Conferenza: II Meeting UMI for Doctoral Students.

Abstract: A Waring decomposition is an expression of a homogeneous polynomial as a sum of powers of linear forms. The minimum possible number of addends in such a decomposition is classically known as the Waring rank, or simply the rank, of the polynomial. We analyze the problem of determining the rank of the powers of quadratic forms, for which there are several examples of decompositions in the literature. We approach this problem from a contemporary perspective and provide some estimates for the rank value.

Conferenza: JM Invariant at 60: A Conference on Tensor Invariants in Geometry and Complexity Theory.

Abstract: An isotropic form is a linear form whose coefficients correspond to an isotropic point. For any natural numbers n and d, the space of homogeneous harmonic polynomial of degree d in n variables is generated by the d-th powers of isotropic forms. This allows us to define the isotropic rank of a homogeneous polynomial h as the minimum natural number r such that h can be written as a linear combination of the d-th powers of r isotropic forms. Using secant varieties and Terracini's Lemma we determine the generic isotropic rank for any value of n and d. This is joint work with Cristiano Bocci and Enrico Carlini.

Conferenza: 5th BYMAT Conference: Bringing Young Mathematicians Together.

Abstract: A Waring decomposition of a homogeneous polynomial is an expression in the form of a sum of powers of linear forms. The minimum possible number of addends in such a decomposition is classically referred to as the Waring rank, or simply the rank, of the polynomial. In this study, we examine the rank of the powers of quadratic forms, for which there are numerous examples of decompositions in the literature. We approach this problem from a contemporary perspective and provide some estimates for the rank value.

Conferenza: Secant v. Cactus.

Abstract: Determining the rank of the powers of quadratic forms is a classical problem. Many examples of special decompositions appear in the literature. We analyze this problem from a modern point of view and we give an estimate of the value of the rank. Moreover, we determine its smoothable rank and its border rank.

Conferenza: Algebraic Geometry with Applications to Tensors and Secants – Kickoff Workshop.

Abstract: We determine the border rank of each power of any quadratic form in three variables. Since the case of forms of rank 1 and 2 is quite simple, we basically focus on the case of non-degenerate quadratic forms. Considering the quadratic form in n variables q_n=x_1^2+...+x_n^2, we first determine the apolar ideal of q_n^s for every s, which turns out to be generated by harmonic polynomials of degree s+1. By this result, we select for each power a specific ideal contained in the apolar ideal of the form q_3^s and, making use of the recent technique of border apolarity, we prove that its border rank is equal to the rank of its central catalecticant map, that is (s+1)(s+2)/2.

Conferenza: XVII Encuentro de Álgebra Computacional y Aplicaciones (EACA 2022).

Abstract: A decomposition of a homogeneous polynomial is a representation of that polynomial as a sum of powers of linear forms; in particular, the minimum number of addends in this sum is said to be the rank of the polynomial. We analyze a way to determine explicit decompositions of a polynomial corresponding to a power of a non-degenerate quadratic form. The main instrument used in this context is the apolarity lemma, which is a classic result relating the summands of a decomposition to its apolar ideal.

Descrizione: parte del ciclo di seminari relativo al programma Applied Algebraic Geometry 2023-2024.

Abstract: Strassen's conjecture is a famous statement first uttered by the German mathematician Volker Strassen in 1973. He conjectured in his article Vermeidung von Divisionen that the multiplicative complexity of the union of two linear systems in two disjoint sets of variables is equal to the sum of the complexities of the two systems. Translated: he conjectured that the rank of the direct sum of two tensors is in general equal to the sum of the ranks of the two tensors. The problem of Strassen's conjecture remained unsolved until 2019, when Yaroslav Shitov proved in his article Counterexamples to Strassen's direct sum conjecture that this conjecture is in fact false, constructing a procedure to find a counterexample for tensors in a large number of variables. We propose to get to the core of this proof and go over the procedure constructed by Yaroslav Shitov, analyzing it in detail and providing graphical representations.

Descrizione: parte del ciclo di seminari Seminari di Geometria.

Abstract: The study of the decompositions of polynomials, also known as Waring decompositions, is a very classical problem. We focus, in particular, on the decomposition of powers of quadratic forms, for which many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, which in this case results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

Descrizione: parte del ciclo di seminari relativo al programma Doc in Progress.

Abstract: The study of the decompositions of polynomials, also known as Waring decompositions, is a very classical problem. We focus, in particular, on the decomposition of powers of quadratic forms, for which many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, which in this case results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

Descrizione: parte del ciclo di seminari relativo al programma Applied Algebraic Geometry 2022-2023.

Abstract: The study of the decompositions of the powers of a quadratic form, also called representations, is a very classical problem. Many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, that in the case of the powers of quadrics results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

Descrizione: parte del ciclo di seminari relativo al programma della Red de Doctorandos en Matemáticas UCM.

Abstract: A decomposition of a homogeneous polynomial is a representation of this one as a sum of powers of linear forms; in particular, the minimum number of addends that can provide such a decomposition is said to be the rank of the polynomial. We analyze a way to determine explicit decompositions of a polynomial corresponding to a power of a non-degenerate quadratic form. The main instrument used in this context is the Apolarity Lemma, which represents a classic result relating the summands of a decomposition of a polynomial to its apolar ideal. The first part of the seminar is focused on some basic notions about Apolarity Theory. The second part is instead centered on a strategy thanks to which we can try to determine some points providing us some suitable decompositions of the considered polynomial.

Description: parte del ciclo di seminari Seminario de Álgebra Geometría y Topología.

