Linear codes in the Hamming metric have played a central role in error correction since the 1950s and have been extensively studied. In contrast, the theory of codes in the sum-rank metric is still in its early stages, with only a few known constructions.
A cornerstone of coding theory in the Hamming metric is the family of Reed–Solomon (RS) codes, which are constructed by evaluating univariate polynomials at distinct elements of a finite field $F_{q}$ . RS codes have optimal parameters, however, their length is by definition limited by the size of $ F_{q}$. Two classical approaches to overcome this limitation, while maintaining control on the parameters, are considering multivariate polynomials, giving rise to Reed–Muller (RM) codes, and evaluating rational function at points on algebraic curves, leading to Algebraic Geometry (AG) codes.
The sum-rank analogue of RS codes is the family of linearized Reed–Solomon (LRS) codes (see U. Martínez-Peñas 2018), which also achieve optimal parameters but face a similar length restriction as RS codes. In this talk, inspired by the similarities between RS and LRS codes,we will introduce analogues of RM and AG codes in the sum-rank metric, known as linearized Reed–Muller (LRM) codes (see E. Berardini and X. Caruso 2025) and linearized Algebraic Geometry (LAG) codes (see E. Berardini and X. Caruso 2024).
We will begin by reviewing key background on sum-rank metric codes and univariate Ore polynomials. Afterwards, we will introduce the theory of multivariate Ore polynomials and their evaluation, leading to the construction of linearized Reed–Muller codes and an analysis of their parameters. Then, we will develop the theory of Riemann–Roch spaces over Ore polynomial rings with coefficients in the function field of a curve, leveraging the classical framework of divisors and Riemann–Roch spaces on curves. Using this foundation, we will construct linearized AG codes, providing lower bounds on their dimension and minimum distance. We will conclude the talk by sketching some related works in progress.[-]

Let $V$ be an $(n+1)$-dimensional vector space over an arbitrary field $\mathbb{K}$ and denote by $\mathrm{PG}(V)$ the corresponding projective space. Define $\Gamma$ as the point-hyperplane geometry of $\mathrm{PG}(V)$, whose points are the pairs $(p, H)$, where $p$ is a point, $H$ is a hyperplane of $\mathrm{PG}(V)$ and $p \in H$ and whose lines are the sets $\ell_{p, *}:=\{(p, U): p \in U\}$ or $\ell_{*, H}=\{(x, H): x \in H\}$. The geometry $\Gamma$ is also known as the long root geometry for the special linear group $\mathrm{SL}(n+1, \mathbb{K})$ and admits an embedding (the Segre embedding of $\Gamma$ ) in the projective space $\mathrm{PG}\left(M_0\right)$, where $M_0$ is the vector space of the traceless square matrices of order $n+1$ with entries in the field $\mathbb{K}$. Since $M_0$ is isomorphic to a hyperplane of the vector space $V \otimes V^*$, we explicitly have

$$
\varepsilon: \Gamma \rightarrow \mathrm{PG}\left(M_0\right), \quad \varepsilon((\langle x\rangle,\langle\xi\rangle))=\langle x \otimes \xi\rangle,
$$

with $x \in V \backslash\{0\}, \xi \in V^* \backslash\{0\}$. The image $\Lambda_1:=\varepsilon(\Gamma)$ of $\varepsilon$ is represented by the pure tensors $x \otimes \xi$ with $x \in V$ and $\xi \in V^*$ such that $\xi(x)=0$.

If the underlying field $\mathbb{K}$ admits non-trivial automorphisms, for $1 \neq \sigma \in \operatorname{Aut}(\mathrm{K})$, then it is possible to define a 'twisted version' $\varepsilon_\sigma$ of $\varepsilon$ as follows

$$
\varepsilon_\sigma: \Gamma \rightarrow \mathrm{PG}\left(V \otimes V^*\right), \varepsilon_\sigma((\langle x\rangle,\langle\xi\rangle))=\left\langle x^\sigma \otimes \xi\right\rangle,
$$

where $x^\sigma:=\left(x_i{ }^\sigma\right)_{i=1}^{n+1}$.
Consequently, the points of $\Lambda_\sigma:=\varepsilon_\sigma(\Gamma)$ are represented by pure tensors of the form $x^\sigma \otimes \xi$, under the condition $\xi(x)=0$.

In the first part of the talk I will address the problem of the universality of the Segre embedding $\varepsilon$ for $\Gamma$ proving that the answer to this question depends on the underlying field $\mathbb{K}$ and generalizing a previous result for $n=2$ (see recent work of I. Cardinali, L. Giuzzi, A. Pasini).

In the second part of the talk, I shall focus on the case where $\mathbb{K}=\mathbb{F}_q$ is a finite field of order $q$. Thus, regarding $\Lambda_1$ and $\Lambda_\sigma$ as projective systems of $\mathrm{PG}\left(M_0\right)$ respectively $\mathrm{PG}\left(V \otimes V^*\right)$, I will consider the linear codes $\mathcal{C}\left(\Lambda_1\right)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ arising from them. I shall determine the parameters of $\mathcal{C}(\Lambda)$ and $\mathcal{C}\left(\Lambda_\sigma\right)$ as well as their weight list. I will also give a (geometrical) characterization of some of the words of these codes having minimum or maximal weight (see recent work of I. Cardinali, L. Giuzzi).[-]

Algebraic curves in positive characteristic and their function fields have been a source of great interest ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet, including the famous application to error-correcting codes given by Goppa's AG codes.

