A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.[-]

Cellular A1-homology is a new homology theory for smooth algebraic varieties over a perfect field, which is often entirely computable and is expected to give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will introduce cellular A1-homology, describe the precise conjectures about cellular A1-homology of smooth projective varieties and discuss how they can be verified for smooth projective rational surfaces. The talk is based on joint work with Fabien Morel.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

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.[-]

We introduce the notion of a plabic tangle, which is the data of a planar bipartite graph drawn inside a disk called an "input disk," together with one or more "output" disks, which lie in faces of the graph. We show that under appropriate hypotheses, a plabic tangle gives rise to a "promotion map," which is a rational map from the Grassmannian of the input disk, to a product of Grassmannians associated to the output disks. We provide a number of examples in which these maps are compatible with the cluster algebra structure on the Grassmannian. Our motivating example is the case of "BCFW promotion," which we used to prove the cluster adjacency conjecture for the amplituhedron. This is based on joint work with Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.[-]