pp. 982–1004. MR2488563 (2010d :14073).DOI information : 10.1016/j.algebra.2008.10.020

A well known theorem by Alexander-Hirschowitz states that all the higher secant varietiesof Vn,d (the d-uple embedding of Pn) have the expected dimension, with few known excep-tions. We study here the same problem for Tn,d, the tangential variety to Vn,d, and prove aconjecture, which is the analogous of Alexander-Hirschowitz theorem, for n ≤ 9. Moreover.we prove that it holds for any n, d if it holds for d = 3. Then we generalize to the case ofOk,n,d, the k-osculating variety to Vn,d, proving, for n = 2, a conjecture that relates thedefectivity of σs(Ok,n,d) to the Hilbert function of certain sets of fat points in Pn.

7. “On the variety parametrizing completely decomposable polynomials.”E. Arrondo, A. Bernardi.J. Pure Appl. Algebra 215 (2011) pp. 201–220.DOI : 10.1016/j.jpaa.2010.04.008

The purpose of this paper is to relate the variety parameterizing completely decomposablehomogeneous polynomials of degree d in n+1 variables on an algebraically closed field, calledSplitd(Pn), with the Grassmannian of n−1 dimensional projective subspaces of Pn+d−1. Wecompute the dimension of some secant varieties to Splitd(Pn) and find a counterexample toa conjecture that wanted its dimension related to the one of the secant variety to G(n−1, n+d − 1). Moreover by using an invariant embedding of the Veronse variety into the Pluckerspace, then we are able to compute the intersection of G(n− 1, n+ d− 1) with Splitd(Pn),some of its secant variety, the tangential variety and the second osculating space to theVeronese variety.

8. “Computing symmetric rank for symmetric tensors.”A. Bernardi, A. Gimigliano, M. Ida.Journal of Symbolic Computation 46 (2011) pp. 34–53.DOI : 10.1016/j.jsc.2010.08.001

We consider the problem of determining the symmetric tensor rank for symmetric tensorswith an algebraic geometry approach. We give algorithms for computing the symmetric rankfor 2× ...× 2 tensors and for tensors of small border rank. From a geometric point of view,we describe the symmetric rank strata for some secant varieties of Veronese varieties.

9. “Higher secant varieties of Pn × Pm embedded in bi-degree (1, d).”A. Bernardi, E. Carlini, M. V. Catalisano.J. Pure Appl. Algebra 215 (2011), pp. 2853–2858.DOI : 10.1016/j.jpaa.2011.04.005

Let X(n,m)(1,d) denote the Segre-Veronese embedding of Pn × Pm via the sections of the sheaf

O(1, d). We study the dimensions of higher secant varieties of X(n,m)(1,d) and we prove that



there is no defective sth secant variety, except possibly for n values of s. Moreover when(m+dd

)is multiple of (m+n+1), the sth secant variety of X

(n,m)(1,d) has the expected dimension

for every s.

10. “Multihomogeneous Polynomial Decomposition using Moment Matrices”A. Bernardi, J. Brachat, P. Comon, B. Mourrain.A. Leykin editor, International Symposium of Symbolic and Algebraic Computation (ISSAC), pp.35–42,San Jose, CA, USA, June, 2011, ACM New York.URL : http://portal.acm.orgcitation.cfm?id=1993886.1993898&coll=DL&dl=ACM&CFID =

30387648&CFTOKEN=71177337

DOI : 10.1145/1993886.1993898.ISBN : 978-1-4503-0675-1.

In the paper, we address the important problem of tensor decompositions which can be seenas a generalisation of Singular Value Decomposition for matrices. We consider general mul-tilinear and multihomogeneous tensors. We show how to reduce the problem to a truncatedmoment matrix problem and give a new criterion for flat extension of Quasi-Hankel ma-trices. We connect this criterion to the commutation characterisation of border bases. Anew algorithm is described which applies for general multihomogeneous tensors, extendingthe approach of J.J. Sylvester on binary forms. An example illustrates the algebraic opera-tions involved in this approach and how the decomposition can be recovered from eigenvectorcomputation.

11. “On the X-rank with respect to linear projections of projective varieties.”E. Ballico, A. Bernardi.Mathematische Nachrichten. 284 No. 17–18, (2011), pp. 2133–2140.DOI : 10.1002/mana.200910275.ISSN : 1522-2616

In this paper we improve the known bound for the X-rank RX(P ) of an element P ∈ PN

in the case in which X ⊂ Pn is a projective variety obtained as a linear projection froma general v-dimensional subspace V ⊂ Pn+v. Then, if X ⊂ Pn is a curve obtained froma projection of a rational normal curve C ⊂ Pn+1 from a point O ⊂ Pn+1, we are ableto describe the precise value of the X-rank for those points P ∈ Pn such that RX(P ) ≤RC(O) − 1 and to improve the general result. Moreover we give a stratification, via theX-rank, of the osculating spaces to projective cuspidal projective curves X. Finally we givea description and a new bound of the X-rank of subspaces both in the general case and withrespect to integral non-degenerate projective curves.

12. “On the X-rank with respect to linearly normal curves.”E. Ballico, A. Bernardi.Collectanea Mathematica : Volume 64, Issue 2 (2013), Page 141–154.DOI : 10.1007//s13348-011-0058-4.

In this paper we study the X-rank of points with respect to smooth linearly normal curvesX ⊂ Pn of genus g and degree n+ g.We prove that, for such a curve X, under certain circumstances, the X-rank of a general



point of X-border rank equal to s is less or equal than n+ 1− s.In the particular case of g = 2 we give a complete description of the X-rank if n = 3, 4 ;while if n ≥ 5 we study the X-rank of points belonging to the tangential variety of X.

13. “Algebraic Geometry tools for the study of entanglement : an application to spin squeezed states”A. Bernardi, I. Carusotto.J. Phys. A : Math. Theor. 45 (2012) 105304 (13pp).DOI : 10.1088/1751-8113/45/10/105304.

A short review of Algebraic Geometry tools for the decomposition of tensors and polynomialsis given from the point of view of applications to quantum and atomic physics. Examplesof application to assemblies of indistinguishable two-level bosonic atoms are discussed usingmodern formulations of the classical Sylvester’s algorithm for the decomposition of homoge-neous polynomials in two variables. In particular, the symmetric rank and symmetric borderrank of spin squeezed states is calculated as well as their Schrodinger-cat-like decompositionas the sum of macroscopically different coherent spin states ; Fock states provide an exampleof states for which the symmetric rank and the symmetric border rank are different.

14. “Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes”E. Ballico, A. Bernardi.Sarajevo Journal of Mathematics. Vol. 8 (20), 33–52 (2012).

