Stable permutations are a class of permutations that arises in the study of the automorphism group of the Cuntz algebra. In this talk, after introducing the Cuntz algebra and surveying the main known results about stable permutations, I will present a characterization of stable permutations in terms of certain associated graphs. As a consequence of this characterization we prove a conjecture in [Advances in Math. 381 (2021) 107590], namely that almost all permutations are not stable, and we characterize explicitly stable 4 and 5-cycles.
  This is a joint work with Roberto Conti and Gleb Nenashev.
  N.B.: please click HERE to attend the talk in streaming.

I will describe the evolution of measures for coupled dynamical systems with/without noise where the number of coupled units is large, but finite. I will compare the evolution for the finite dimensional system with its thermodynamic limit, which is described by a nonlinear self-consistent transfer operator. In particular, I will give sufficient conditions for the equilibrium states of the thermodynamic limit to be “self-sustaining” for the finite dimensional system: These states are characterized by being “almost” invariant for the finite system, and although might be far from any stationary state, they describe the statistical behavior of the system for long transients whose duration scales exponentially with the number of coupled units.

We consider eigenvalue problems and bifurcation of positive solutions for elliptic equations with indefinite weights and with Neumann boundary conditions. We give complete results concerning the existence and non- existence of positive solutions for the superlinear coercive and non-coercive problems, showing a surprising complementarity of the respective results. Joint work with Bernhard Ruf (Università degli Studi di Milano).
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

  A few years ago, Bonnafé-Rouquier defined 'Calogero-Moser cells' through the representation theory of rational Cherednik algebras. These cells partition the elements of a complex reflection group G, but are currently difficult to calculate except in small rank examples. In the special case when G is a finite Coxeter group, the cells are conjectured to be the same as Kazhdan-Lusztig cells. In other words, conjecturally 'Calogero-Moser cells' generalise Kazhdan-Lusztig cell theory from Coxeter groups to complex reflection groups. I will discuss a confirmation of this conjecture for G being the symmetric group. The proof uses ideas from integrable systems (Gaudin algebras), algebraic geometry (moduli of points on genus zero curves), and combinatorics (crystals).
  This is joint work with A.Brochier and N.White.
  N.B.: please click HERE to attend the talk in streaming.

  The original quantum groups - in particular, quantized universal enveloping algebras, in short QUEA's - have been introduced as depending on just one "continuous" parameter. Later on, multiparameter quantum groups - in particular, multiparameter QUEA's - have been introduced in differente ways, with the new, "discrete" parameters either affecting the coalgebra structure or the algebra structure (while leaving the dual structure unchanged). Both cases can be realized as special type deformations - namely, either by twist, or by 2-cocycle deformation - of Drinfeld's celebrated QUEA U_h(g). In this talk I will introduce a new, far-reaching family of multiparameter QUEA's that encompasses and generalizes the previous ones, while also being stable with respect to both deformation by twists and deformations by cocycles. Taking semiclassical limits, these new multiparameter QUEA's give rise to a new family of multiparameter Lie bialgebras, that in turn is stable under both by twist and deformations by 2-cocycles (in the Lie bialgebraic sense).
  This is a joint work with Gastón Andrés García - cf. arXiv:2203.11023 (2022).
  N.B.: please click HERE to attend the talk in streaming.

Piecewise polynomials over triangles and tetrahedrons have applications in several branches ranging from finite element analysis, surfaces in computer aided design... The smoothness on tetrahedrons is obtained either by high degrees of polynomials or using smaller degrees when splitting the tetraehedron into smaller pieces. Here we consider the Alfeld split which generalizes the Clough-Tocher split of a triangle. Simplex splines with arbitrary knots are the natural generalization of univariate B-splines to several variables. We consider degrees d = 2s-1 in dimension s and construct a partition of unity basis for the space S¹_2s-1;s on the Alfeld split, consisting of simplex-splines. We also show a Marsden like identity for s ≤ 5. Joint work with Jean-Louis Merrien. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

The problem of prescribing conformally the scalar curvature on a closed manifold of negative Yamabe invariant is always solvable, if the function to be prescribed is strictly negative, while sufficient and necessary conditions are known in the case that function is non positive. Still in the case of a negative Yamabe invariant, Rauzy (Trans. Amer. Math. Soc. 1995) showed solvability, if the function to be prescribed is not too positive, as quantified by Aubin-Bismuth (J. Funct. Anal. 1997) later on. In this talk we will review these results variationally and shed some light on the case, when Rauzy’s conditions fail. This talk is joint work with Martin Mayer.  
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

Many physical phenomena are well described as the propagation of waves: the motion of the sea, the transmission of sound, electromagnetic waves (light, radio waves). Their mathematical description is often extremely complicated, and characterized by the coexistence of stable and chaotic behaviors. I will discuss some models of wave propagation by nonlinear Partial Differential Equations illustrating briefly the main difficulties as well as some mathematical methods used to study them.

A small refreshment will be served at the end of the colloquium.

MS Teams Link for the streaming

Note: This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

We consider a partially damped nonlinear beam-wave system of evolution PDE's modeling the dynamics of a degenerate plate. The plate can move both vertically and torsionally and, consequently, the solution has two components. We show that the component from the damped beam equation always vanishes asymptotically while the component from the (undamped) wave equation does not. In case of small energies we show that the first component vanishes at exponential rate. Our results highlight that partial damping is not enough to steer the whole solution to rest and that the partially damped system can be less stable than the undamped system. Hence, the model and the behavior of the solution enter in the framework of the so-called "indirect damping" and "destabilization paradox". These phenomena are valorized by a physical interpretation leading to possible new explanations of the Tacoma Narrows Bridge collapse. Joint work with Abdelaziz Soufyane (University of Sharjah, UAE)
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

In the last 10-15 years, “bootstrap” was applied to problems as disparate as critical exponents of second-order phase transitions, scattering of elementary particles, chaotic dynamical systems, and Laplacian spectra on hyperbolic manifolds. This produced many new computer-assisted bounds on various quantities of interest. I will explain common features of these problems, and what is this bootstrap method which applies to all of them.