[Seminario cofinanziato dal PRIN2022 Interactions between Geometric Structures and Function Theories]

I will explain some joint work in progress with R. Svaldi and S. Velazquez on the construction of moduli spaces of foliations which is modeled on the approach to the construction of the moduli space of curves.  Time permitting I will explain some applications of this work to moduli problems of related structures (such as fibrations).

[Seminario cofinanziato dal PRIN2022 Interactions between Geometric Structures and Function Theories]

This talk offers an introductory exploration of Cauchy-Riemann Geometry (or CR geometry), a mathematical area bridging Differential Geometry, Complex Analysis and Partial Differential Equations theory. Starting from the so-called tangential Cauchy-Riemann equations, we will introduce the foundational concepts of CR structures, which are particular complex subbundles on real manifolds.

We will then focus on the interplay between CR geometry and other branches of Differential Geometry such as contact, Sub-Riemannian and Lorentzian geometries.

Finally we will consider the counterpart of harmonic maps within the CR category, namely subelliptic harmonic maps. The principal part of the Euler-Lagrange equations system describing these maps is the sublaplacian, a second order elliptic degenerate yet subelliptic operator: this will bring into the picture the relevance of subelliptic PDEs, which play within CR geometry the same strong role held by elliptic theory in the Riemannian one.

Some of the results presented in this last section are part of a joint work with S. Dragomir and E. Loubeau.

The classical theorem of Poincaré-Hopf about the index of vector fields on smooth manifolds has given rise to a vast literature concerning indices of vector fields and Baum-Bott residues for singular holomorphic foliations. In this talk we will review some of the foundational aspects of this theory having as a spear-

head the GSV-index, an invariant that generalizes the Poincaré-Hopf index to the case of singular varietes.

Let $UT_n^{(-)}$ be the Lie algebra of the $n\times n$ upper triangular matrices over a field. The description of the group gradings on this álgebra is an important problem. When considering the associative algebra $UT_n$, it was shown by Valenti and Zaicev that every grading is isomorphic to elementar one (in other words all matrices $e_{ij}$ are homogeneous in the grading). The elementary gradings on $UT_n$ were described some 20 years ago by Di Vincenzo, PK, Valenti. Surprisingly (or not so) in the case of Lie algebras there appear gradings that are not isomorphic to elementary ones. We classify all group gradings on $UT_n^{(-)}$. Similar methods and results hold for the Jordan algebra $UT_n^{(+)}$. We also draw conclusions concerning the graded polynomial identities these gradings satisfy. This is a joint work with F. Yukihide, parts of it published around 2018, and also recently, in 2023, in streamlined and more general form.

[Seminario cofinanziato dal PRIN2022 Interactions between Geometric Structures and Function Theories]

Studiamo le funzioni normali (sezioni del fibrato Jacobiano) definite sullo spazio dei moduli di curve piane (puntate). 

Utilizzando tecniche introdotte da Griffiths Green e Voisin dimostriamo che una  funzione normale con  supporto non banale ma sufficientemente "piccolo", non può essere "localmente costante".

Come applicazione, diamo una dimostrazione variazionale del seguente teorema di Zu:

Teorema: Se C è una curva piana molto generale di grado d e C' è una qualsiasi curva di grado d', allora la cardinalità i(C,C') dell'intersezione tra C e C' è >d-3.

Mostriamo inoltre che se d>3 e i(C,C')=d-2 allora C' è una retta bitangente o di flesso. Per d=4 questo è un risultato di Chen Rield e Yeong.

Il lavoro e' frutto di una collaborazione con Lorenzo Fassina.

We will present a self-contained overview on the geometry of the geodesic flow of Lie groups with left-invariant pseudo-Riemannian or holomorphic-Riemannian metrics. As well as recalling some beautiful and seminal classical theory, we will present some recent developments in this topic. This talk will be based on a series of joint works with S. Chaib, A. Elshafei, H.Reis, M. Sanchez and A. Zeghib.

‘Supersymmetry multiplets’ are a class of representations of the super-Poincaré algebra that underpin all supersymmetric field theories in physics. In this talk, I will explain how the pure spinor superfield formalism offers a systematic method for deriving supersymmetry multiplets from geometric data associated with certain algebraic varieties — namely, nilpotence varieties, which parametrize square-zero elements in a Lie superalgebra. After discussing several examples, I will outline, time permitting, a generalization of the formalism within the framework of derived geometry, in connection with Koszul duality.