We give a partial “ quasi-stratification ” of the secant varieties of the order d Veronesevariety Xm,d of Pm. It covers the set σt(Xm,d)† of all points lying on the linear span ofcurvilinear subschemes of Xm,d, but two “ quasi-strata ” may overlap. For low border ranktwo different “ quasi-strata ” are disjoint and we compute the symmetric rank of theirelements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties.To get a stratification we attach to each P ∈ σt(Xm,d)† the minimal label of a quasi-stratumcontaining it.

15. “Decomposition of homogeneous polynomials with low rank.”E. Ballico, A. Bernardi.Math. Z. (2012) 271 :1141–1149.DOI : 10.1007/s00209-011-0907-6.ISSN : 0025-5874.

Let F be a homogeneous polynomial of degree d in m+ 1 variables defined over an algebrai-cally closed field of characteristic zero and suppose that F belongs to the s-th secant varieties

of the standard Veronese variety Xm,d ⊂ P(m+dd )−1 but that its minimal decomposition as

a sum of d-th powers of linear forms M1, . . . ,Mr is F = Md1 + · · · + Md

r with r > s. Weshow that if s+ r ≤ 2d+ 1 then such a decomposition of F can be split in two parts : one ofthem is made by linear forms that can be written using only two variables, the other part isuniquely determined once one has fixed the first part. We also obtain a uniqueness theoremfor the minimal decomposition of F if the rank is at most d and a mild condition is satisfied.

16. “Higher secant varieties of Pn × P1 embedded in bi-degree (a, b)”.E. Ballico, A. Bernardi, M. V. Catalisano.Communications in Algebra. 40 (2012) 3822–3840.DOI : 10.1080./00927872.2011.595748.



In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of Pn×P1 via the section of the sheaf O(a, b) for any n, a, b ∈ Z+. Werelate this result to the Grassmann Defectivity of Veronese varieties and we classify all theGrassmann (1, s− 1)-defective Veronese varieties.

17. “Symmetric tensor rank with a tangent vector : a generic uniqueness theorem”.E. Ballico, A. Bernardi.Proceedings of the American Mathematical Society 140, 10, (2012), 3377–3384.DOI : 10.1090/S0002-9939-2012-11191-8.

Let Xm,d ⊂ PN , N :=(m+dm

)− 1, be the order d Veronese embedding of Pm. Let τ(Xm,d) ⊂

PN , be the tangent developable of Xm,d. For each integer t ≥ 2 let τ(Xm,d, t) ⊆ PN , bethe join of τ(Xm,d) and t − 2 copies of Xm,d. Here we prove that if m ≥ 2, d ≥ 7 and

t ≤ 1+b(m+d−2

m

)/(m+1)c, then for a general P ∈ τ(Xm,d, t) there are uniquely determined

P1, . . . , Pt−2 ∈ Xm,d and a unique tangent vector ν of Xm,d such that P is in the linearspan of ν ∪ {P1, . . . , Pt−2}, i.e. a degree d linear form f (a symmetric tensor T of order d)associated to P may be written as

f = Ld−1t−1Lt +

t−2∑i=1

Ldi , (T = v

⊗(d−1)t−1 vt +

t−2∑i=1

v⊗di )

with Li linear forms on Pm (vi vectors over a vector field of dimension m+ 1 respectively),1 ≤ i ≤ t, that are uniquely determined (up to a constant).

18. “Grassman secants, identifiability, and linear systems of tensors.”E. Ballico, A. Bernardi, M. V. Catalisano, L.Chiantini.Linear Algebra and its Applications 438 (2013) 121–135DOI : 10.1016/j.laa.2012.07.045

For any irreducible non-degenerate variety X ⊂ Pr, we relate the dimension of the s-thsecant varieties of the Segre embedding of Pk×X to the dimension of the (k, s)-Grassmannsecant variety GSX(k, s) of X. We also give a criterion for the s-identifiability of X.

19. “General Tensor Decomposition, Moment Matrices and Applications.”A. Bernardi, J. Brachart, P. Comon, B. Mourrain.J. Symbolic Comput. Special Issue : ISSAC-2011. 52 (2013) 51–71.DOI : 10.1016/j.jsc.2012.05.012.

In the paper, we address the important problem of tensor decompositions which can be seenas a generalisation of Singular Value Decomposition for matrices. We consider general mul-tilinear and multihomogeneous tensors. We show how to reduce the problem to a truncatedmoment matrix problem and give a new criterion for flat extension of Quasi-Hankel ma-trices. We connect this criterion to the commutation characterisation of border bases. Anew algorithm is described which applies for general multihomogeneous tensors, extendingthe approach of J.J. Sylvester on binary forms. An example illustrates the algebraic opera-tions involved in this approach and how the decomposition can be recovered from eigenvectorcomputation.



20. “The cactus rank of cubic forms”A. Bernardi, K. Ranestad.J. Symbolic Compu. (2013) 291–297DOI : 10.1016/j.jsc.2012.08.001

We prove that the smallest degree of an apolar 0-dimensional scheme of a general cubicform in n+ 1 variables is at most 2n+ 2, when n ≥ 8, and therefore smaller than the rankof the form. For the general reducible cubic form the smallest degree of an apolar subschemeis n+ 2, while the rank is at least 2n.

21. “Real and complex rank for real symmetric tensors with low ranks”E. Ballico, A. Bernardi.Algebra, vol. 2013, Article ID 794054, 5 pages, 2013.DOI :10.1155/2013/794054

We study the case of real homogeneous polynomial P whose minimal real and complexdecompositions in terms of powers of linear forms are different. In particularly we will showthat, if the sum of the complex and the real ranks of P is smaller or equal than 3 deg(P )−1,then the difference of the two decompositions is completely determined either on a line oron a conic.

22. “Unique decomposition for a polynomial of low rank”E. Ballico, A. Bernardi.Ann. Polon. Math. 108 (2013), 219–224.DOI : 10.4064/ap108-3-2.

Let F be a homogeneous polynomial of degree d in m+1 variables defined over an algebrai-cally closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of

the d-uple Veronese embedding of Pm into P(m+dd )−1 but that its minimal decomposition as

a sum of d-th powers of linear forms requires more than s addenda. We show that if s ≤ dthen F can be uniquely written as F = Md

1 + · · · + Mdt + Q, where M1, . . . ,Mt are linear

forms with t ≤ (d− 1)/2, and Q a binary form such that Q =∑q

i=1 ld−dii mi with li’s linear

forms and mi’s forms of degree di such that∑

(di + 1) = s− t.

23. “Minimal decomposition of binary forms with respect to tangential projections.”E. Ballico, A. Bernardi.Journal of Algebra and its Applications, 12, 6 (2013) 1350010 (8 pages).DOI : 10.1142/S0219498813500102

Let C ⊂ Pn+1 be a rational normal curve and let X ⊂ Pn be one of its tangential projection.We describe the X-rank of a point P ∈ Pn in terms of the schemes evincing the C-rank orthe border C-rank of the preimage of P .

