Abstract: Brill-Noether theory investigates, on smooth algebraic curves, the existence of line bundles of fixed degree whose space of global sections has dimension at least a prescribed number. The picture is well understood for general curves of any fixed genus, and in recent years considerable attention has shifted toward the Brill-Noether theory of general covers of the projective line of fixed degree and genus. I will report on joint work with Andreas Leopold Knutsen in which we settle the Brill-Noether theory of linear series of type \(g^1_d\) on general covers of an elliptic curve. I will also outline our strategy for studying linear series \(g^r_d\) for arbitrary r.

Abstract: Chabauty’s method, first presented 1941, is still being used to compute or bound the rational points of specific curves when the rank of the Mordell Weil group of their Jacobian is strictly smaller than their genus. The quadratic variant of this method extends the applicability when the Jacobian has non-trivial N´eron-Severi group. This makes it a good candidate to study rational points on the ‘non-split’ modular curves, as done for X+ ns(13) = X+ s (13) in 2019. We’ll explain this p-adic method, in the geometric version, and in particular how to apply it to modular curves without computing explicit equations, only using the moduli interpretation. This is joint work with Bas Edixhoven, and ongoing with Sachi Hashimoto, Davide Lombardo, Nicolas Mascot and Pierre Parent

Abstract: Mazur—Rubin asked to what extent one can reconstruct a curve C over a number field K from its set of points over \(\overline{K}\), viewed as a Galois set. They asked the same question specifically about the set of fields where C acquires new points and gave evidence for a positive answer for curves of genus 0. In this talk we will present upcoming work with Zev Klagsbrun where we provide a positive answer for a generic pair of elliptic curves with full 2-torsion over a number field. The method of proof uses the combination of additive combinatorics and descent introduced in joint work of the speaker and Koymans in 2024. I will overview several other recent results obtained, by a number of authors, with that method.

Abstract: Deciding whether a subvariety of an algebraic variety is contractible is a deep problem of algebraic geometry. Even when the subvariety is a single smooth rational curve C, the question is extremely subtle. In this talk, I will assume moreover that the ambient variety is a Calabi-Yau threefold. When C is contractible, its Donovan-Wemyss contraction algebra (which pro-represents the deformation theory of C) governs much of the geometry. Our expectation is that deformation theory not only controls contractibility but detects it, even when C is not known to contract. To investigate the deformation theory of C, we use technology developed by Brown and Wemyss to describe a local model for C.I will introduce the key ideas and tools appearing in this problem, the leading conjectures, and I will describe the (partial) results I obtained so far in collaboration with G. Brown and M. Wemyss.

Abstract: Classically, spaces of coinvariants are constructed by assigning representations of affine Lie algebras—and more generally, of vertex operator algebras—to pointed algebraic curves. These constructions yield sheaves on the moduli space of curves via descent along an infinite-dimensional torsor. In this talk, I will explain how an analogous descent procedure produces sheaves on the moduli space of abelian varieties. The key input is a perspective in which the moduli space of abelian varieties arises as the quotient of an infinite-dimensional moduli space introduced by Arbarello and De Concini. I will thus show how the methods familiar from the theory of conformal blocks on curves can be extended to abelian varieties.

Abstract: The Poisson boundary is a measure-theoretic object attached to a group equipped with a probability measure, and is closely related to the notion of harmonic function on the group. In many cases, the group is also endowed with a topological boundary arising from its geometric structure, and a recurring research theme is to identify the Poisson boundary with the topological boundary. For instance, for lattices in hyperbolic space, it is natural to ask if the (visual, or Gromov-)boundary of the hyperbolic space is a model for the Poisson boundary. In this talk, we solve the identification problem for the Poisson boundary of a random walk with finite entropy on a hyperbolic group and on a discrete subgroup of a semisimple Lie group. The main technical novelty is that we do not require any moment assumption on the measure. Joint with K. Chawla, B. Forghani, and J. Frisch.

Abstract: The Zilber–Pink conjectures predict far-reaching constraints on unlikely intersections across several arithmetic–geometric settings, extending classical results such as Manin–Mumford and Mordell–Lang. In this talk, I will present joint work in progress with F. Barroero, L. Capuano and T. Ge on unlikely intersections in the relative setting of abelian schemes, with the aim of generalizing previous results for curves due to Masser–Zannier and Barroero–Capuano. To do this, we investigate unlikely intersections in relation to a parameter known as the “degeneracy threshold”, introduced by Ge to study boundedness of heights in this context. By extending ideas of Habegger–Pila from the constant case to the relative framework, this approach yields new partial results toward Pink’s conjecture for higher-dimensional subvarieties.

Abstract: The search for extremal metrics is a classical and still very active topic in Kähler geometry; the existence of these metrics is linked to algebro-geometric stability notions that have frequently proved useful in constructing moduli spaces. A typical example is given by the Yau-Tian-Donaldson correspondence on a Fano variety: it admits a Kähler- Einstein metric if and only if it is K-polystable. The goal of the talk is to present the invariance of extremal Kähler manifolds under a suitable class of bimeromorphic mophisms such as for instance the blow-ups along sub- manifolds, generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam. Focusing mostly on the simplest case of cscK (constant scalar curvature Kähler) metric on manifolds with no nontrivial holomorphic vector fields, I will describe the original variational approach performed and the main ideas of the proof. In the last part of the talk, I will explain how such approach applies more generally to a much larger class of extremal metrics and to any equivariant resolution of singularities of Fano type of a compact Kähler klt space. This is a joint work with S. Boucksom and M. Jonsson.

Abstract: A convex projective manifold is the quotient of a properly convex open subset of real projective space under a discrete group of projective transformations preserving it. Such manifolds come naturally equipped with a Hilbert metric whose key features are hidden in the structure of the boundary of the convex set and are reflected in the algebraic properties of the discrete group. This interaction between geometry, dynamics, and algebra has been fruitfully explored, leading to a rich and well-developed theory. However, in the closed case, most of the known examples of convex preojective manifolds are strictly convex. This puts serious constraints on the type of groups and geometries that appear. Together with Pierre-Louis Blayac, we constructed many non-strictly convex examples in arbitrary dimension that exhibit various new phenomena.