Various techniques have been given to produce free resolutions of monomial ideals in polynomial rings, including the Taylor Resolution and the Barile-Macchia Resolution.  We will discuss how to modify these methods to produce resolutions of monomial ideals in rational normal surfaces.  This talk is based on joint work with Margherita Barile.

In order to investigate the shape of a lower Bruhat interval in a Coxeter group, we introduce the notion of Lehmer complex of a Bruhat interval. It is obtained by using the so-called Lehmer code, which provide a description of a lower Bruhat interval as a multicomplex. In this talk, the basic notions involved are defined and some results regarding the Poincaré polynomials of lower Bruhat intervals in the symmetric group are presented. In particular, it is established a characterization of all possible Poincaré polynomials of smooth Schubert varieties in a flag variety, by setting a correspondence between them and a well-known set of partitions. Moreover, we improve a result due to Bjorner and Frankl and a classical result due to Stanley, both concerning M-sequences. 

This is a joint work with Paolo Sentinelli.

We will review the state of the art and some recent results on the rationality and unirationality of conic bundles and hypersurfaces. We will discuss some open problems, and the unirationality of surface conic bundles with eight singular fibers.

There are several important theorems concerning the subject of matrix embeddings. Many of these results aim to address the following fundamental question: When does a given ring have an embedding into n × n matrices over some commutative ring? An obvious necessary condition is that the ring must satisfy the polynomial identities of n × n matrices, but this condition is not sufficient for n > 2. Procesi proved that an algebra R with trace can be embedded into n × n matrices over some commutative ring if and only if it satisfies the Cayley-Hamilton identity of degree n. Here we recall that the Cayley–Hamilton polynomial of a matrix a can be written as a polynomial whose coefficients are polynomials in the traces of ak, k ≥ 1. Such algebras are known as Cayley-Hamilton algebras. Interestingly, the image of this embedding aligns with a ring of invariants, hinting at potential geometrical applications. The main goal of this talk will be to discuss these results in algebras far beyond Cayley-Hamilton algebras, presenting recent developments and findings in this area. This research is supported by the Sa ̃o Paulo Research Foundation (FAPESP), grant No 2023/04011-9.

In [6], Sheekey introduced a class of linearized polynomials called scattered, bringing to light their connection to maximum rank-metric codes. Although these polynomials have already appeared in various contexts, such as in [1] and [5], this link with coding theory has significantly advanced their study and investigation, sparking considerable interest in recent years. In this talk, we will explore their relationship with certain subsets of the finite projective line over the finite field of order qn known as maximum scattered linear sets, as well as with codes made up of square matrices of order n equipped with the rank metric. We will present the known examples of scattered polynomials up to date and discuss some of their key properties.

We will also address the classification of maximum scattered linear sets of the finite projective line for small values of n and discuss some characterization results for the examples known

so far. Finally, we will retrace how each scattered polynomial gives rise to a translation plane, as

discussed in [2] and in [4]. This talk is based on the recent survey [3].

[1] A. Blokhuis, M. Lavrauw. Scattered spaces with respect to a spread in PG(n, q). Geom. Dedicata 81(1) (2000), 231-243.

[2] V. Casarino, G. Longobardi, C. Zanella. Scattered linear sets in a finite projective line and translation planes, Linear Algebra Appl. 650 (2022), 286-298.

[3] G. Longobardi. Scattered polynomials: an overview on their properties, connections and applications, submitted to Art Discrete Appl. Math.

[4] G. Longobardi, C. Zanella. A standard form for scattered linearized polynomials and properties of the related translation planes, J. Algebr. Comb. 59(4) (2024), 917-937.

[5] G. Lunardon, O. Polverino. Blocking sets and derivable partial spreads. J. Algebr. Combin. 14 (2001), 49-56.

[6] J. Sheekey. A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3) (2016), 475-488.

The aim of the talk is to present some recent developments in the theory of minimal graded free resolutions of ideals defined in a polynomial ring over a field. In this connection, some authors introduced the expression “Barile-Macchia resolution”  to denote a notion that is rapidly changing its meaning.

The topics addressed are: commutative algebra, graph theory, homological algebra.

We begin with the notions of holomorphic foliations and the positivity. The essential point of this talk, consists in understanding the role played by a positivity condition on some object associated to a foliation on the global behaviour of it.

By using conjugate and disconjugate theorems for second-order linear differential equations, we establish an improvement of the Bonnet--Myers theorem for complete Riemannian manifolds via m-Bakry--Émery Ricci curvature. In contrast to the classical theorem of S.B. Myers (Duke Math. J. 8 (1941), 401--404), our result does not require non-negativity of the $m$-Bakry--Émery Ricci curvature and is new even the $m$-Bakry--Émery Ricci curvature is reduced to the Ricci curvature.

We study which properties of finite-dimensional nilpotent algebras are typical. For example, any r-generated nilpotent algebra of nilpotent class c is isomorphic to a factor-algebra of a free nilpotent algebra F=F(r,c) of rank r>1 and class c>1 by an ideal J. We prove that  a typical algebra is obtained iff J belongs to the annihilator (center) of F. We also prove that the typical automorphisms of F are scalar modulo F2. The algebras are not necessarily associative or Lie.

One of the most important features of identical relations in algebras is that in some important classes, two algebras are isomorphic if and only if they satisfy the same identities. The first decisive result in this area was proven half a century ago by Yuri P. Razmyslov. Recently, his virtually forgotten result reemerged in the literature because it appeared to be stronger than a number of much more recent theorems. The goal of the talk is to describe the current state of research in this intriguing area.

[seminario cofinanziato dal PRIN2022 Interactions between Geometric Structures and Function Theories ]

The first variation of the Landau Hall functional on a Riemannian manifold leads to the notion of magnetic curves. Computing the second variation, we obtain the equation of a Jacobi-type field along a magnetic curve. In this talk we focus on the contact magnetic trajectories in Sasakian and cosymplectic manifolds (as ambient space) emphasising the main differences between the two cases. We will give several examples.

This talk is based on some joint papers:

• M.I. Munteanu, \emph{Magnetic geodesics in (almost) cosymplectic Lie groups of dimension 3}, Mathematics, (Special Issue: Topics in Differential Geometry), 10 (2022) 4,  art. 544.

• J. Inoguchi, M.I. Munteanu, \emph{Magnetic Jacobi fields in 3-dimensional Sasakian space forms}, J. Geom. Analysis, 32 (2022) 3, art. 96.

• M.I. Munteanu, A.I. Nistor, \emph{Magnetic Jacobi fields in cosymplectic 3-dimensional manifolds}, Mathematics, (Special Issue: Differential Geometry: structures on manifolds and their applications), 9 (2021) 24 art. 3220.

• J. Inoguchi, M.I. Munteanu, \emph{Magnetic Jacobi fields in Sasakian space forms}, Mediterranean J. Mathematics, 20 (2023) art. 29.

The Lvov-Kaplansky conjecture states that the image of a multilinear polynomial evaluated on the matrix algebra over a field is a vector space. This result has only been proven for matrices of order 2, with some restriction on the base field. In this talk, we will present the topic of images of polynomials in algebras, and recent results on some variations of the Lvov-Kaplansky conjecture, considering different (not necessarily associative) algebras as well as algebras with additional structures.

I will survey some basic facts about "regular functions" in the quaternionic setting.

First, I will compare the non-equivalent notion of regularity and the I will specialise in the theory of slice regularity (possibly in relation to monogenicity). Then I will discuss covering properties and algebraic structure for slice regular functions.

Let G denote an Abelian group of odd order, written additively. A 3-term arithmetic progression of G, 3-AP for short, is a set of three distinct elements of G of the form g, g+d, g+2d, for some elements g, d of G. 

Let S denote a subset of G. Then S is called 3-AP-free if it does not contain a 3-AP. Moreover, S is called complete 3-AP-free if S is 3-AP-free and it is not contained in a larger 3-AP-free set. We will say that S is 3-AP saturating if for each element x of G \ S there is a 3-AP of G consisting of x and two elements of S. Clearly, S is complete 3-AP-free if and only if it is 3-AP-free and 3-AP saturating.

A classical problem in additive combinatorics is to obtain good upper and lower bounds for the maximal size of a 3-AP-free set. In the special case when G=GF(3)n (GF(q) denotes the finite field of order q) then the 3-AP-free sets are exactly the affine point sets without a collinear triple. In general, point sets of V=GF(q)n without a collinear triple are called (affine) caps. In finite geometry it is a classical problem to find small complete caps (a cap is complete if it is not contained in a larger cap) and small saturating sets (a point set S is saturating if for each point x of V \ S there is a collinear triple ofV consisting of x and two elements of S).

In this talk I will present constructions of complete 3-AP-free sets and 3-AP saturating sets whose size is close to the trivial lower bound. We will be mostly concerning the cases when G is a finite vector space or a cyclic group.

