Abstract: We talk about "unlikely intersections" whenever we have a non-empty intersection between algebraic varieties that, for dimensional reasons, we do not expect to intersect. This expectation lies behind several landmark results and conjectures in Diophantine geometry, including Faltings’ Theorem (formerly the Mordell Conjecture), the Manin-Mumford Conjecture, the André-Oort Conjecture (proved by Pila, Shankar and Tsimerman), and the still open Zilber-Pink Conjecture. In this talk, I will give an introduction to unlikely intersections, focusing first on algebraic tori and abelian varieties, and then on families of abelian varieties. I will survey some key results by Masser-Zannier and Barroero-Capuano in this setting, and finally present results from my PhD thesis establishing some partial progress on the Zilber-Pink Conjecture for curves in abelian schemes.

Abstract: We present the construction of distinguished non-Kähler metrics on non-compact Calabi-Yau 3-folds. These metrics solve a system of equations known as the IIB system which arises in theoretical physics and is related to recent attempts to define notions of "canonical" metrics on non-Kähler Calabi-Yau manifolds. The examples we construct include infinitely many complete metrics obtained by deforming an asymptotically conical Calabi-Yau 3-folds in the direction of a non-trivial Äppli class and families of solutions on the ordinary double point and its smoothing that enjoy a cohomogeneity one symmetry (i.e. there is a symmetry group that acts with 1-dimensional orbit space) and produce a "conifold transition" in non-Kähler Calabi-Yau geometry. The talk is based on joint work with Mario Garcia-Fernandez.

Abstract: In 1972, Serre proved that the Galois representations arising from the p-power torsion points of non-CM elliptic curves over \(\mathbb{Q}\) have open image in \(\operatorname{GL}_2(\mathbb{Z}_p)\), and Mazur later initiated a vast programme to determine all such possible images explicitly -- for fixed p, it is known that there are only finitely many possibilities. Much progress has been made for small primes p, but a complete classification remains open beyond \(p \in \{2,3,13,17\}\). In this talk, I will describe recent progress on this problem for p = 7, based on a surprising correspondence between rational points on modular curves and primitive integer solutions to certain generalised Fermat equations of signature (2,3,7), such as \( a^2 + 28b^3 = 27c^7. \) We show that these Diophantine equations can be reduced to determining the rational points on a finite collection of genus-3 curves. As a consequence, we are able for example to determine the rational points on a modular curve of genus 69 and establish that the 7-adic Galois images of elliptic curves over \(\mathbb{Q}\) are determined by their reduction modulo \(7^2\).

Abstract: In join work with Manon Parent we have investigated how small the \(L^2\)-norm of an exponential polynomial can get if we fix the norm of the its coefficient vector. We prove lower and upper bounds of this minimum in terms of the degree of the polynomial and we apply our methods to a variant of a problem of Hilbert. We prove that on any given interval the infimum of the \(L^2\)-norm of exponential polynomials with integer coefficients on this interval is 0.

Abstract: In 1919, while studying birational automorphisms of the projective plane, A. Coble introduced a new family of surfaces, today known as Coble surfaces. We start by showing some basic facts, and underling the properties which link these rational surfaces to the world of Enriques surfaces. Next, we present some recent results on automorphisms of a Coble surface, with a particular attention to the case of involutions. We show that, under suitable generic assumptions, any involution on a Coble surface is a lift of a Bertini involution. The content of this talk is based on my PhD thesis, written under the supervision of Prof. A. Verra.

Abstract: A variety is retract rational if the solutions of its defining equation can be parametrized by rational functions. Examples are provided by rational or stably rational varieties. We explain some history on the problem of deciding whether a given variety is (retract) rational, concentrating mostly on the case of cubics. We will then explain how recent joint work with Engel and de Gaay Fortman on the failure of the integral Hodge conjecture for abelian varieties implies by earlier work of Voisin that very general cubic threefolds over the field of complex numbers are not retract rational, hence not stably rational.

Abstract: we explore the intersection of the Hassett divisor \(C_8\), parametrizing smooth cubic fourfolds X containing a plane P with other divisors \(C_i\). Notably we study the irreducible components of the intersections with \(C_{12}\) and \(C_{20}\). These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of P with the other surface. Then we consider the problem of rationality.