24. “Tensor ranks on tangent developable of Segre varieties”.E. Ballico, A. Bernardi.Linear and Multilinear Algebra, 61 (7) , pp. 881–894 (2013).DOI : 10.1080/03081087.2012.716430.

We describe the stratification by tensor rank of the points belonging to the tangent develo-pable of any Segre variety. We give algorithms to compute the rank and a decomposition



of a tensor belonging to the secant variety of lines of any Segre variety. We prove Co-mon’s conjecture on the rank of symmetric tensors for those tensors belonging to tangentialvarieties to Veronese varieties.

25. “Stratification of the fourth secant variety of Veronese variety via the symmetric rank.”E. Ballico, A. Bernardi.Advances in Pure and Applied Mathematics 4 (2) 215–250 (2013).DOI : 10.1515/apam-2013-0015.

If X ⊂ Pn is a projective non degenerate variety, the X-rank of a point P ∈ Pn is definedto be the minimum integer r such that P belongs to the span of r points of X. We describethe complete stratification of the fourth secant variety of any Veronese variety X via theX-rank. This result has an equivalent translation in terms both of symmetric tensors andhomogeneous polynomials. It allows to classify all the possible integers r that can occur inthe minimal decomposition of either a symmetric tensor or a homogeneous polynomial ofX-border rank 4 (i.e. contained in the fourth secant variety) as a linear combination ofeither completely decomposable tensors or powers of linear forms respectively.

26. “A comparison of different notions of ranks of symmetric tensors”A. Bernardi, J. Brachat, B. Mourrain.Linear Algebra and Its Applications, 460, 2014, 205–230.Doi :10.1016/j.laa.2014.07.036

We introduce various notions of rank for a symmetric tensor, namely : rank, border rank,catalecticant rank, generalized rank, scheme length, border scheme length, extension rankand smoothable rank. We analyze the stratification induced by these ranks. The mutualrelations between these stratifications, allow us to describe the hierarchy among all the ranks.We show that strict inequalities are possible between rank, border rank, extension rank andcatalecticant rank. Moreover we show that scheme length, generalized rank and extensionrank coincide.

4.5 Preprints

27. “Curvilinear schemes and maximum rank of forms”E. Ballico, A. Bernardi.Preprint : http://arxiv.org/abs/1210.8171

We define the curvilinear rank of a degree d form P in n + 1 variables as the minimumlength of a curvilinear scheme, contained in the d-th Veronese embedding of Pn, whosespan contains the projective class of P . Then, we give a bound for rank of any homogenouspolynomial, in dependance on its curvilinear rank.

28. “Computing the cactus rank of a general form”A. Bernardi, J. Jelisiejew, P. Macias Marques, K. RanestadPreprin : http://arxiv.org/abs/1211.7306

We present an aproach to computing the cactus rank for a general homogeneous polynomialF ∈ K[x0, . . . , xn] and use it to show that the cactus rank of a generic cubic form is 12(resp. 15, 18, 20, 22) when n = 6, (resp. n = 7, 8, 9, 10).


Curriculum Vitae de Alessandra Bernardi 5 INTERETS DE RECHERCHE

29. Lecture notes on Waring problems, Secant varieties and Sylvester algorithm. A. Bernardi. Draftavailable at http://www.mimuw.edu.pl/ jabu/conf/2013/lukecin.html.

These are the Lecture notes on “Waring problems, Secant varieties and Sylvester algo-rithm” for the 36th Autumn School in Algebraic Geometry “Power sum decomposition andapolarity, a geometric approach” September 1st-7th, 2013 Lukecin, Poland.

5 Interets de Recherche

Mes interets de recherche sont dans le domaine de la geometrie algebrique. En particulier :Varietes Secantes ; Schemas zero-dimensionnels et leurs postulations ; Grassmanniennes ; Varietes pa-

rametrant des formes et/ou des tenseurs dans le champ complexe ou reel ; Rang de tenseurs symetriqueset de tenseurs structures ; Unicite de la decomposition d’un tenseur en tenseurs de rang 1.

5.1 Resume du projet Marie Curie

Boursiere Marie Curie a l’INRIA - Mediterranee, Sophia Antipolis, Nice (France).Scientifique en charge : Prof. Bernard Mourrain.Projet : FP7-PEOPLE-2009-IEF - 252367 - DECONSTRUCT : “Decomposition of Structured Tensors,Algorithms and Characterization”.Debut du projet : 1 Novembre 2010.Duree : 2 ans.

RESUME : Tensors play a wide role in numerous application areas as Signal Processing forTelecommunications, Arithmetic Complexity or Data Analysis. In some applications tensorsmay be completely symmetric, or symmetric only in some modes, or may not be symmetric.In most of these applications, the decomposition of a tensor into a sum of rank-1 terms isrelevant, since tensors of interest have a reduced rank. Most of them are structured i.e. theyare either symmetric or enjoy some index-invariance. Lastly, they are often real, which raisesopen problems concerning the existence and calculation of the decompositions. These issuesbuild the basic bricks of the research program we propose. The classes of tensors describedabove have a geometric translations in terms of classical algebraic varieties : Segre, Veronese,Segre-Veronese varieties and Grassmannians and their secant varieties. A complete descriptionof equations for those secant varieties and their dimensions is still not known (only dimensionsof secant varieties to Veronsean are classified), although they have been studied by algebraicand differential geometers and algebraists for a long period up to now. The aim of this researchproject is :— to attack both the description of the ideal of those secant varieties and their dimensions,

starting from low dimensions and low degrees,— to propose algorithms able to compute the rank of structured tensors.Workshops in Palo Alto (CA-USA, 2008) and in Nice (FR, 2009) showed that Italie andFrance are among the most active in Europe in the field of tensor decompositions. Both thecoordinator of this project and the hosting organization have already obtained results in thisfield regarding equations and algorithms. Hence this program is crucial for the developmentof those research areas in the European Community, along with the numerous internationalcollaborations already existing. The impact of this project will be visible in both academicand industrial worlds.


Curriculum Vitae de Alessandra Bernardi 6 RECHERCHE A L’ETRANGER

6 Recherche a l’Etranger

1. 17 Septembre - 17 Decembre, 2004, Queen’s University (Kingston, Ontario, Canada), invitee parle prof. A.V. Geramita ;

L’activite de recherche durant cette visite etait :— “varietes secantes de varietes osculantes a des varietes de Veronese” avec le prof. A.V.Geramita ;— techniques de la theorie des representations pour l’etude du probleme de le generation

des ideaux des varietes secantes a des varietes de Segre avec les profs. M. Roth et I.Dimitrov.

Les resultats obtenus lors de cette visite ont ete inclus dans la these de doctorat.

