Seminars at Unicampania

Title: Quadratic sets of the Klein quadric

Quadratic sets in projective spaces were introduced by Buekenhout with the intention to characterize the quadrics as certain point sets in these geometries. We will define a similar notion of a quadratic set of a given quadric Q for which the standard examples are the intersections of Q with the quadrics of the ambient projective space of Q. We will focus our discussion on the Klein quadric Q^+(5,q) which is a particular quadric in the 5-dimensional projective space PG(5,q). We will discuss classification and (non-)existing results for such quadratic sets as well as some applications of them to open problems involving certain line sets in PG(3,q) and hyperovals of Q^+(5,q).

Title: On the weight-distribution bound for eigenfunctions of strongly regular graphs

The weight-distribution bound (WDB) is a lower bound for the cardinality of support of an eigenfunction of a distance-regular graph corresponding to the smallest eigenvalue. This bound was introduced by Krotov, Mogilnykh and Potapov in 2016.

Moreover, it was shown that in case when the cardinality of support of an eigenfunction achieves WDB, then the subgraph induced by the support is a bipartite distance-regular graph.

There are the following two reasons why it is especially interesting to consider eigenfunctions of strongly regular graphs achieving WDB.

First, the subgraph induced by their support is a complete bipartite graph. 

Second, since the complement of a strongly regular graph is again a strongly regular graph, WDB is also defined for the positive non-principal eigenvalue of a strongly regular graph.

Strongly regular graphs occupy a central place in algebraic combinatorics due to their rich connections with many other areas, including finite geometry.

In this talk, I plan to discuss previous developments on eigenfunctions of geometric strongly regular graphs achieving WDB and formulate some open problems, which naturally arise in this framework.

Title: Optimal Full-Length Cyclic Orbit Flag Codes and Their Distance Distribution

Let q be a prime power and n > 1 an integer. A flag code is a set of sequences of nested Fq -subspaces of Fqn (flags), where Fq denotes the finite field with q elements. In this talk, we focus on cyclic orbit flag codes, namely those obtained as orbits of flags under the multiplicative action of Fqn* on the flag variety. More specifically, we restrict our attention to full-length orbits and characterize the flags that generate optimal full-length cyclic orbit flag codes, that is, codes achieving the maximum possible minimum distance. For these optimal codes, we also determine their distance distribution, which depends solely on q, n, and the dimensions of the subspaces in the generating flag.

All these ideas are collected in the work: C. Alonso-Gonzàlez and M.A. Navarro-Pérez, Distance distribution of cyclic orbit flag codes. Designs, Codes and Cryptography, Vol. 93 (2025), 4433–4459.

Title: On the Hermitian Veronesean

The Hermitian Veronesean in PG(3,q^2) is a well-known algebraic curve which lies on the Hermitian surface. It possesses many interesting properties and is related to other objects in finite geometry such as pseudo-ovals, special sets and ovoids. In this talk, I will discuss some of these connections, as well as recent characterisation results obtained in collaboration with John Bamberg, University of Western Australia.

Title: Cercare la chiave che non esiste: NP, coNP e crittografia

La possibile uguaglianza tra le classi di complessità NP e coNP è uno dei più grandi problemi aperti della teoria della complessità. In questo seminario divulgativo (in italiano) spiegherò con esempi semplici ma rigorosi il senso di questa uguaglianza, con enfasi sulle sue applicazioni nella teoria dei polinomi e, soprattutto, nella crittografia.

Title: Continuously Non-Malleable Codes: A Tutorial

Non-malleable codes (Dziembowski, Pietrzak, Wichs, ICS10) allow to encode a message into a codeword in such a way that limited tampering against the codeword either yields the original message or a completely unrelated value. Continuous non-malleability further strengthen the above in the setting where the attacker can carry multiple tampering attacks.

In this talk, I will review the main definitions and constructions of continuously non-malleable codes, highlighting their applications to cryptography.

Title: Crittoanalisi per difenderci dai ransomware

In questo seminario, rivolto principalmente a studenti di  matematica e informatica, riassumo il ruolo della  crittografia nei ransomware. 