Abstract: A decomposition of a homogeneous polynomial is a representation of that polynomial as a sum of powers of linear forms; in particular, the minumum number of addends in this sum is said to be the rank of the polynomial. We analyze a way to determine explicit decompositions of a polynomial corresponding to a power of a non degenerate quadratic form. The main instrument used in this context is the Apolarity Lemma, that is a classic result relating the summands of a decomposition to its apolar ideal. The first part of the seminar is focused on some basic notions about Apolarity Theory. The second part is instead centered on a strategy thanks to which we can try to determine some points providing us some suitable decompositions of the considered polynomial.

Description: seminar activity preparatory to the master's degree programme in Mathematics.

Abstract: Lo studio di un complesso doppio da un punto di vista algebrico può essere molto importante in geometria complessa. Esso permette infatti di ottenere importanti informazioni di tipo coomologico sulle varietà olomorfe. Un risultato ottenuto da J. Stelzig garantisce che un generico complesso doppio limitato può essere scomposto in modo essenzialmente unico come somma diretta di complessi più piccoli, di tipo elementare e che non possono essere scomposti ulteriormente. Si tratta di quei complessi doppi detti indecomponibili, che nel corrispondente diagramma assumono la forma di un quadrato o di uno zigzag. La dimostrazione si basa su un procedimento che sfrutta la costruzione di una particolare filtrazione ascendente su un complesso doppio arbitrario, le cui componenti vengono poi ulteriormente scomposte per arrivare, tramite ulteriori filtrazioni, alla forma cercata. Agendo in questo modo si fornisce implicitamente anche un modo pratico per rappresentare graficamente la struttura del complesso. L'obiettivo proposto è quello di utilizzare tale risultato per studiare degli esempi concreti di strutture di complessi doppi su varietà olomorfe. In particolare, dopo aver introdotto alcune nozioni basilari di geometria complessa, come il concetto di varietà olomorfa o di forma dierenziale complessa e i principali risultati correlati, esaminiamo la varietà di Iwasawa e le sue piccole deformazioni, di cui analizziamo, anche gracamente, alcuni esempi espliciti dei relativi complessi doppi delle forme dierenziali invarianti a sinistra. Mentre nel primo caso la struttura del complesso è relativamente semplice, non si può dire lo stesso per le piccole deformazioni, che presentano una forma in generale molto più complicata. Gli esempi che prendiamo in considerazione rientrano nella classe delle cosiddette nilmanifold, cioè varietà differenziabili diffeomorfe a un quoziente di un gruppo di Lie connesso, semplicemente connesso e nilpotente per un sottogruppo discreto e co-compatto. Grazie a una serie di risultati relativi, se vengono soddisfatte opportune ipotesi, la struttura delle principali coomologie risulta interamente determinata dal sottocomplesso nito delle forme invarianti a sinistra. Dunque, dal momento che le varietà che analizziamo soddisfano tali proprietà, una volta disegnato il grafico del complesso relativo, risulta più semplice determinare tutte le informazioni che riguardano le principali coomologie studiate, quali le coomologie di Dolbeault, di Bott-Chern e di Aeppli; in particolare, può essere vericata la semicontinuità superiore delle loro dimensione rispetto alle piccole deformazioni. Inoltre, da un'analisi dei graci ottenuti è possibile ottenere facilmente informazioni sulla successione spettrale di Frölicher e sulle proprietà relative; in particolare eettuiamo qualche considerazione sulla sua degenerazione e osserviamo come le dimensioni delle pagine successive, eccetto la prima, siano in generale non semicontinue superiormente rispetto alle piccole deformazioni.

Conferenza: Homemade Algebraic Geometry: Celebrating Enrique Arrondo's 60th birthday.

Abstract: We determine the border rank of the powers of any quadratic form in three variables. Thanks to the determination of the apolar ideal of any s-th power of a quadratic form, which is the homogeneous ideal generated by the harmonic polynomials of degree s+1, we select  a specific ideal contained in its apolar ideal and, after verifying some properties, we prove through the apolarity lemma that the border rank of the s-th power of a ternary quadratic form is equal to the rank of its middle catalecticant matrix, that is (s+1)(s+2)/2, for every natural number s.

Conferenza: Commutative Algebra Towards Applications.

Abstract: Determining the rank of the powers of quadratic forms is a classical problem. Many examples of special decompositions appear in the literature. We analyze this problem from a modern point of view and we give an estimate of the value of the rank.

Descrizione: corso di lettura relativo ad argomenti fondamentali di geometria algebrica.

Ruolo: co-organizzatore, assieme a Jessica Alessandrì (Università degli Studi dell'Aquila), Stefano Canino (Politecnico di Torino), Martina Costa Cesari (Università degli Studi di Padova), Vincenzo Galgano (Università degli Studi di Trento) and Saverio Andrea Secci (Università degli Studi di Torino); amministratore della pagina web relativa.

Descrizione: ciclo di incontri periodici basati su attività seminariali relative ad argomenti di geometria algebrica e in particolare allo studio dei tensori e delle varietà secanti.

Ruolo: co-organizzatore, assieme ad Alessandra Bernardi (Università degli Studi di Trento), Enrico Carlini (Politecnico di Torino), Gianfranco Casnati (Politecnico di Torino), Luca Chiantini (Università degli Studi di Siena), Alessandro Gimigliano (Alma Mater Studiorum – Università di Bologna), Massimiliano Mella (Università degli Studi di Ferrara), Giorgio Ottaviani (Università degli Studi di Firenze);  amministratore della pagina web relativa.