Le isogenie sono morfismi di varietà abeliane. La loro teoria algoritmica è sviluppata da oltre 30 anni, motivata in parte dall'algoritmo di Schoof-Elkies-Atkin per il conteggio di punti, algoritmo fondamentale in crittografia ellittica. I progressi algoritmici hanno portato negli ultimi 20 anni allo sviluppo di una nuova branca della crittografia, detta a base d'isogenie. L'oggetto centrale di questa disciplina non è più una curva ellittica isolata, bensì un grafo di curve ellittiche legate da isogenie. I grafi d'isogenie esibiscono diverse strutture combinatorie interessanti (foreste, grafi di Cayeley, grafi espansori), e offrono dei problemi computazionalmente difficili come la ricerca di cammini. Su queste basi, siamo oggi in misura di costruire un vasto spettro di primitive crittografiche: cifratura e firma digitale resistenti agli attacchi quantistici, crittografia a orologeria, sistemi a soglia, ecc. In questo talk, darò un'introduzione alla teoria delle isogenie di curve ellittiche su corpi finiti, e spiegherò come la crittografia è costruita a partire da esse.

   In the paper [De Concini C., Lusztig G., <em>Homology of the zero-set of a nilpotent vector field on a flag manifold</em>, J. Amer. Math. Soc. <strong>1</strong> (1988), no. 1, 15-34] it was proven that the so called Springer fiber <em>B_n</em> for any nilpotent element <em>n</em> in a complex simple Lie algebra <strong>g</strong> has homological properties that suggest that <em>B_n</em> should have a paving by affine spaces. <br>   This last statement was proved to hold in the case in which <strong>g</strong> is classical, but remained open for exceptional groups in types <em>E</em>₇ and <em>E</em>₈. <br>   In a joint project with Maffei we are trying to fill the gap. At this point our efforts have been successful in type <em>E</em>₇ and "almost" in type <em>E</em>₈, where one is reduced to show it only in one case. <br>   The goal of the talk is to survey the problem and give an idea on how to show our new results. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

  Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the well-known Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to the cup product) and from a suitable loop in the categories of extensions (leading to the Lie bracket). We show how Schwede's construction admits a vast generalisation to general monoidal categories with coefficients of the Ext groups taken in (weak) left and right monoidal (or Drinfel'd) centres. In case of the category of left modules over bialgebroids and coefficients given by commuting pairs of braided (co)commutative (co)monoids in these categorical centres, we provide an explicit description of the algebraic structure obtained this way, and a complete proof that this leads to a Gerstenhaber algebra is then obtained from an operadic approach. This, in particular, considerably generalises the classical construction given by Gerstenhaber himself. Conjecturally, the algebraic structure we describe should produce a Gerstenhaber algebra for an arbitrary monoidal category enriched over abelian groups, but even the bilinearity of the cup product and of the Lie-type bracket defined by the abstract construction in terms of extension categories remain elusive in this general setting. <br> This is a joint work with Niels Kowalzig, cf. arXiv:2112.11552. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

The so called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE. In this talk, we will focus on metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of ''maximal’’ dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we cover this specific dimension constructing and studying singular solutions of critical dimension. <br> <b>NB</b>:<i>This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006</i>

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will review some results and techniques related to the latter setting. I will specifically concentrate on the case of Hilbert scheme of points on K3 surfaces and (as a work in progress) on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macri' and X. Zhao.

In this talk, we will survey known results on deformations of rational curves inside hyperk&aumlhler manifolds, and then provide suitable generalizations for singular hyperk&aumlhler varieties. As an application, we will construct uniruled divisors in many hyperk&aumlhler varieties. This is joint work with Ch. Lehn and G. Pacienza.

&nbsp; Dynamical Galois groups are invariants associated to dynamical systems generated by the iteration of a self-rational map of <strong>P</strong>¹. These are still very mysterious objects, and it is conjectured that abelian groups only appear in very special cases. We will show how the problem is deeply related to a dynamical property of these rational maps (namely that of being post-critically finite) and we will explain how to approach and prove certain non-trivial cases of the conjecture. <br> &nbsp; This is based on joint works with A. Ostafe, C. Pagano and U. Zannier.

&nbsp; It is notoriously difficult to compute class numbers of cyclotomic fields. <br> &nbsp; In this expository lecture we describe an experimental approach to this problem. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.

We show that every C2 Lagrangian invariant torus W of a Tonelli Hamiltonian defined in the n-torus containing an orbit with totally irrational homology class is a graph of the canonical projection. This result extends the graph property obtained by Bangert and Bialy-Polterovich for Lagrangian minimizing tori without periodic orbits in the unit tangent bundle of a Riemannian metric in the two torus. Motivated by the famous Hedlund's examples of Riemannian metrics in the n-torus with n closed, homology independent, minimizing geodesics having minimizing tunnels, we also show the C1-generic nonexistence of Lagrangian invariant tori with "large" homology. <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1642532731244?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

In the last years there have been a renewed interest for a conjecture by Griffiths stated in 1969. The conjecture characterises the positive characteristic forms for positive (in the sense of Griffiths) holomorphic Hermitian vector bundles: those should be the exactly the forms belonging to the positive cone spanned by Schur forms. After recalling the various definitions of positivity for holomorphic Hermitian vector bundles and (p,p)-forms, we shall explain a recent result, obtained in collaboration with my PhD student F. Fagioli, which partially confirms Griffiths' conjecture. The result is obtained as an application of a pointwise, differential-geometric version of a Gysin type formula for the push-forward of the curvature of tautological bundles over the flag bundle.