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

Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective K3 surfaces. In the talk I will present a joint work with G. Mongardi in which we show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.

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.

In this talk I will explain how to extract an 'anyon theory' (braided tensor category) from a gapped ground state of an infinite two-dimensional lattice spin system. Just as in the DHR formalism from AQFT, the anyon types correspond to certain superselection sectors of the observable algebra of the spin system. We apply this formalism to Kitaev's quantum double model for finite gauge group G, and find that the anyon types correspond precisely to the representations of the quantum double algebra of G.
The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

The Cauchy-Liouville theorem (1844) states that any bounded entire function of a complex variable is necessarily constant. In the realm of PDE's, by a Liouville-type theorem, one usually means a statement asserting the nonexistence of solutions in the whole space (or a suitable unbounded domain). Numerous results of this kind have appeared over the years and many far-reaching applications have arisen, conferring Liouville-type theorems an important role in the theory of PDE's and revealing strong connections with other mathematical areas (calculus of variations, geometry, fluid dynamics, optimal stochastic control). After a brief historical detour (minimal surfaces - Lagrange, Bernstein, de Giorgi, Bombieri,… and regularity theory for linear elliptic systems - Giaquinta, Necas, ...), we will recall the developments of the 1980-2000's on nonlinear elliptic problems, leading to powerful tools for existence and a priori estimates for Dirichlet problems (Gidas, Spruck, Caffarelli, ...), based on the combination of Liouville type theorems and renormalization techniques. In a more recent period, this line of research has also led to much progress in the study of singularities of solutions, both for stationary (elliptic) and evolution PDEs. In particular, in the case of power like nonlinearities, we will recall the equivalence between Liouville type theorems and universal estimates, based on a method of doubling-rescaling (joint work with P. Polacik and P. Quittner, 2007). Then we will present recent developments which show that these renormalization techniques can be applied to nonlinearities without any scale invariance, even asymptotically, with applications to initial and final blowup rates or decay rates in space and/or time.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. We apply this general paradigm to Calabi-Yau varieties Z in characteristic p. First, we show that if Z has torsion-free crystalline cohomology and degenerating Hodge-de Rham spectral sequence (and for p=2 a lift to W/4), then its mixed characteristic deformations are unobstructed. This generalises the BTT theorem from characteristic 0 to characteristic p. If Z is ordinary, we show that it moreover admits a canonical (and algebraisable) lift to characteristic zero, thereby extending Serre-Tate theory from abelian varieties to Calabi-Yau varieties. This is joint work with Taelman, and generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl–Shepherd-Barron, Iacono-Manetti, Schröer, Serre-Tate, and Ward.

Circular polygons are closed plane curves formed by concatenating a finite number of circular arcs so that, at the points where two arcs meet, their tangents agree. These curves are strictly convex and C1, but not C2. We study the billiard dynamics in domains bounded by circular polygons. We prove that there is a set accumulating on the boundary of the domain in which the return dynamics is semiconjugate to a transitive shift on infinitely many symbols. Consequently the return dynamics has infinite topological entropy. In addition we give an exponential lower bound on the number of periodic orbits of large period, and we prove the existence of trajectories along which the angle of reflection tends to zero with optimal linear speed. These results are based on joint work with Rafael Ramírez-Ros.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)