This is a joint work and work in progress with Zoltán Lóránt Nagy. 

We prove the existence of residues related to tangency points of maps generically transverse to locally free holomorphic distribution. We provide some formulas relating these residues to the characteristic classes of these distributions. This is a joint work with Mauricio Corrêa.

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with Ulrich and Vasconcelos, but due to their nature, no numerical invariant was immediately available.

The present work introduces the Bourbaki degree as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree of the given graded module. Since the main intent is a study of plane curve singularities via this new numerical invariant, accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form f ∈ k[x, y, z] – i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a tiny facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds. In particular, we revisit results of du Plessis and Wall, and of Dimca and co-authors.

(Joint work with Marcos Jardim and Abbas N. Nejad)

Linear Degenerations of the Flag Variety arise as very natural generalizations of the Complete Flag Variety and their geometrical properties very often appear to be linked with interesting combinatorial patterns.

In the first part of the talk I am going to introduce these objects, discuss their remarkable properties and  state the main problems in this interesting research area. 

In the second part of the talk I will focus on a special class of linear degenerations, the Flat Degenerations, that have the remarkable property of being equidimensional algebraic varieties of the same dimension as the Complete Flag Variety.

Finally,  if there is enough time, I will show how suitable torus actions and quiver representation theory can be used in this setting to prove some new results about the smooth locus in Flat Linear Degenerations.

The notion of Zeta function for groups was introduced in a seminal paper from Grunewald, Segal and Smith and proved to be a powerful tool to study the subgroup growth in some classes of groups. In this seminar I will introduce this Zeta function presenting some general properties for this object. I will then focus  on some results obtained for class two nilpotent groups. In particular, I will describe some combinatorial tecniques used to tackle this problem, namely the study of series associated to polyhedral integer cones.

This is a joint work with Christopher Voll and Marlies Vantomme.

This is a joint work with S. Coughlan, Y. Hu e T. Zhang. In the theory of the algebraic varieties of general type the key invariants are the (geometric) genus and the volume. I will discuss the known relations among these two numbers, and then discuss the case when the volume is minimal respect to the genus. In particular I will present the classical results of M. Noether-F-Enriques-E. Horikawa in dimension 2 and then present the analogous result we achieved in dimension 3.

The Cayley–Bacharach condition is a very classical property that found its roots in ancient and classical geometry. In this talk we retrace the main steps through the history that have led to the modern formulation. We show how be- ing Cayley–Bacharach with respect to the complete linear system of hypersurfaces of given degree forces a set of points in the projective space to lie on a reduced curve of low degree. In particular, starting from a result for points in the projec- tive plane due to Lopez and Pirola, we present some partial extensions to any Pn. Moreover, in a joint work with F. Bastianelli, we study the Cayley–Bacharach con- dition for points on Grassmannians; we rephrase the Cayley–Bacharach condition as a property for linear subspaces and we prove that this property affects their geometry. Namely, we get an upper bound for the dimension of the linear span of linear subspaces satisfying Cayley–Bacharach conditions. Finally we apply these results to different topics. The main one is the computation of the covering go- nality of the 3-fold and 4-fold symmetric product of a smooth complex projective curve of genus at least 4 and 5 respectively. Other applications concern linear se- ries on curves lying on smooth surfaces in P3 and the so-called correspondences with null trace.

A group il called finitely generated if every element can be expressed in terms of a finite subset of elements of the group. We will be discussing hyperbolic groups, a class of finitely generated groups including fundamental groups of closed negatively curved manifolds, as well as some generalizations.

Nella prima parte di questo seminario introdurremo le algebre di Lie differenziali graduate (dgLa) e spiegheremo il loro uso in teoria delle deformazioni, illustrando molti esempi classici in cui esse vengono usate in maniera vantaggiosa. Nella seconda parte ci concentreremo sullo studio dei luoghi di Brill-Noether. Sia C una curva liscia proiettiva, i luoghi di Brill-Noether W_d^k sono spazi dei sistemi lineari su C di grado d e di dimensione proiettiva almeno k. Sono classicamente note proprietà di tali luoghi, quali la connessione, la dimensione, i luoghi singolari e ci sono vari lavori abbastanza recenti in cui tali risultati vengono generalizzati al caso di fibrati vettoriali di qualunque rango e a varietà lisce proiettive di qualunque dimensione. In un lavoro con Donatella Iacono, ci occupiamo di studiare i luoghi di fibrati vettoriali aventi almeno un numero fissato di sezioni indipendenti su una varietà proiettiva liscia. Usando le algebre di Lie differenziali graduate siamo in grado di riottenere e generalizzare alcuni risultati che riguardano lo spazio e il cono tangente e i luoghi singolari di tali spazi.

