Il contesto naturale della congettura di Greenberg è la teoria di Iwasawa. In questo colloquio introdurrò la teoria di Iwasawa e la congettura di Greenberg, per poi presentare risultati recenti ottenuti in collaborazione con Pietro Mercuri.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

The solution of systems of non-autonomous linear ordinary differential equations is crucial in various applications, such as nuclear magnetic resonance spectroscopy. We introduced a new solution expression in terms of a generalization of the Volterra composition. Such an expression is linear in a particular algebraic structure of distributions, which can be mapped onto a subalgebra of infinite matrices. It is possible to exploit the new expression to devise fast numerical methods for linear non-autonomous ODEs. As a first example, we present a new method for the operator solution of the generalized Rosen-Zener model, a system of linear non-autonomous ODEs from quantum mechanics. The new method’s computing time scales linearly with the model’s size in the numerical experiments. A second example is the analysis of temporal network, where the new expression might lead to novel extension of subgraph centrality indexes.

Enumerating genus g curves passing through g points in an abelian surface is a natural problem, whose difficulty highly depends on the degree of the curves. For "primitive" degrees, we have an easy explicit answer. For "divisible" classes, such a resolution is quite demanding and often out of reach. Yet, the invariants for divisible classes easily express in terms of the invariants for primitive classes through the multiple cover formula, conjectured by G. Oberdieck a few years ago. In this talk, we'll show how tropical geometry enables to prove the formula without any kind of concrete enumeration.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

We discuss the unique continuation property for linear differential operators of the form sum of squares of vector fields satisfying Hörmander's bracket generating condition. We provide some negative and some positive results.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Mean curvature flow is the gradient flow of the area functional where an embedded hypersurface evolves in direction of its mean curvature vector. This constitutes a natural geometric heat equation for hypersurfaces, which ideally will evolve the embedding into a nicer shape. But due to the nonlinear nature of the equation singularities are guaranteed to form. Nevertheless, a key observation in geometry and physics is that generic solutions, obtained by small perturbations, can exhibit simpler singularities. In this direction, a conjecture of Huisken posits that a generic mean curvature flow encounters only the simplest singularities. We will discuss work together with Chodosh, Choi and Mantoulidis which together with recent work of Bamler-Kleiner establishes this conjecture for embedded hypersurfaces in R³

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

The famous Beurling theorem provides a concrete characterization of closed invariant subspaces for the shift on the Hardy space H^2 in the unit disc, stating that every such space is of the form fH^2 , where f is an inner function. This result can also be interpreted in an operator sense by saying that every closed subspace invariant for the shift is the image of H^2 via an isometry. From this perspective, Beurling’s theorem has been extended by Lax, Halmos, and Rovnyak to shifts of any index, proving that a closed subspace is invariant for a shift if and only if it is the image of the space via a quasi-isometry that commutes with the shift (the so-called Beurling-Lax theorem).
In this talk, I will present a generalization of the “concrete” form of Beurling’s theorem for the shift on the direct finite sum of H^2. I will show that every closed invariant subspace is given, up to multiplication by an inner function, by the intersection of what we call “determinantal spaces”—which, roughly speaking, are the preimages of shift-invariant subspaces of H^2 via a linear operator commuting with the shift and that are constructed through a determinant of certain matrices with entries given by holomorphic bounded functions. The concreteness of such a structure theorem allows us to prove by rather simple algebraic manipulation, as in the classical Beurling theorem, that the only non-trivial maximal closed shift-invariant subspaces are of codimension one. Using the universality of the (backward) shift in the class of operators with defect less than or equal to the index of the shift, this gives a proof of the following result: every bounded linear operator from a Hilbert space into itself whose defect is finite has a non-trivial closed invariant subspace.

The talk is based on a joint work with Eva Gallardo-Gutierrez

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

We investigate the problems of unirationality and rationality for conic bundles (S o mathbb{P}^1) over a C1 field k, which can be described as the zero locus of a hypersurface in the projectivization of a rank-3 vector bundle over (mathbb{P}^1). Conic bundles can be classified by the degree d of the discriminant, i.e. the number of points on the base where the corresponding fiber is not a smooth conic. Unirationality for d < 8 was already established by Kollár and Mella in 2014, while the case for general d remains open. In this work we focus on the next case d=8, and explicitly show that for all four possible types of such conic bundles S, the set of unirational ones forms an open subset of the parameter space. We also examine algebraic constraints, depending on the base field k, under which a general S is not rational over k.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

Consider the Restricted Planar Elliptic Three Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. For eccentricity of Jupiter $e_0$ small enough, we show that there exists a family of probability measures supported at the $3 : 1$ mean motion resonance such that the push forward under the associated Hamiltonian flow has the following property. At the timescale $te_0^{-2}$, the distribution of the Jacobi constant of the Asteroid weakly converges to an Ito diffusion process on the line as $e_0 o 0$. This resonance corresponds to the biggest of the Kirkwood gap on the Asteroid belt in the Solar System. This is a joint work with V. Kaloshin, P. Martin and P. Roldan.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Maps to projective space are given by basepoint-free linear series, thus these are key to understanding the extrinsic geometry of algebraic curves. How does a linear series degenerate when the underlying curve degenerates and becomes nodal? Eisenbud and Harris gave a satisfactory answer to this question when the nodal curve is of compact type. I will report on a joint work in progress with Luca Battistella and Jonathan Wise, in which we review this question from a moduli-theoretic and logarithmic perspective. The logarithmic prospective helps understanding the rich polyhedral and combinatorial structures underlying degenerations of linear series; these are linked with the theory of matroids and Bruhat-Tits buildings.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027), Prin 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

  Structure properties of free Lie algebras are a fundamental tool in group theory and its many applications. However, it is not always easy in practice to recognize that a Lie algebra is free. The talk will survey various results that allow to conclude to freeness, and various concrete examples.
  Based on joint work with L. Foissy.