I ransomware sono attacchi informatici che puntano  (tra le altre cose) a cifrare documenti nei computer delle vittime, rendendoli inutilizzabili. 

L’unico modo per recuperare i documenti originali è  pagare il riscatto ”ransom” ai cyber criminali, che  restituiscono (forse)  la chiave di decifratura. 

Sebbene molto diffusi, i ransomware sono soggetti a  vincoli pratici che si riflettono sulle proprietà matematiche degli algoritmi crittografici che usano. Queste proprietà possono talvolta essere dedotte e sfruttate per decrittare i file senza piegarsi al ricatto. 

Completerà il seminario un accenno alla mia esperienza  personale in questo campo.

Title: Rank-metric Lattices -- slides 

Higher-Weight Dowling Lattices (HWDL in short) are special families of geometric lattices introduced by Dowling in connection with coding theory. The elements of HWDLs are the F_q-linear subspaces of F_q^n having a basis of vectors with Hamming weight bounded from above, ordered by inclusion. These lattices were further studied, among others, by Bonin, Kung, and more recently by Ravagnani.

In this talk, we define and investigate structural properties of the q-analogues of HWDLs, which we call rank-metric lattices (RML in short). Their elements are the F_q^m-linear subspaces of F_{q^m}^n having a basis of vectors with rank weight bounded from above, ordered by inclusion. We determine which RMLs are supersolvable and compute their characteristic polynomials. In the second part of the talk, we establish a connection between RMLs and the problem of distinguishing between inequivalent rank-metric codes.

The new results in this talk are joint work with A. Ravagnani.

Title: Rank-metric codes with high degree of symmetry -- slides 

Rank-metric codes are linear spaces of matrices over a field, endowed with the metric induced by the rank. Their study was initiated by Delsarte, who originally introduced rank-metric codes for a purely combinatorial interest. It is only in 2008 that these codes raised the interest of coding theorist, due to their application in network coding proposed by Silva-Kschischang-Koetter. Since then, the theory of rank-metric codes has exponentially grown, bridging researchers in mathematics, engineering and computer science.

In this talk, I will focus on rank-metric codes with restrictions, starting from symmetric ones. Over fields of characteristic different from 2, symmetric matrices can be identified with degree 2 homogeneous polynomials. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. I will show how to extend the study of symmetric rank-metric codes to higher degree of symmetry. One can equip the space of homogeneous polynomials of degree d with the metric induced by the essential rank, which is the minimal number of linear forms needed to express a polynomial. In this context, I will outline a generalized construction of symmetric Gabidulin codes to polynomials of degree d over field of characteristic 0 or larger than d.

This is a joint work with Arthur Bik.

Title: Lectures on “Eigenvalue techniques in graph theory” -- notes

In this talk we will discuss eigenvalue bounds in algebraic graph theory with an emphasis on how to apply them in the context of finite geometry. The goal is to give complete proofs of these bounds. A familiarity with linear algebra and some basic notions of graph theory will be assumed, but no background in algebraic graph theory is required.

We will start by proving the expander mixing lemma. Then we will discuss how it can be applied to prove some results in projective planes, which can often also be proven through purely combinatorial arguments. Next we will discuss some more advanced tools to calculate the eigenvalues of graphs. This leads us to the theory of association schemes. We will show how eigenvalue bounds can surpass purely combinatorial arguments in some more complicated graphs than those coming from association schemes.

Title: On the non-existence of Cameron-Liebler sets of k- spaces in PG(n,q)

See the link.

Title: A graph co-spectral to NO+(8,2)

The graph NO+(8,2) is a strongly regular graph with parameters (120,63,30,36). In their recent book, Brouwer and Van Maldeghem mention the existence of a non-isomorphic, strongly regular graph with the same parameters, admitting Sym(7) as automorphism group. In this talk we discuss a geometric construction of this alternative graph. This is joint work with Sam Adriaensen, Robert Bailey and Morgan Rodgers.

Title: Skew polynomial rings and their applications