Janet Vassilev: Toric face rings are a generalization of both Stanley-Reisner rings and affine semigroup rings. Our goal in this talk is two-fold: (1) We will describe the ring of differential operators of a toric face ring $R$ using the rings of differential operators on affine semigroup rings which are algebra retracts of $R$, and (2) we will give a complete characterization of monomial ideals in an affine semigroup ring which are fixed by a differential operator. This is joint work coming from two projects with Berkesch, Chan, Klein, Matusevich and Page and with Miller and Taylor.

Dimiter Vassilev: The general themes of the talk are Dirichlet forms, fractional operators and associated Sobolev type spaces on groups of homogeneous type. Our results lead to explicit integral formulas of the infinitesimal generators, which are the studied fractional operators, and embedding theorems between the relevant spaces. The considered groups are not assumed to be Carnot groups or to satisfy a H\"ormander type conditions. Finally, we will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.

One of the important issues in differential geometry is to study the relation between curvature and topology of Riemannian manifolds. In this talk, I would like to introduce various generalizations and improvements of the classical Myers theorem via some modifications of the Ricci curvature.If time permits, I would also like to discuss some relations between topology and geometric structures on Riemannian manifolds such as Myers-type theorems on Sasaki manifolds.

A number of different polynomial functions over finite fields have relevant applications in applied areas of Mathematics, as Cryptography or Coding Theory. Among them, APN functions, PN functions, APN permutations, permutation polynomials have been widely studied in the recent years.

In order to investigate non-existence of such functions or to provide constructions of infinite families, algebraic varieties over finite fields are a useful tool. In this direction, a key ingredient is an estimate of the number of rational points of such algebraic varieties and therefore Hasse-Weil type theorems (Lang-Weil, Serre,. . . ) play a fundamental role.

The aim of this talk is to summarize recent results in this direction.

In the tentative to move from the Kähler to the non-Kähler setting, one can formulate several problems concerning Hermitian metrics on complex manifolds with special curvature properties. Among these problems, we mention the existence of Hermitian metrics with constant scalar curvature with respect to the Chern connection, and the generalizations of the Kähler-Einstein condition to the non-Kähler setting. They are usually translated and attacked as analytic pdes.

In this context, the Chern-Ricci flow plays an useful role. The Chern-Ricci flow is a parabolic evolution equation for Hermitian metrics that extends the Kähler-Ricci flow to Hermitian manifolds. It is expected that the behaviour of solutions of the Chern-Ricci flow deeply reflects the underlying complex structure. In particular, understanding the behaviour of the Chern-Ricci flow on non-Kähler compact complex surfaces is particularly interesting, due to the fact that minimal class VII surfaces are not yet completely classified.

In this talk, we study the problem of uniform convergence of the normalized Chern-Ricci flow on Inoue-Bombieri surfaces with Gauduchon metrics.

The talk is based on a joint work with Valentino Tosatti, and on collaborations and discussions with Simone Calamai, Francesco Pediconi, Cristiano Spotti, and others.

A projective variety X of dimension n is said to be rational if it is birationally isomorphic to the n-dimensional projective space P^n, i.e. if it contains an open dense subset isomorphic to an open dense subset of P^n. 

Due to our knowledge about projective spaces, it is important to understand whether a given variety is rational or satisfies some property of rational varieties.  

In addition, there has been a great deal of recent interest and progress in studying the so-called “measures of irrationality”, i.e. birational invariants that somehow measure how a given variety is far from satisfying properties of rational varieties.

In this talk, I will discuss these invariants and I will focus on various recent results concerning measures of irrationality for hypersurfaces in P^n.

The theory of distributions and foliations has its origins in the studies of nineteenth-century mathematicians such as Grassmann, Jacobi, Clebsch, Cartan and Frobenius. They were motivated by the fundamental work due to Pfaff, who proposed a geometric approach to the study of differential equations. Distributions and foliations play an important role in several subjects from differential, complex, algebraic and Poisson geometries. The qualitative study of foliations induced by complex polynomial differential equations was investigated by Poincaré, Darboux and Painlevé. In this talk I will discuss some Algebro-geometric methods in  the study of holomorphic distributions and foliations, their moduli spaces and topological invariants.