2. 9 Janvier - 1 Mars 2005, a l’Universidad Complutense de Madrid (Madrid, Spain), invitee par leprof. E. Arrondo ;

Pendant cette visite, l’activite de recherche etait l’etude des varietes secantes a des varietesparametrant des formes qui peuvent s’ecrire comme produit de formes lineaires. Nous avonsproduit un contrexemple a la conjecture de Ehrenborg (1999) qui pretendait que la dimensionde certaines varietes secantes a des grassmanniennes etait la meme que certaines varietessecantes a des varietes parametrant de formes qui se scindent comme le produit de formeslineaires. Les resultats obtenus pendant cette visite ont ete inclus dans la these de doctorat.

3. 26 Septembre - 15 Octobre, 2005, a la Texas A&M University (College Station, Texas, USA),Research Assistant pour le cours de Doctorat “MATH 689-computational complexity geometry”tenu par le prof. J.M. Landsberg ;

Pendant cette visite, en plus de l’activite d’enseignement, j’ai entrepris l’etude de la dimen-sion des premieres varietes secantes aux varietes adjointes (varietes associees a des algebresde Lie) avec le prof. J. M. Landsberg.

4. 27 Octobre - 15 Decembre 2005 a la Texas A&M University (College Station, Texas, USA), Re-search Assistant pour le cours de Doctorat “MATH 689-computational complexity geometry” tenupar le prof. J.M. Landsberg ;

Pendant cette visite, en plus de l’activite d’enseignement, j’ai continue l’etude de la di-mension des premieres varietes secantes aux varietes adjointes demarree lors de la visiteprecedente.

5. 2 Mars - 15 Avril 2006 a la Texas A&M University (College Station, Texas, USA), ResearchAssistant pour le cours de Doctorat “MATH 689-computational complexity geometry” tenu parle prof. J.M. Landsberg ;

Pendant cette visite, en plus de l’activite d’enseignement, j’ai continue l’etude de la di-mension des premieres varietes secantes aux varietes adjointes demarree lors de la visiteprecedente.

6. 28 Juin - 27 Septembre 2006 a la Universidad Complutense de Madrid (Madrid, Spain), inviteepar le professor E. Arrondo dans le cadre du “Programa de visitantes distinguidos e investigadoresextranjeros en la UCM” finance par le “GRUPO SANTANDER”.

Durant cette visite, j’ai etudie en collaboration avec le prof. E. Arrondo le lieu d’intersectionentre des grassmanniennes et des



— varietes de Veronese,— varietes secantes a des varietes de Veronese,— varietes tangentielles et varietes de Veronese,— varietes osculantes a des varietes de Veronese,— varietes parametrant des formes qui se decomposent en produit de formes lineaires.

7. 7-20 Juillet 2008 au MSRI (Mathematical Sciences Research Institute), Berkeley (California - USA)invitee en tant que Teaching Assistant pour le Graduate Workshop “Geometry and representationtheory of tensors for computer science, statistics and other areas”.

Cet atelier etait organise par J.M. Landsberg (Texas A&M - Texas - USA), Lek-HengLim (UC Berkeley - California - USA) et Jason Morton (UC Berkeley - California -USA) avec le but d’introduire des doctorants aux concepts plus importants de la geometrieet de de la theorie des representations. La complexite calculatoire, la theorie des statis-tiques de l’apprentissage, le traitement du signal, l’analyse des donnees scientifiques ontete recemment formules en des termes geometriques via la theorie des representations.Le probleme specifique attaque consistait en la “multiplication des matrices”. Pendant ladeuxieme semaine de l’atelier, il etait possible de travailler sur des problemes ouverts.

8. 21 - 27 Juillet 2008 au AIM (American Institute of Mathematics) de Paolo Alto (California - USA)pour le Workshop “Geometry and representation theory of tensors for computer science, statisticsand other areas”.

Cet atelier etait organise par J.M. Landsberg (Texas A&M - Texas - USA), Lek-Heng Lim(UC Berkeley - California - USA), Jason Morton (UC Berkeley - California - USA) andJerzy Weyman (Northeastern University - Boston - MA - USA).Il etait dedie a l’etude de problemes en calcul quantique, en theorie de la complexite, entheorie des statistiques de l’apprentissage, traitement du signal et analyse de donnees. Danstous ces domaines, il apparaıt des varietes dans des espaces de tenseurs qui sont invariantessous changement de coordonnees.Le but de cet atelier etait de traduire des problemes du monde applique dans un langagemathematique et, si possible, de les resoudre.

9. 1 - 7 Septembre, 2008 au Departement de Mathematiques at the Mathematical department ofUniversidad Complutense de Madrid (Espagne), invitee par le prof. E. Arrondo.

Durant cette visite, j’ai etudie en collaboration avec le prof. E. Arrondo le lieu d’intersectionentre des grassmanniennes et des— varietes de Veronese,— varietes secantes a des varietes de Veronese,— varietes tangentielles et varietes de Veronese,— varietes osculantes a des varietes de Veronese,— varietes parametrant des formes qui se decomposent en produit de formes lineaires.

10. 21 - 23 Janvier 2009 au Laboratoire d’Informatique, Signaux et Systemes de Sophia-Antipolis(Francia), invitee par le prof. P. Comon.

Lors de cette visite, nous avons redige un projet de recherche sur— la conjecture de l’equivalence entre le rang symetrique et le rang tout court d’un tenseur

symetrique ;



— combien de rangs generiques peuvent exister pour un tenseur reel (il a ete conjecturequ’il n’en existe que deux) ;

— le fermeture de l’ensemble des tenseurs de rang inferieur ou egal a un entier donne.

11. 1 - 5 Fevrier 2009 au Departement de Mathematiques de l’Universidad Complutense de Madrid(Espagne), invitee par le prof. E. Arrondo.

Lors de cette visite, ont ete acheves l’etude de l’intersection entre des grassmanniennes et :— varietes parametrant des formes qui se decomposent en produit de formes lineaires,— certaines parmi leurs varietes secantes,— varietes de Veronese,— varietes tangentielles et varietes de Veronese,— deuxiemes varietes osculantes a des varietes de Veronese,— certaines varietes secantes a des varietes de Veronese.

12. Novembre 2010 - Novembre 2012 a l’INRIA-Mediterranee (Sophia Antipolis, France) en tantque borsiere post-doctorale Marie Curie. Projet : FP7-PEOPLE-2009-IEF - 252367 - DECONS-TRUCT : “Decomposition of Structured Tensors, Algorithms and Characterization”.

Le resume de ce projet est presente dans la Section 5.1 de ce Curriculum.

13. 17 Janvier - 26 Fevrier 2011 au Mittag-Leffler Institut (The Royal Swedish academy of sciences)invitee par A. Dickenstein, S. Di Rocco, R. Piene, K. Ranestad and B. Sturmfels a participer entant que Visitor au Spring Semester 2011 “Algebraic geometry with a view towards applications”.

