Roth's theorem states that a subset $A$ of $\{1, \ldots, N\}$ of positive density contains a positive $N^2$-proportion of (non-trivial) three arithmetic progressions, given by pairs $(a, d)$ with $d \neq 0$ such that $a, a+d, a+2 d$ all lie in $A$. In recent breakthrough work by Kelley and Meka, the known bounds have been improved drastically. One of the core ingredients of the their proof is a version of the almost periodicity result due to Croot and Sisask. The latter has been obtained in a non-quantitative form by Conant and Pillay for amenable groups using continuous logic.
In joint work with Daniel Palacín, we will present a model-theoretic version (in classical first-order logic) of the almost-periodicity result for a general group equipped with a Keisler measure under some mild assumptions and show how to use this result to obtain a non-quantitative proof of Roth's result. One of the main ideas of the proof is an adaptation of a result of Pillay, Scanlon and Wagner on the behaviour of generic types in a definable group in a simple theory.[-]

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 mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]

The goal of this short course is to explain the concept of “triality”, which is an isomorphism between a large class of of (generalized) tame frieze patterns, certain spaces of linear difference equations, and the moduli space of configurations of points in the projective space. This approach will be used in several directions, in particular:
• to define “good” coordinates on moduli spaces related to cluster algebras and symplectic geometry
• to find simple proofs of some properties of friezes, such as periodicity
• to connect the subject to dynamical systems
• to create new types of friezes
• to count friezes of certain types.
The presentation is based on several joint papers with Sophie Morier-Genoud, Sergei Tabachnikov, and also Charles Conley, and Richard Schwartz. Coxeter friezes and geometry of the projective line. I will start with the classical Coxeter's frieze patterns and connect them to configurations of point in the 1-dimensional projective space P1. As a consequence, a (pre)symplectic structure on the space of Coxeter's friezes will be described. The basic notions of projective geometry, such as the cross-ratio and Schwarzian derivative will be recalled/explained and used.[-]

The goal of this talk is to explore the connections between various frieze patterns and representation theory of associative algebras. We begin with the classical Conway- Coxeter friezes over positive integers and their correspondence with Jacobian algebras of type A, where entries in the frieze count the number of submodules of indecompos- able representations. This can also be reinterpreted in terms of applying the Caldero- Chapoton map, providing a close connection to Fomin-Zelevinsky's cluster algebras. Extending these ideas beyond the classical case, we will also discuss higher dimen- sional friezes, called (tame) SLk friezes, as well as their relation to cluster algebras on coordinate rings of Grassmannians Gr(k,n) and their categorification. Furthermore, SLk friezes are a special type of SLk tilings, integer tilings of the plane satisfying the condition that every k x k square has determinant 1. We will present a characterization of SLk tilings in terms of pairs of bi-infinite sequences in Zk and discuss applications to duality and positivity.[-]

This short course is about modelling SL2-tilings and Coxeter frieze patterns with Farey complexes. The first talk concerns tame frieze patterns over the integers. We introduce the Farey tessellation of the hyperbolic plane, drawing inspiration from the theory of dessins d'enfants. The geometric and numeric properties of the Farey tessellation shed light on known results on classifying frieze patterns and they provide a framework for new results. This approach originated in work of Morier-Genoud, Ovsienko, and Tabachnikov; we will discuss their ideas and generalisations. There will be diagrams aplenty, several exercises, and a few open questions.[-]