Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will discuss the classical question of whether Severi varieties are irreducible and its relation to the irreducibility of other moduli spaces of curves. I will indicate how tropical methods can be used to answer such irreducibility questions. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.

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

  The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode). Based on joint work with Myriam Mahaman.

  In this presentation, I will discuss the combinatorics of the noncrossing partition posets associated with Coxeter groups of rank three. In particular, I will describe the techniques used to prove the lattice property and lexicographic shellability. These properties can then be used to solve several problems on the corresponding Artin groups, such as the K(π,1) conjecture, the word problem, the center problem, and the isomorphism between standard and dual Artin groups.
  This is joint work with Emanuele Delucchi and Mario Salvetti.

We present a survey of nonlinear elliptic equations with nonlocal interactions. These equations describe the collective behavior of self-interacting many-body systems at different scales, from atoms and molecules to the formation of stars and galaxies. What sets these models apart from classical nonlinear PDEs is the presence of nonlocal terms in the equations, introduced via Coulomb-type interactions or a fractional Laplacian term, or both. We provide an overview of typical problems with repulsive interactions originating from Density Functional Theory; and Choquard type problems with attractive gravitational interactions. We also outline recent results in problems featuring competing attractive/ repulsive terms, which create particularly complex structures.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

I will report on recent developments concerning the control of a class of short-range perturbations of the Hamiltonian of an Ising chain. An example covered by our analysis is the celebrated XXZ chain. The talk is based on a joint work with S. Del Vecchio, J. Fröhlich, and A. Pizzo.

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