J’ai ete invitee par les organisateurs du semestre en tant qu’experte de varietes secantes, durang des tenseurs, et de leurs applications. J’ai demarre des collaborations avec K. Ranestad(voir la publication 20), avec J. D. Hauenstein sur un algorithme numerique efficace pour lecalcul du rang d’un polynome generique, avec E. S. Allman et J. A. Rhodes sur un problemede phylogenetique qui fait intervenir le rang d’un tenseur.

14. 29 Mai - 10 Juin 2011 au Mittag-Leffler Institut (The Royal Swedish academy of sciences) inviteepar K. Ranestad.

Pendant cette visite, j’ai continue deux collaborations. L’une avec K. Ranestad sur lerang des cubiques. L’autre avec J. D. Hauenstein sur le developpement d’un algorithmenumerique efficace pour le calcul du rang d’un polynome generique.

15. 15-19 Octobre 2012 au Departamento de Matematica Universidade de Evora (Portugal) inviteepar P. M. Marques.

Pendant cette visite nous avons termine l’article 28 decrite dans la section 4.

16. 22-25 Octobre 2012 au Universite de Pau (France) invitee par D. Faenzi.

Pendant cette visite nous avons etudie l’ideal des premieres varietes secantes aux varietesadjointes.

17. 21-31 Juillet 2013 au University of Idaho (USA) invitee H. Abo.

Pendant cette visite nous avons etudie deux problems des ideaux de certains varietes dualand de defectivite faible de Segre variete de Grassmannian et un espace projective.


Curriculum Vitae de Alessandra Bernardi 8 ACTIVITE D’ENSEIGNEMENT EN ITALIE

7 Activite d’enseignement a l’Etranger

— Research Assistant pour le cours de l’Ecole Doctorale en Mathematiques : MATH 689-computationalcomplexity geometry, I Semestre A.A. 2005-2006, Texas A&M University, College Station(Texas, USA) ; responsable du cours, prof. J. M. Landsberg (College Station, Texas, USA).

— Teaching Assistant pour les profs. J.M. Landsberg (Texas A&M - Texas - USA), Lek-Heng Lim(UC Berkeley - California - USA) et Jason Morton (UC Berkeley - California - USA) lors duGraduate Workshop “Geometry and representation theory of tensors for computerscience, statistics and other areas”, au MSRI (Mathematical Sciences Research Institute),Berkeley (California - USA).Juillet 2008.

— Main Speaker a la “36th Autumn School in Algebraic Geometry : Power sum decomposition andapolarity, a geometric approach”. Lukecin, Poland, September 1-7, 2013.

8 Activite d’enseignement en Italie

8.1 Travaux Diriges

1. geometrie et Algebre I, niveau L1, Travaux Diriges, I semestre, A.A. 2002-2003. Responsabledu cours prof. G. Bolondi, Faculte d’Ingenieurie Mathematique et Physique, Politecnico di Milano.

2. Elements d’Analyse Mathematique, Algebre et Geometrie, niveau L1, Travaux Diriges,I semestre, A.A. 2003-2004, responsable du cours F. Colombo, faculte de Ingenieurie Mecanique,Politecnico di Milano.

3. Geometrie et Algebre Lineaire, (eleves A-K), niveau L1, Travaux Diriges, I semestre, A.A.2003-2004, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universitede Bologna.

4. Geometrie et Algebre Lineaire, (eleves L-Z), niveau L1, Travaux Diriges, I semestre, A.A.2003-2004, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universitede Bologna.

5. Analyse Mathematique B (eleves Ingenieurs Civils), niveau L1, Travaux Diriges, II se-mestre, A. A. 2004-2005, responsable du cours prof. G. Verzini, faculte d’Ingenieurie Civile, Poli-tecnico di Milano.

6. Geometrie et Algebre Lineaire, (eleves A-K), niveau L1, Travaux Diriges, I semestre, A.A.2006-2007, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universitede Bologna.


8. Geometrie et Algebre Lineaire, (eleves A-K), niveau L1, Travaux Diriges, I semestre, A.A.2007-2008, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universite


Curriculum Vitae de Alessandra Bernardi 9 PROJETS DE RECHERCHE

de Bologna.


10. Geometrie et Algebre (Cours Integre, modules de Algebre et Geometrie), II semestre,A.A. 2007-2008, responsable du cours prof. A. Gimigliano, parcours de Sciences de la FormationPrimaire, faculte de Sciences de la Formation, Universite de Bologna.

11. Analyse Mathematique I, Geometrie et Algebre (Cours Integre, modules de Algebreet Geometrie), (eleves A-K), niveau L1, II semestre, A.A. 2007-2008, responsable du coursprof. A. Gimigliano, parcours de Sciences de la Formation Primaire, faculte de Sciences de laFormation, Universite de Bologna.

12. Analyse Mathematique I, Geometrie et Algebre Lineaire (Cours Integre, modules deAlgebre et Geometrie), (eleves A-K), niveau L1, Travaux Diriges, I semestre, A.A. 2008-2009, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universite deBologna.

13. Analyse Mathematique I, Geometrie et Algebre Lineaire (Cours Integre, modules deAlgebre et Geometrie), (eleves A-K), niveau L1, Travaux Diriges, I semestre, A.A. 2009-2010, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universite deBologna.

14. Analyse Mathematique I, Geometrie et Algebre Lineaire (Cours Integre, modulesde Algebre et Geometrie), (eleves L-Z), niveau L1, Travaux Diriges, I semestre, A.A. 2009-2010, responsable du cours prof. A. Gimigliano, faculte d’Ingenieurie Gestionnelle, Universite deBologna.

8.2 Charge de cours

1. Algebra 2 Mathematiques. 2012-2013. Charge de cours. Universites de Torino (Italie).

2. Geometria 1, Cours A et Cours B. Mathematiques. 2012-2013. Charge de cours. Universites deTorino (Italie).

9 Projets de recherche

9.1 Projets de recherche finances ecrits par Alessandra Bernardi

— Titre du projet : “Programa de visitantes distinguidos e investigadores extranjeros en la UCM” :“Varieties parameterizing forms.”Finance par : GRUPO SANTANDER.Duree : 3 mois ; periode : 28 Juin - 27 Septembre 2006.Responsable local : Prof. Enrique Arrondo (Universidad Complutense de Madrid - Espagne).

— Titre du projet : “Dimension et ideaux des secantes a de varietes qui parametrisent des formeset/ou des tenseurs. Generation d’algorithmes pour le calcul du rang structure de leurs elements.”



Finance par : CIRM – FBK (Trento, Italie).Duree : 1 an ; periode : Novembre 2009 – Novembre 2010.Responsable local : Prof. Edoardo Ballico (Universite de Trento).