Let $\mathcal{X}$ be a projective, geometrically irreducible, non-singular algebraic curve defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$. Let $\mathbb{K}(\mathcal{X})$ be the field of rational functions on $\mathcal{X}$ (i.e. the function field of $\mathcal{X}$ over $\mathbb{K}$ ). The $\mathbb{K}$-automorphism group $\operatorname{Aut}(\mathcal{X})$ of $\mathcal{X}$ is defined as the automorphism group of $\mathbb{K}(\mathcal{X})$ fixing $\mathbb{K}$ element-wise. The group $\operatorname{Aut}(\mathcal{X})$ has a faithful action on the set of points of $\mathcal{X}$.

By a classical result by Schnid (1938), Aut( $\mathcal{X}$ ) is finite whenever the genus $g$ of $\mathcal{X}$ is at least two. Furthermore it is known that every finite group occurs in this way, since, for any ground field $\mathbb{K}$ and any finite group $G$, there exists an algebraic curve $\mathcal{X}$ defined over $\mathbb{K}$ such that $\operatorname{Aut}(\mathcal{X}) \cong G$ (see for example the work of Valentini-Madden, 1982).

This result raised a general problem for groups and curves, namely, that of determining the finite groups that can be realized as the $\mathbb{K}$-automorphism group of some curve with a given invariant. The most important such invariant is the genus $g$ of the curve. In positive characteristic, another important invariant is the so-called $p$-rank of the curve, which is the integer $0 \leq \gamma \leq g$ such that the Jacobian of $\mathcal{X}$ has $p^7 p$-torsion points.

Several results on the interaction between the automorphism group, the genus and the $p$-rank of a curve can be found in the literature. A remarkable example is the work of Nakajima (1987) who showed that the value of the $p$-rank deeply influences the order of a $p$-Sylow subgroup of $A u t(\mathcal{X})$. Extremal examples with respect to Nakajima's bound are known from the work of Korchmáros-Giulietti (2017) and Stichtenoth (1973). The following open problem arose naturally:

Open Problem 1: How large can a d-group of aulomorphisms $G$ of an algebraic curve $\mathcal{X}$ of genus $g \geq 2$ be when $d \neq p$ is a prime number? Is there a method to construct extremal cxamples as for the case $d=p$ ?

In his work Nakajima also analyzed the case of curves for which the $p$-rank is the largest possible (the so-called ordinary curves), namely $\gamma=g$, proving that they can have at most $84\left(g^2-g\right)$ automorphisms. Since no extremal examples for this bound were found by Naka.jima, also the following open problem arose naturally:

Open Problem 2: Is Nakajima's bound $\mid$ Aut $(\mathcal{X}) \mid \leq 84\left(g^2-g\right)$, sharp for an ordinary curve $\mathcal{X}$ of genus $g \geq 2$ ?

Hurwitz (1893) showed that if $\mathcal{X}$ is defined over $\mathbb{C}$ then $|A u t(\mathcal{X})| \leq 84(g-1)$, which is
known as the Hurwitz bound. This bound is sharp, i.e., there exist algebraic curves over $\mathbb{C}$ of arbitrarily high genus $g$ whose automorphism group has order exactly $84(g-1)$. Well-known examples are the Klein quartic and the Fricke-Macbeath curve.

Roquette (1970) showed that Hurwitz bound also holds in positive characteristic $p$, if $p$ does not divide $|\operatorname{Aut}(\mathcal{X})|$. A general bound in positive characteristic is $|\operatorname{Aul}(\mathcal{X})| \leq 16 g^4$ with one exception: the so-called Hermitian curve. This result is due to Stichtenoth (1973). The quartic bound $|A u t(\mathcal{X})| \leq 16 g^4$ was improved by Henn (1978). Henn's result shows that if $|A u t(\mathcal{X})|>8 g^3$ then $\mathcal{X}$ is $\mathbb{K}$-isomorphic to one of 4 explicit exceptional curves, all having $p$-rank equal to zero. A third natural open problem arose as a consequence of this result:

Open Problem 3: Is it possible to find a (optimal) function $f(g)$ such that the existence of an automorphism group $G$ of $\mathcal{X}$ with $|G|>f(g)$ implies that $\mathcal{X}$ has p-rank zero?

Henn's result clearly implies that $f(g) \leq 8 g^3$, but it is pleausible to believe that a quadratic bound with respect to $g$ could also be found.

In this talk, we will describe our main contributions to the three problems mentioned above and more generally in understanding the relation between automorphism groups of algebraic curves in positive characteristic and the other invariants mentioned above. If time allows, applications of these results in determining isomorphism classes of algebraic curves over finite fields will also be discussed.[-]

Supersingular elliptic curve isogeny graphs have isomorphism classes of supersingular elliptic curves over a finite field as their vertices and isogenies of some fixed degree between them as their edges. Due to their apparent "random" nature, supersingular isogeny graphs - which are optimal expander graphs - have been used as a setting for certain cryptographic schemes that are resistant to attacks by quantum computers. Hidden structures in these graphs may have implications to the security of these systems. In this talk, we analyze a number of graph theoretic structural properties of supersingular isogeny graphs over a finite field $\mathbb{F}_{p^2}$ and their subgraphs induced by the vertices defined over $\mathbb{F}_p$. This is joint work with Sarah Arpin (Virginia Tech) and our jointly supervised undergraduate student Taha Hedayat (University of Calgary).[-]

Let $E/F$ be an elliptic curve over a number field $F$. For a prime $p$, the extension $F(E[p^k])/F$ generated by the coordinates of the $p^k$-torsion points, is finite and Galois. We consider when the coincidence $F(E[p^k])=F(E[p^{k+1}])$ holds. Daniels and Lozano-Robledo classified such coincidences when $F=\mathbb{Q}$. In this talk, we will describe some results over a general number field $F$ and give additional possible coincidencesin this larger case.[-]

Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco.[-]

I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension of K. We will give examples where the strategy works well![-]