The study of the (non)-existence of integral Hopf orders was originally motivated by Kaplansky's sixth conjecture, which is a generalization of Frobenius theorem in the Hopf algebra setting. In fact, Larson proved that a Hopf algebra which admits an integral Hopf order satisfies the conjecture.
  The aim of this talk is to give a partial answer to the following question: "Does a semisimple complex Hopf algebra admit an integral Hopf order?"
  In particular, we will present several families of semisimple Hopf algebras which do not admit an integral Hopf order. These Hopf algebras will be constructed as Drinfeld twists of group algebras.
  This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra and on my Ph.D. thesis.

Circa dieci anni fa, V. Jones introduceva diverse rappresentazioni unitarie dei gruppi di Thompson e vari sottogruppi interessanti sono emersi come stabilizzatori di vettori in queste rappresentazioni. In questo seminario presenterò il lavoro svolto su questo argomento.

We will report on joint works with Paola Frediani, Juan Carlos Naranjo and Gian Pietro Pirola. We study the second fundamental form of the Siegel metric in A_5 restricted to the moduli space of the intermediate jacobians cubic threefolds and the second fundamental form in A_9 restricted to the moduli of the the intermediate jacobians of (2,2) threefolds in P^2xP^2 . In both case there is a natural composition with a multiplication map. For cubic threefold this composition results to be zero, while for (2,2) threefolds it gives a not zero holomorphic section of a bundle.

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).