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

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)

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 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.

  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 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).

C*-algebraic bundles (nowadays simply called Fell bundles) were introduced by Fell at the end of the sixties as yet another tool to deal with the representation theory of locally compact groups. The reason why the related literature is still growing is probably due to the fact that they fit pretty well with various aspects of the theory of (twisted) actions (and coactions) of groups (and groupoids) on C*-algebras. For instance, many familiar constructions like group C*-algebras and crossed products can be viewed as cross sectional C*-algebras of suitable Fell bundles. In the talk we will introduce the concept of positive definite "multiplier" between Fell bundles and discuss some consequences and applications. Especially, a notion of amenability for Fell bundles naturally appears. Other applications are concerned with the construction of certain functors from the category of positive definite multipliers to the category of completely positive maps between C*-algebras and with the existence of certain C*-correspondences associated to left actions of Fell bundles on right Hilbert bundles. (Joint work with E. Bedos)

Sia X una varietà di Fano liscia, complessa, di dimensione 4, e rho(X) il suo numero di Picard. Inizieremo discutendo il seguente risultato: se rho(X)>12, allora X è un prodotto di superfici di del Pezzo; se rho(X)=12, allora X ha una contrazione razionale X-->Y dove Y ha dimensione 3. Una contrazione razionale è una mappa data da una successione di flips seguita da un morfismo suriettivo a fibre connesse, vedremo degli esempi espliciti. Poi discuteremo le proprietà geometriche delle Fano 4-folds che hanno una contrazione razionale su una 3-fold. Un obiettivo è di determinare il massimo numero di Picard di X, ed eventualmente di classicare i casi con numero di Picard grande. Un altro obiettivo è di usare questa descrizione geometrica per costruire nuovi esempi con rho grande; questo è un progetto in corso con Saverio Secci.

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

In the setting of many-body quantum mechanics, I am going to present a rigorous and recently developed version of Bogoliubov theory. Furthermore, I am going to show how this theory can be applied, on the one hand to study the low-energy spectrum of dilute Bose gases (i.e. to determine the low-lying eigenvalues of their Hamilton operator) and, on the other hand, to approximate their time-evolution, capturing fluctuations around the nonlinear Gross-Pitaevskii equation describing the dynamics of the Bose-Einstein condensate.
NB:This colloquium is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

We explain our recent work on the classification of surfaces of general type with p_g=q=2 or p_g=q=1. Our approach is based on cohomological rank functions, the Chen-Jiang decomposition/Fujita decomposition and Severi type inequalities. This talk is based on a joint work with Jiabin Du and Guoyun Zhang and a joint work in progress with Hsueh-Yung Lin.