— Title of the project : “Decomposition of Structured Tensors, Algorithms and Characterization”.Finance par : Union Europeenne dans le cadre du programme FP7-PEOPLE-2009-IEF - 252367,Marie Curie Fellow.Duree : 2 ans ; periode : Novembre 2010 – Novembre 2012.Responsable local : Prof. Bernard Mourrain (INRIA-Mediterranee, Sophia Antipolis, France).

— Title of the project : “Algebra e Geometria Algebrica”.Finance par : Dipartimento di Matematica “Giuseppe Peano”, Universtia di Torino.Duree : 2 ans ; periode : Decembre 2013 – Decembre 2014.Responsable local : Alessandra Bernardi.

9.2 Bourses de recherche obtenues par Alessandra Bernardi

— Titre du projet : “Varietes des secantes de varietes qui parametrisent des formes”.Finance par : Dipartimento di Matematica dell’ Alma Mater Studiorum Universita di Bologna.Duree : 12 mois, ensuite renouvele pour autres 12 mois ; periode Novembre 2005 - Novembre2006 et Novembre 2006 - Novembre 2007.Responsable local : Prof. Alessandro Gimigliano.

— Titre du projet : “Varietes des secantes de varietes qui parametrisent des formes”.Finance par : Dipartimento di Matematica dell’ Alma Mater Studiorum Universita di Bologna.Duree : 12 mois, ensuite renouvele pour autres 12 mois ; periode Novembre 2007 - Novembre2009.Responsable local : Prof. Alessandro Gimigliano.

9.3 Autres projets de recherche

— Titre du projet : “Questions de Geometrie, Topologie, et Algebre”.Finance par : Universita degli Studi di Milano, Departement de Matematiques “Federigo Enri-ques”.Duree : 1 an ; periode : 2002.Responsable : Prof. Antonio Lanteri (Universita degli Studi di Milano).





— Titre du projet : “Geometrie sur les varietes algebriques”.Cofinance par : MIUR (Ministere Italien de l’Universite et de la Recherche) et par l’Universitede Milano.Duree : 2 ans ; periode : 2002-2004.Responsables : Prof. Antonio Lanteri (Universita degli studi di Milano), Prof. Alessandro Verra(Universita di Roma III).


— Titre du projet : “Geometrie sur les varietes algebriques”.Cofinance par : MIUR (Ministere Italien de l’Universite et de la Recherche) et par l’Universitede Milano.Duree : 2 ans ; periode : 2005-2006.Responsables : Prof. Lambertus Van Germen (Univerisita degli Studi di Milano), Prof. Alessan-dro Verra (Universita di Roma III).

— Titre du projet : “Projet PRIN 2004 (Projets d’interet national)”.Finance par : fonds du gouvernement national.Duree : 2 ans ; periode : 2004-2005.Responsable : Prof. Angelo Vistoli (Universita degli Studi di Bologna).

— Titre du projet : “RFO 2006 funds (Recherche Fondamentale Orientee).”Finance par : Universita degli studi di Bologna.Duree : 1 an ; periode : 2006.Responsable : Prof. Mirella Manaresi (Universita degli Studi di Bologna).

— Titre du projet : “Projet PRIN 2006 (Projets d’interet national)”.Finance par : fonds du gouvernement national.Duree : 2 ans ; periode : 2006-2007.Responsable : Prof. Mirella Manaresi (Universita degli Studi di Bologna).





Curriculum Vitae de Alessandra Bernardi 10 EXPOSES INVITES

— Titre du projet : “Projet PRIN 2008 (Projets d’interet national)”.Finance par : fonds du gouvernement national.Duree : 2 ans ; periode : 2008-2009.Responsable : Prof. Mirella Manaresi (Universita degli Studi di Bologna).

10 Exposes invites

1. “Osculating varieties of Veronesean and their higher secant varieties”, 10 Decembre 2004 - CMS2004 Winter Meeting, Montreal (Quebec, Canada).

2. “Secant varieties to osculating varieties of Veronesean” , 18 Fevrier 2005 - Departamento deAlgebra, Universidad Complutense de Madrid. (Madrid, Espagne).

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

4. “Varietes des secantes a des varietes qui parametrisent des formes obtenues comme produit deformes lineaires”, 29 Mai 2006, Giornate di Geometria Algebrica e argomenti correlati VIII, Di-partimento di Matematica, Universita di Trieste (Italie).

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

6. “Secant varieties to osculating varieties of Veronese Varieties” , 4 Septembre 2008 - Departamentode Algebra, Universidad Complutense de Madrid, (Madrid, Espagne).

7. “Representation de varietes algebriques”, 28 October 2008, Dipartimento di Matematica, Univer-sita degli studi di Bologna (Italie).

8. “Varietes qui parametrisent polynomes completement decomposables”, 13 Mars 2009, Diparti-mento di Matematica, Universita degli Studi di Firenze (Italie).

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

10. “Du probleme de Waring aux telecommunications”, 10 December 2009, Universita degli studi diTrento (Italie).

11. “From the Waring problem to tensor rank through secant varieties”, 18 Mars 2010, SAGA WinterSchool, Auron, Nice (France).

12. “Du probleme de Waring aux telecommunications”, 20 Avril 2010, Universita degli studi di Ancona(Italie).

13. “Un peu de science dans l’iconographie russe”, 17 Juin 2010, Universita degli studi di Trento(Italie).

14. “Decomposition of Homogeneous Polynomials”, 15 September 2010, Workshop on Tensor Decom-positions and Applications (TDA 2010). 13-17 Septembre 2010. Monopoli, Bari (Italie).

15. “Varietes des secantes a des varietes qui parametrisent des tenseurs : actualite du probleme deWaring, aspects geometriques correles et applications”, 7 Octobre 2010. Trieste (Italie).


Curriculum Vitae de Alessandra Bernardi 11 PRESENTATIONS

16. “Applications recentes de resultats classiques sur les varietes secantes a des varietes qui pa-rametrisent des tenseurs. Du probleme de Waring au rang des tenseurs”, 22 Novembre 2010,Conference “Progressi Recenti in Geometria Reale e Complessa”, Levico Terme (Trento, Italie),17–22 octobre, 2010.

17. “Secant varieties and Rank of tensors”, 1 Fevrier 2011, Mittag-Leffler Institute, Spring Semester :“Algebraic Geometry with a view towards applications”, 17 Janvier – 15 Juin 2011 (Suede).

18. “Polynomial and Tensor Decompositions”, 22 March 2011, GALAAD–INRIA, Sophia AntipolisMediterranee (France).

19. “Decomposition of Structured Tensors, Algorithms and Characterization”, 9 Mai 2011, MultimediaGeometry Seminars, Universita degli Studi di Trento (Italie).

