I will talk about a new cohomology theory for algebraic varieties in positive characteristic, called edged crystalline cohomology. This is a generalisation of crystalline cohomology and depends on the choice of a "decay-function''. Linear decay-functions correspond to integral versions of rigid cohomology, while logarithmic decay-functions produce the conjectured family of log-decay crystalline cohomology theories, parametrised by positive real numbers. During the talk, I will explain the construction of this theory after a brief recall of the classical crystalline and rigid cohomology theories.

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

Hénon maps were introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. They are among the most studied discrete-time dynamical systems that exhibit chaotic behaviour. Complex Hénon maps have been extensively studied over the last three decades, in parallel with the development of pluripotential theory. I will present a recent result obtained with Tien-Cuong Dinh, where we show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence of a recent result by Björklund-Gorodnik, the Central Limit Theorem holds for all Hölder observables.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

We consider a random discrete time system in which the evolution of a stochastic differential equation is sampled at a sequence of discrete times. We set up a functional analytic framework for which we can prove the existence of a spectral gap and estimate the behavior of the leading eigenvalue of the related transfer operator as the system is perturbed by putting a ”hole” in it that cor- responds to a rare event. By doing so, we derive the distribution of the hitting times corresponding to the rare event and the extreme value theory associated with it.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Vertex algebras are algebraic structures coming from two dimensional conformal field theory. This talk is about their relation with moduli spaces of Riemann surfaces.
I will first review some background material. In particular, I will recall that a vertex algebra is a graded vector space V with additional structures, and these structures force the Hilbert-Poincaré series of V, conveniently normalized, to be a modular form.
I will then associate to any holomorphic vertex algebra a collection of Teichmüller modular forms (= sections of powers of the lambda class on the moduli space of Riemann surfaces), whose expansion near the boundary gives back some information about the correlation functions of the vertex algebra. This is a generalization of the Hilbert-Poincaré series of V, it uses moduli spaces of Riemann surfaces of arbitrarily high genus, and it is sometime called partition function of the vertex algebra. I will also explain some partial results towards the reconstruction of the vertex algebra out of these Teichmüller modular forms.
Using the above mentioned construction, we can use vertex algebras to study problems about the moduli space of Riemann surfaces, such as the Schottky problem, the computation of the slope of the effective cone, and the computation of the dimension of the space of sections of powers of the lambda class. On the other hand, this construction allows us to use the geometry of the moduli space of Riemann surfaces to classify vertex algebras; in particular, I will discuss how conjectures and known results about the slope of the effective cone can be used to study the unicity of the moonshine vertex algebras.
This is a work in progress with Sebastiano Carpi.
The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Scattering amplitudes can be extracted from time-ordered $n$-point functions by means of the well known LSZ reduction formula, even in non-perturbative Quantum Field Theories, such as Quantum Chromodynamics (QCD). However, in the context of Lattice QCD, one can access only Euclidean $n$-point functions sampled at discrete points and with finite (but systematically improvable) precision and accuracy. This makes the problem of analytically continuing back to Minkowski space-time ill-posed. I will present here one particular strategy which allows to extract scattering amplitudes from Euclidean correlators, while avoiding analytic continuation, technically turning an ill-posed problem into a merely ill-conditioned one.
Working in the axiomatic framework of the Haag-Ruelle scattering theory, we show that scattering amplitudes can be approximated arbitrarily well in terms of linear combinations of Euclidean correlators at discrete time separations. The essential feature of the proposed approximants is that one can calculate them, at least in principle, from Lattice-QCD data. In this talk, after reviewing the basic ideas behind Haag-Ruelle scattering theory, I will sketch the derivation of the approximations formulae, and discuss extensively how they can be used in practical numerical calculations. Also, similarities and differences with other methods, e.g. Lüscher's formalism, will be reviewed.
MS Teams link to seminar streaming:
https://teams.microsoft.com/l/meetup-join/19%3a9d30e9bd907b4500bbe1387b3dcad011%40thread.tacv2/1715274588963?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%229b393957-c38b-46ad-8d9a-31172e44b55f%22%7d

We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential consequences for the long-time dynamics in the two-dimensional Euler equations.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

A fruitful way to produce examples of IHS varieties is to consider terminalizations of symplectic quotients of symplectic varieties. In a work in collaboration with A. Grossi, M. Mauri and E. Mazzon we classify all terminalizations of quotients of Hilbert schemes of K3 surfaces and generalized Kummer varieties by the action of symplectic automorphisms induced by the underlying surface. Furthermore, we determine their second Betti number, the fundamental group of their singular locus and, in the Kummer case, we determine the singularities of their universal quasi-étale cover. Finally, we compare our deformation types with the examples known in literature, placing our work in the classification program proposed by Menet.

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

  Beilinson-Drinfeld's Grassmannian on an algebraic curve is an important object in Representation Theory and in the Geometric Langlands Program. I will describe some analogs of this construction when the curve is replaced by a surface, together with related preliminary results.
  This is partly a joint work with Benjamin Hennion (Orsay) and Valerio Melani (Florence), and partly a joint work in progress with Andrea Maffei (Pisa) and Valerio Melani (Florence).

  A complex flag manifold F = G/Q decomposes into finitely many real orbits under the action of a real form of G. Their embeddings into F define CR-manifold structures on them. We give a complete classification of all closed simple homogeneous CR-manifolds that have finitely nondegenerate Levi forms.

A standard approach to the construction of smooth low degree polynomial splines over an unstructured triangulation is based on splitting of triangles in such a way that the refined triangulation allows the imposition of smoothness constraints without dependence on geometry. A well-established splitting technique is the Powell-Sabin 6-refinement, which can be used to define C1 quadratic splines as well as splines of higher degree and smoothness. In this talk we review the construction of splines over a Powell-Sabin 6-refinement with a special emphasis on C1 cubic splines. We present B-spline-like functions that enjoy favorable properties such as local support, stability, nonnegativity, and a partition of unity. In particular, we discuss what super-smoothness properties these functions possess and how they depend on geometric properties of the underlying refinement. Based on this we explain how to establish approximation spaces that are suitable for completely unstructured triangulations, partially structured triangulations, and triangulations with a high level of symmetry, e.g., three-directional triangulations. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).