Abstract: The known results about the stable irrationality of very general smooth Fano complete intersections \(X^n\subset\mathbb P^{n+c}\) of dimension \(n\geq 3\) and fixed type \((d_1,\ldots, d_c)\) pose the natural problem of understanding under which conditions there (might) exist rational examples of that type/dimension, e.g. index one; index two; quartic fourfolds and fivefolds; etc. After introducing the problem and after briefly recalling the state of the art, we shall present some results and geometric motivations pointing towards the statement of a possible solution.

Abstract: TBA

Abstract: Using classical techniques and results from Bridgeland's stability condition theory, we will discuss a conjecture that predicts the maximum genus of a curve on a Calabi-Yau 3-fold.

Abstract: Let G be an algebraic group acting on a finite-type scheme X; a natural question is whether there exists a categorical quotient \(X // G\) of this action as a finite-type scheme. When G is reductive, this is answered by D. Mumford’s Geometric Invariant Theory (GIT), a keystone of contemporary algebraic geometry. In the case where G is no-longer reductive, the problem is more subtle; if the unipotent radical of G admits an internal grading, one may instead invoke results from Non-Reductive GIT, as initially developed by F. Kirwan et. al. In this talk, we will explain how the existence of quotients by internally-graded groups in non-reductive GIT follows from more general existence results for moduli spaces of algebraic stacks of filtered objects, in the sense of D. Halpern-Leistner. Time permitting, we will also indicate how this more general perspective can be used to construct quasi- projective moduli spaces of unstable principal bundles with reductive structure group G, which (when G = GL(n)) simplifies the non-reductive GIT constructions of moduli spaces of unstable vector bundles of J. Jackson, Y. Qiao and V. Hoskins–J. Jackson. This talk is based on joint ongoing work with Ludvig Modin (Hannover)

The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2025" is a Winter meeting in Algebra and Geometry in Rome that takes place in the rione Testaccio. For more info here is the Link to the Official Webpage

Abstract: Negli ultimi 50 anni la ricerca in Geometria Proiettiva ha prodotto alcuni tra i migliori risultati della matematica contemporanea, riconosciuti attraverso l’assegnazione di numerose Fields Medals e altri premi prestigiosi. Nella conferenza presenterò alcuni aspetti del programma sui Minimal Model e della Teoria di Mori, ponendo in evidenza l’influenza del lavoro di Gino Fano e, più in generale, dei ricercatori italiani coinvolti in questa splendida avventura di ricerca.

Abstract: A recent influential article by Liu, Liu, and Xu gave a new example of a 42-dimensional (singular) irreducible symplectic variety with a Lagrangian fibration. This is obtained by compactifying a construction by Iliev and Manivel arising from cubic fourfolds and fivefolds, together with techniques developed recently by Saccà. In work in progress with O’Grady and Saccà, we consider the case of Gushel-Mukai manifolds, by obtaining an example in dimension 20. The approach is similar to Liu-Liu-Xu: we compactify another construction by Iliev-Manivel, but with technical complications due to the Gushel-Mukai setting.

Abstract: Arakelov geometry offers a framework to develop an arithmetic counterpart of the usual intersection theory. In fact, for varieties defined over the ring of integers of a number field, and inspired by the geometric case, one can define a suitable notion of arithmetic Chow groups and of an arithmetic intersection product. In this talk, I will begin with a brief introduction to Arakelov geometry in the flavour of Gillet-Soulé, and then present a joint work with R. Gualdi (Universitat Politècnica de Catalunya). We prove an arithmetic analogue of the classical Shioda-Tate formula, relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety to some of its geometric invariants. In doing so, we also characterize numerically trivial arithmetic divisors, partially confirming a conjecture by Gillet and Soulé.

Abstract:

Abstract: It is a long-standing question to resolve the singularities of algebraic varieties by Nash-blowups. This is a very natural process associating to smooth points their tangent space and to then take the Zariski-closure of the associated graph. This works for curves and, when combined with normalization, also for surfaces. It was a big surprise when Castillo, Duarte, Leyton-Alvaro and Liendo constructed in 2024 a four- (and then three-) dimensional toric counter-example to resolution. We will briefly describe their construction and then relate it to the concept of algebraic curvatures of varieties. These are invariants of varieties at smooth points transferring and generalizing the differential-geometric notion of curvature to the algebraic geometry setting.

Abstract:

Abstract:

Abstract:

Abstract: