Abstract: By Faltings' theorem, the set of rational points on a curve of genus 2 or more is finite. We describe how p-adic heights ("quadratic Chabauty") can be used to determine this set for certain curves of genus 2 or more, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves and highlight several examples. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

Abstract: Halphen surfaces are the blow- up of the plane at nine points. On one hand they are rational surfaces, on the other hand they carry natural linear systems which embed them in any projective space as a limit of general smooth K3 surfaces. We will describe their geometry, focusing on their Severy variety, whose connectedness leads to the irreducibility of the Severi variety of a general K3 surface (joint work with M. Lelli Chiesa).

Abstract: Given a finite simple Lie algebra \(g\) one may consider the associated loop algebra \(g((t))\) for which, given any complex number \(k\), there is an attached central extension, called \(\hat{g}_k\). These are known as affine Kac-Moody algebras, and \(k\) is usually referred to as "the level". It turns out that there is a special level, called \(k_c\) the critical level, for which the representation theory of \(\hat{g}_{k_c}\) is astoundingly different from that of all other levels. The representation theory of Affine Kac-Moody algebras at the critical level plays a crucial role in the Geometric Langlands Program pioneered by Gaitsgory et. al.. In the talk we will review some crucial theorems in this theory, such as the description of the center by Feigin and Frenkel as the algebra of functions on the space of Opers and the theorem of Frenkel and Gaitsgory on the description of spherical representations. We will also give a peek on some generalizations of the above results in the setting of multiple singularities.

Abstract: Given an abelian scheme over char 0 and an irreducible subvariety \(X\), one can define the \(t\)-th degeneracy locus of \(X\) for each integer \(t\). This geometric concept of degeneracy loci has recently seen many applications in Diophantine Geometry, notably when \(t\) is 0 and 1, in the recent developments on the uniformity of the number of rational points on curves, on the solutions of the Uniform Mordell-Lang Conjecture and of the Relative Manin-Mumford Conjecture. In this talk, I will define the degeneracy loci in the universal abelian variety, and explain how they are used in the applications mentioned above.

Abstract: The purpose of this talk is to discuss the significance of (co)homology in geometry/topology, the significance of various extra structures on it, leading to a discussion of the recent P=W conjecture and its verification for abelian varieties.

Abstract: A hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics. The theory in dimension two, namely K3 surfaces, is well understood. The aim of the seminar is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from a point of view of their classification; in particular, about existence of Lagrangian fibrations. We will present some results in dimension four, obtained in collaboration with Olivier Debarre, Daniel Huybrechts and Claire Voisin.

Abstract: Siegel modular forms are holomorphic functions on the Siegel upper half-space \(H_g\) that satisfy certain automorphy relations, and can be seen as sections of line bundles on the moduli space \(A_g\) of principally polarized abelian varieties. There are criteria to determine whether a certain modular form f cannot be written as a product of two non-constant modular forms, but what about the irreducibility of the modular form f as a mere holomorphic function on \(H_g\) (universal irreducibility)? We will show that, for \(g>2\), the two notions agree and so every Siegel modular form can be written as a product of finitely many irreducible analytic functions on \(H_g\), which are themselves modular forms. We will also discuss a few examples. This is a joint work with Riccardo Salvati Manni.

Abstract: We introduce the notion of formality of algebraic and topological structures, that is typically used to show that some spectral sequence collapses. The formality of configuration spaces is relevant for a spectral sequence computing the homology of the space of knots. Their rational formality was proved by Arnold and Kontsevich. We show that in positive characteristic formality fails and we construct a new non trivial differential. This is joint work with Andrea Marino.

Abstract: We introduce the notion of Higgs bundles on a smooth projective surface \(S\). After sketching the spectral construction, we'll see that these bundles can be transformed into torsion sheaves on a Calabi-Yau threefold \(X\), which are called "spectral sheaves". We'll discuss their deformation/obstruction theory and get a glimpse of their moduli.

Abstract: The study of continuous actions of groups like \(\mathrm{GL}(n;\mathbb{R})\) and \(\mathrm{GL}(n;\mathbb{C})\) on Banach spaces leads to interesting algebraic questions concerning modules over polynomial rings. I want to talk on progress here.

Abstract: In this talk we will introduce Ulrich sheaves on projective algebraic varieties. The corresponding notion for modules over rings originated from the work of Ulrich, but only after a paper by Eisenbud and Schreyer its geometric side received many attention due to the connection with determinantal and Pfaffian representations of (Chow forms of) varieties. The main questions are existence of such sheaves and, if so, their minimal rank. Very few is known in dimension at least 3; therefore, after some example for curves and surfaces, we will focus on Fano 3-folds.