20. “Varietes des secantes : dimension, ideaux et rang des tenseurs”, 23 May 2011, Politecnico diTorino (Italie).

21. “Tensor decompositions : achievements and developments”, 26 October 2011, Universita degli studidi Trento (Italie).

22. “Rang de tenseur”, 16 Novembre 2011, Universita degli studi di Torino (Italie).

23. “Ranks of Tensors, related varieties and applications”, 18 November 2011, “Genova-Torino-MilanoSeminar : some topics in Commutative Algebra and Algebraic Geometry”, Milano (Italie), 17–18novembre , 2011.

24. “Decomposition of partially symmetric tensors”, 2 December 2011, Universita degli studi di Firenze(Italie).

25. “Tensor Decomposition : a link between Algebraic Geometry and Applications”, 4 Avril 2012,Universita degli studi di Bologna (Italie).

26. “Various approaches for polynomial decomposition”, 23 Octobre 2012, Universite de Pau (France).

27. “A generalization of Sylvester Algorithm”, 4 December 4 2012, Universidad Complutense de Ma-drid (Espagne).

28. “Algebraic Geometry in Signal processing, Phylogenetic and Quantum Physics”, Colloquium Po-litecnico di Torino, 30 Mai 2013, Politecnico di Torino (Italie).

29. “Tensor Ranks”, 1 August 2013, 2013 SIAM Conference on Applied Algebraic Geometry, FortCollins (Colorado, USA).

30. Main Speaker at “36th Autumn School in Algebraic Geometry : Power sum decomposition andapolarity, a geometric approach”. September 1-7, 2013, Lukecin, Poland.

11 Presentations

31. “Sulle funzioni convesse” (“Sur les fonctions convexes”), 27 Fevrier 2002, Dipartimento di Mate-matica, Universita di Trieste.


Curriculum Vitae de Alessandra Bernardi 11 PRESENTATIONS

32. “Dimostrazione del teorema di Darboux” (“Demonstration du theoreme de Darboux”), 27 Sep-tember 2002, Dipartimento di Matematica, Universita di Trieste.

33. “Programma di Sarkisov in dimensione 2 per la classificazione degli Spazi Fibrati di Mori secondo laTeoria di Mori” (“Programme de Sarkisov en dimension 2 pour la classification des Espaces Fibresde Mori suivant la theorie de Mori”), 18 Juillet 2002, Dipartimento di Matematica, Universta diMilano.

34. “Esposizione dell’articolo di G. Canuto Curve associate e Formule di Pluker nelle Grassmaniane,apparso su “Inventiones Mathematicae”, 53, 77-90 (1979)” (“Presentation de l’article par G. Ca-nuto Courbes associees et Formules de Pluker dans des Grassmanniennes, paru dans “InventionesMathematicae”, 53, 77-90 (1979)”) , 15 Janvier 2003, Dipartimento di Matematica, Universita diMilano.

35. “How one’s can calculate all the differential invariants of Seg(Pn × Pn) ∩H, where H is a generichyperplane. Understand this as a homogeneous variety of Sln+1C”, 13 Fevrier 2003, Dipartimentodi Matematica, Universita di Trieste.

36. “Un’introduzione al problema dello studio della Postulazione dei Punti Grassi” (“Une introduc-tion au probleme de l’etude de la postulation des points epais”), 19 Mars 2003, Dipartimento diMatematica, Universita degli Studi di Milano.

37. “Un’introduzione al problema dello studio della Postulazione dei Punti Grassi e recenti applica-zioni” (“Une introduction au probleme de l’etude de la postulation des points epais et applicationsrecentes”), 23 Mai 2003, Dipartimento di Matematica, Universita di Pavia.

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

39. “Secant varieties to the Osculating varieties of the Veronesean”, 13 Octobre 2004, MathematicalDepartment, Queen’s University (Kingston, Ontario, Canada).

40. “Varieta delle secanti alle Veronese e applicazioni algebriche” (“Varietes des secantes aux Veroneseet applications algebriques”), 26 Janvier 2005 - Departamento de Algebra Universidad Complu-tense de Madrid (Madrid, Espagne).

41. “Varieta delle secanti alle varieta tangenziali ed osculanti a varieta di Veronese” (“Varietes dessecantes aux varietes tangentielles et osculantes a ds varietes de Veronese”) , 2 Fevrier 2005,Departamento de Algebra, Universidad Complutense de Madrid (Madrid, Espagne).

42. “Construction of Cominuscule Varieties” , 6 Octobre 2005, Mathematical Department, Texas A&MUniversity (College Station, Texas, USA).

43. “An introduction to Representation Theory”, 2 Novembre 2005, Mathematical Department, TexasA&M University (College Station, Texas, USA).

44. “An introduction to de Rham Cohomology”, 17 Novembre 2005, Mathematical Department, TexasA&M University (College Station, Texas, USA).

45. “An introduction to de Rham Cohomology”, 18 November 2005, Mathematical Department, TexasA&M University (College Station, Texas, USA).


Curriculum Vitae de Alessandra Bernardi12 PARTICIPATION A D’AUTRES CONFERENCES ET ECOLES

46. “An introduction to de Rham Cohomology”, 22 November 2005, Mathematical Department, TexasA&M University (College Station, Texas, USA.

47. “On Alexander-Hirshowitz theorem via Lemma d’Horace”, 1 Decembre 2005, Mathematical De-partment, Texas A&M University (College Station, Texas, USA).

48. “Lemme d’Horace differentiel”, 5 Decembre 2005, Mathematical Department, Texas A&M Uni-versity (College Station, Texas, USA).

49. “Dall’Algebra Lineare a questioni irrisolte” (“De l’algebre lineaire a des questions ouvertes”), 15Mai 2008, Seminario per le cours d’Algebre Superieure, Dipartimento di Matematica, Universitadi Bologna.

50. “Ideale delle varieta di Segre-Veronese” (“L’ideal des varietes de Segre-Veronese”), 12 Juin 2008,DIMA Universita degli studi di Genova.

51. “Rango e rango simmetrico di tensori simmetrici” (“Rang et rang symetrique de tenseurs symetriques”),3 Mars 2009 - Dipartimento di Matematica, Universita degli Studi di Bologna.

52. “Algorithms for computing the rank of a tensor”, 11 Fevrier 2011, Mittag-Leffler Institute, SpringSemester : “Algebraic Geometry with a view towards applications” 17 Janaury – 15 June 2011.

53. “Tenseurs”, 8 Mars 2011, GALAAD–INRIA Mediterranee, Sophia Antipolis, France.

12 Participation a d’autres conferences et ecoles

— “Summer School Perugia”, Perugia (Italie), 29 Juillet - 1 Septembre 2001.

— “Pragmatic 2003”, Catania (Italie), 9 - 28 Juin 2003.

— “Interpolation problem and Projective embeddings”, Torino (Italie), 15 - 20 Septembre 2003.

