Univariate Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices will be discussed. Their computation is based on the homotopy continuation concept that transforms Gaussian quadrature rules from the so called source space to the target space. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odd-degree space, and building the source space as a union of such discontinuous elements, we derive rules for target spline spaces with higher continuity across the elements. We demonstrate the concept by computing numerically Gaussian rules for spline spaces of various degrees, particularly those with non-uniform knot vectors and non-uniform knot multiplicities. We also discuss convergence of the spline rules over finite domains to their asymptotic counterparts, that is, the analogues of the half-point rule of Hughes et al., that are exact and Gaussian over the infinite domain. Finally, the application of spline Gaussian rules in the context of isogeometric analysis on subdivision surfaces will be discussed, showing the advantages and limitations of the tensor product Gaussian rules. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

A classic yet delicate fact of Morse theory states that the unstable manifolds of a Morse-Smale gradient-flow on a closed manifold M are the open cells of a CW-decomposition of M. I will describe a self-contained proof by Abbondandolo and myself. The key tool is a "system of invariant stable foliations", which is analogous to the object introduced by Palis and Smale in their proof of structural stability of Morse Smale diffeomorphisms and flows, but with finer regularity and geometric properties.
[Stable foliations and CW structure induced by a Morse-Smale gradient flow, A.Abbondandolo,P.Majer] https://www.worldscientific.com/doi/10.1142/S1793525321500527
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

  Vinberg representations are representations of algebraic groups that arise from a cyclic grading of a semisimple Lie algebra. In the literature they are mainly known as theta-groups or Vinberg pairs. A distinguishing feature of these representations is that it is possible to classify the orbits of the algebraic group. We sketch how this can be done when the base field is the complex numbers. This mainly uses results of Vinberg of the 70's. Then we describe techniques for classifying the orbits when the base field is the real numbers. This talk is based on joint work with Mikhail Borovoi, Hong Van Le, Heiko Dietrich, Marcos Origlia, Alessio Marrani.

  The combinatorial invariance conjecture asserts that the Kazhdan-Lusztig (KL) polynomial of an interval [u,v] in Bruhat order can be determined just from the knowledge of the poset isomorphism type of [u,v]. Recent work of Blundell, Buesing, Davies, Velicković, and Williamson posed a conjectural recurrence for KL polynomials depending only on the poset structure of [u,v]. Their formula uses a new combinatorial structure, called a hypercube decomposition, that can be found in any interval of the symmetric group. We give a new, simpler, formula based on hypercube decompositions and prove it holds for "elementary" intervals: an interval [u,v] is elementary if it is isomorphic as a poset to an interval with linearly independent bottom edges. As a result, we prove combinatorial invariance for Kazhdan-Lusztig R-polynomials of elementary intervals in the symmetric group, generalizing the previously known case of lower intervals.
  This is a joint work with Christian Gaetz.

Since 1976 J.E. Roberts introduced a non-Abelian 1-cohomology of charge-transporters on the Haag-Kaster networks, and as early as 1990 he proved that this cohomology gives a category equivalent to the one of the DHR sectors of the (Haag dual) net of the observables on the Minkowski d=1+3.
In the DHR framework, the covariance of the sectors by the geometric symmetry is introduced through the vacuum representation and morphisms. Quite recently, with G. Ruzzi and E. Vasselli, motivated by theories on a globally hyperbolic spacetime and by sectors with electric charges, as in the analysis by Buchholz and Roberts, we introduced a novel cohomology covariant under the geometric symmetry, for simply connected spacetimes. I will discuss these recent results and some open problems about non-simply connected spacetimes.

Over the last few decades, the study of the nonlinear Schrödinger equation on $mathbb{R}^N$ has been investigated by numerous researchers. However, very few results are known when the domain is non-Euclidean. In this talk, we will see some recent results regarding the existence and multiplicity of solutions for the nonlinear Schrödinger equation on non-compact Riemannian manifolds. In particular, we will focus our attention to the interplay between the necessary assumptions on the potential in the Schrödinger operator and those on the manifold.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

In this talk, I will go back to the origin of the minimal exponent and give a brief history on how it naturally arises in the context of integration over vanishing cycles (Arnold-Varchenko), counting integer solutions of congruence equations (Igusa) and Archimedean zeta functions (Atiyah, Bernstein, Loeser). Then I will talk about some joint work in progress with Dougal Davis (on birational formula of higher multiplier ideals via Beilinson’s formula from Jansen’s conjecture in geometry representation theory) and Ming Hao Quek (on birational characterization of minimal exponents via toric geometry and multi weighted blow-ups).

  Scale groups are closed subgroups of the group of isometries of a regular tree that fixes an end of the tree and are vertex-transitive. They play an important role in the study of locally compact totally di-sconnected groups as was recently observed by P-E. Caprace and G. Willis. In the 80’s they were studied by A. Figa-Talamanca and C. Nebbia in the context of abstract harmonic analysis and amenability. It is a miracle that they are closely related to fractal groups, a special subclass of self-similar groups.
In my talk I will discuss two ways of building scale groups. One is based on the use of scale-invariant groups studied by V. Nekrashevych and G. Pete, and a second is based on the use of liftable fractal groups. The examples based on both approaches will be demonstrated using such groups as Basilica, Hanoi Tower Group, and a group of intermediate growth (between polynomial and exponential). Additionally, the group of isometries of the ring of p-adics and the group of dilations of the field of p-adics will be mentioned in relation with the discussed topics.

The role of automorphic forms as intertwiners between various representations of free group factors was discovered a long time ago by Vaughan Jones, starting with a remarkable formula relating Peterson scalar product with the intrinsical trace. The intertwiner associated to an automorphic form is an eclectic object, not much can be computed, but the Muray von Neuman dimension can be used to get hints on its image. Vaughan Jones used that to settle the problem of finding analytic functions vanishing on the orbit under the modular group of a point in the upper half plane. In past work of the speaker, it was put in evidence that this is related to equivariant Berezin quantization. This leads to a different representation of free group factors and to the existence of a quantum dynamics whose associated unbounded Hochschild 2- cocycle is related to the isomorphism problem. I will explain some concrete formulae and some new interpretation of the associated quantum dynamics

Dynamical systems subject to perturbations that decay over time are relevant in the description of many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, as well as in celestial mechanics. For this reason, in the present talk, we consider a time-dependent perturbation of a Hamiltonian dynamical system having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of orbits converging in time to the quasiperiodic solutions associated with the unperturbed system. This result generalizes the work of Canadell and de la Llave, where exponential decay in time was considered, and the one of Fortunati and Wiggins, where arithmetic, non-degeneracy conditions, and exponential decay in time are assumed. We apply this result to the example of the planar three-body problem perturbed by a given comet coming from and going back to infinity asymptotically along a hyperbolic Keplerian orbit (modeled as a time-dependent perturbation).
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)