— “Workshop on Algebraic curves, monodromy and related topics”, Milano (Italie), 1-2 Avril 2004 ;

— “International school on Projective Geometry”, Anacapri (Italie), 1-5 Juin 2004.

— “Projective Varieties with unexpected geometric properties”, Siena (Italie), 8-12 Juin 2004.

— “School/Workshop on Computational Algebra for Algebraic Geometry and statistics”, Torino (Ita-lie), 6 - 11 Septembre 2004.

— “Rt. 81 conference in honor of Graham Evans and Workshop on Resolution (for young resear-chers)”, Cornell University of Ithaca, New York, USA, 1-3 Octobre 2004.

— “CMS 2004 Winter meeting”, Montreal, Quebec (Canada), 10-13 Decembre 2004.

— “AGaFE, Geometry of Algebraic Varieties”, Ferrara, 22-25 Juin 2005.

— “Texas Geometry and Topology conference”, Austin, Texas, USA, 30 Septembre - 2 Octobre 2005.

— “Geometric and Probabilistic Methods in group theory and dynamical systems”, 4-6 Novembre2005, Texas A&M University, College Station, Texas, USA.

— “Harvey/Polking conference, Singularities in Analysis, Geometry and Topology”, 11-13 Novembre


Curriculum Vitae de Alessandra Bernardi12 PARTICIPATION A D’AUTRES CONFERENCES ET ECOLES

2005, Rice University, Austin, Texas, USA.

— “Giornate di Geometria Algebrica e argomenti correlati VIII” (“Journees de Geometrie Algebriqueet sujets correles”, Trieste, 26-29 Mai 2006.

— Graduate Workshop : “Geometry and representation theory of tensors for computer science, sta-tistics and other areas”, MSRI - Mathematical Sciences Research Institute - (Berkeley, California- USA), 7 - 20 Juillet 2008.

— “Geometry and representation theory of tensors for computer science, statistics and other areas”,AIM - American Institute of Mathematics - (Paolo Alto, California - USA), 21 - 27 Juillet 2008.

— “INDAM workshop : Geometry of projective varieties” (Roma), 30 Septembre - 4 Octobre 2008.

— “Workshop on tensors and interpolation”, 10-11-12 Juin 2009, INRIA-JAD-CNRS Nice, France.

— “Conference on Classical and recent aspects in the study of projective varieties. In honour ofLucian Badescu on the occasion of his 65th birthday”, Genova (Italie), 21-22 Janvier 2010.

— “SAGA Winter School”, Auron, Nice, France March, 15-19 Mars, 2010.

— “INdAM Conference “Complex Geometry””, Levico Terme, Trento, Italie, 31 Mai - 4 Juin 2010.

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

— “Summer school : Geometry of tensors and applications”, Sophus Lie Conference Center, Nordf-jordeid - Norvege, 14-18 Juin 2010.

— “International Conference on Perspectives on Algebraic Varieties”, Levico Terme, Trento (Italie),5-10 Septembre 2010.

— “Workshop on Tensor Decompositions and Applications (TDA 2010)”. 13–17 Septembre 2010.Monopoli, Bari, Italie.

— “Progressi Recenti in Geometria Reale e Complessa”, Levico Terme (Trento, Italie), 17–22 Octobre2010.

— “Algebraic Geometry in the sciences”, (Oslo, Norvege), 10–14 Janvier 2011.

— Spring semester 2011 “Algebraic Geometry with a view towards applications”, (Mittag-LefflerInstitute, Stokholm, Suede) 17 Janvier 17 – 26 Fevrier 2011.

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

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

— “MEGA 2011 : Effective Methods in Algebraic Geometry” , Stockholm University (Stockholm,Suede), 30 Mai - 3 Juin, 2011.

— “Genova-Torino-Milano Seminar : some topics in Commutative Algebra and Algebraic Geometry”,17–18 Novembre 2011, Milano (Italie).

— “G.T.M. Seminar Genova, Torino, Milano Seminar : Some Topics in Commutative Algebra andAlgebraic Geometry”, Juin 28–29, 2012, Torino (Italie).


Curriculum Vitae de Alessandra Bernardi 13 TRAVAIL DE RAPPORTEUR

— “Groebner Bases, Curves, Codes and Cryptography”, 30 Juillet – 10 Aout, 2012, Trento (Italie).

— “School (and Workshop) on Invariant Theory and Projective Geometry”, 17 – 22 Septembre 2012,Trento (Italie).

— “3rd SAGA Workshop”, 9–11 Octobre 2012, Trento (Italie).

— [Invitee] “Tensor Ranks”, 1 Aout 2013, 2013 “SIAM Conference on Applied Algebraic Geometry”,Fort Collins (Colorado, USA).

— [Invitee] Main Speaker at “36th Autumn School in Algebraic Geometry : Power sum decomposi-tion and apolarity, a geometric approach”. 1-7 September 2013, Lukecin (Pologne).

— Genova-Torino-Milano Seminar : Some Topics in Commutative Algebra and Algebraic Geometry28-29 Janvier 2014 (Politecnico di Milano, Italie).

— Vector Bundles Days II, Pau-Trieste Workshop on Vector Bundles and Related Topics. On theoccasion of Emilia Mezzetti’s 60th birthday. 29-31Janvier 2014 (Trieste, Italie).

— [Invitee] [Decline pour des raisons familiales] Tensors and Optimization Conference 19–22 Mai2014, “SIAM Optimization Meeting”, Town and Country Resort and Convention Center, SanDiego (California, USA).

— [Invitee] [Decline pour des raisons familiales] Computational Nonlinear Algebra, “ICERM Confe-rence”, 2–6 Juin 2014, Brown University (Providence, USA).

— [Invitee] [In Programma] November 2014 – December 2014. Long-Term Participant al SimonsInstitute for Theory of Computing, Berkeley (California, USA), nell’ambito dell programma au-tunnale 2014 “Algorithms and Complexity in Algebraic Geometry”, Invitata da P. Burgisser, JMLandsberg, K. Mulmuley, B. Sturmfels.

13 Travail de Rapporteur

13.1 Reviewer

Reviewer dans les domaines 14N05, 13D40, 14J26, 14M15, 13P10 (MSC 2000) pour : AMS Mathema-tical Reviews et Zentralblatt MATH.

13.2 Rapporteur

Rapporteur pour les revues suivantes :— Algebra & Number Theory (ANT).— Experimental Mathematics (EM).— Indiana University Mathematics Journal (IUMJ).— International Journal of Engineering, Science and Technology (IJEST).— Journal of Symbolic Computation (JSC).— Mediterranean Journal of Mathematics (MedJM).— Numerical Linear Algebra with Applications (NLA).— Proceedings of the American Mathematical Society (Proc. Amer. Math. Soc.).


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