In this talk we present the state of an ongoing project concerning the construction of local nets of real subspaces on causal homogeneous spaces M = G/H by starting with a unitary representation of G. Here G/H can be a causal symmetric spaces (non-compactly or compactly causal) or a causal flag manifold. In Physics, the case of timeoriented Lorentzian spaces, these three types correspond to de Sitter space, anti-de Sitter space and the causal compactification of Minkowski space, or some covering of such a space.
For a given unitary representation, we characterize in purely representation theoretic terms: a) the existence of a corresponding net on M with the Bisognano-Wichmann and the Reeh-Schlieder property. b) locality properties of the nets, including in particular spacelike and timelike duality on Minkowski space. This is joint work with V. Morinelli and G. Olafsson

Scalar conservation laws $\partial_t u + \operatorname{div} f(u) = 0$ are nonlinear first-order PDEs that model the transport of a conserved quantity $u$, with a flux $f$ that depends on $u$. A prototype, in one space dimension, is Burgers' equation, where $f(u) = u^2/2$. Even for smooth initial data, their solutions develop discontinuities (shocks), so one works with weak solutions, which may jump across a codimension-one set, the jump set. However, weak solutions are not unique. Uniqueness is restored by selecting entropy solutions, for which every convex function of the solution (an entropy) can only be dissipated. While the nonlinearity of the flux creates shocks, it also has a regularizing effect. In this talk, we discuss the fine regularity it provides outside the jump set. In several space dimensions, we show that entropy solutions are continuous there, assuming only that the flux is nonlinear in a qualitative sense. We then consider the broader class of weak solutions with finite entropy production, which can also create entropy, as long as the entropy production is a finite measure. For these, continuity fails already in one space dimension. For every nonlinear flux, we construct such solutions with zero initial datum whose essential infimum and supremum on every open set are two fixed states, so they are essentially continuous at no point. For Burgers' equation, we also construct a solution with finite entropy production that has a large set of non-Lebesgue points outside the jump set.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We present a variational definition of essential spectrum that suits nonlinear operators like the Dirichlet p-Laplacian. We extend to this framework Persson's Theorem, which relates the bottom of the spectrum to the sharp Poincaré constant at infinity; then, we discuss some consequences on the fundamental spectral gap. Based on joint works with Luca Briani, Lorenzo Brasco, and Francesca Agnese Prinari.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Classical Iwasawa theory studies the growth of class groups in towers of number fields. A parallel theory has emerged for towers of graph coverings, with graph Jacobians playing the role of class groups and Ihara zeta functions providing analytic counterparts. In this joint talk, we will introduce this analogy and survey growth formulas and main conjectures for graph coverings. Particular emphasis will be placed on our joint work concerning the $q$-primary parts of graph Jacobians in $\mathbb{Z}_p^d$-towers, where $q\ne p$. The first part, presented by Antonio Lei, will give the background, examples, and statements of the main results. In the second part, Katharina Müller will explain the principal ideas behind the proofs. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

I will present a probabilistic construction of the three-dimensional hyperbolic Wess–Zumino–Witten (H₃ WZW) model, starting directly from its path integral. Even though this construction works on arbitrary closed Riemann surfaces, in the presence of smooth background gauge fields, I will focus on the simple case of the Riemann sphere with trivial bundle. Within this framework, I will establish a rigorous correspondence between correlation functions of the H₃ WZW model and those of Liouville conformal field theory, extending formulas by Ribault–Teschner and Hikida–Schomerus.
This is joint work with Colin Guillarmou and Antti Kupiainen.

  We construct braided Hopf algebroids in a braided monoidal category over a field k and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids associated to braided Hopf Galois extensions.
  We introduce braided Lie-Rinehart algebras, which generalize the definition in a paper by Aschieri, Landi, and Pagani, and study their universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the modules of a triangular Hopf algebra. We then focus on the case of braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the H-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras.
  Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the k-order universal enveloping algebra of the braided right invariant vector fields and the k-th jet space of a braided Hopf algebroid, and furthermore a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid, under a finiteness condition.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

  An algebra A is called PI if there is some polynomial in non-commuting variables that vanishes under every evaluation on elements of A. Simple examples of PI algebras include finite-dimensional algebras and commutative algebras. In general, it is a hard problem to recognize when an algebra is PI. In this talk, I will describe a simple characterization of the quivers with a PI path algebra. I will then explain how to describe the full T-ideal of their polynomial identities in terms of a preorder incidence algebra that is naturally associated with the quiver.
  This is based on joint work with A. Berele, G. Cerulli Irelli, and E. Pascucci.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

Orthogonal Shimura varieties arise naturally as moduli spaces in algebraic geometry, one example being the moduli space of quasi-polarized K3 surfaces. Their geometry is closely related to the theory of automorphic forms. I will discuss two instances of this relationship. First, the obstruction space for extending a line bundle on an orthogonal Shimura variety to its Baily-Borel compactification can be identified with a certain space of theta series for a symplectic group. By showing that these theta series span the space of cusp forms we conclude that the Picard group of this Baily-Borel compactification has rank one and is spanned by a multiple of the hodge line bundle if the underlying lattice splits two hyperbolic planes globally and three locally. In the second part, I will discuss a result in which Eisenstein series for orthogonal groups are used to study the birational geometry of orthogonal Shimura varieties. We construct pluricanonical forms from Eisenstein series, yielding new results on when such varieties are of general type.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

In this talk we will focus for N\geq 2 on fractional W^(1-1/N,N) maps from an (N−1)-dimensional sphere into itself as traces of maps with finite energies modelled on the N-Dirichlet integral. We consider energy minimization among maps with trace of prescribed topological degree. Our goal is to prove existence and quantitative stability for minimizers in homotopy classes. Due to the lack of compact Sobolev embeddings and conformal invariance, minimizing sequences may fail to converge strongly, leading to a concentration-compactness alternative and bubbling phenomena. When they occur, these concentration effects are investigated using PDEs tools like blow-up analysis and measure-theoretic tools such as Cartesian currents. Finally, we discuss genericity results under perturbations of the energy functional and we present some open problems. This is a joint ongoing work with Xavier Lamy (Toulouse).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

The classical many body problem with constant masses is non-integrable, therefore, to investigate the dynamics of planetary systems, perturbation theory based on the exact solution of the two-body problem is usually applied. However, in the case of variable masses, the two-body problem is generally non-integrable as well. Consequently, to apply perturbation theory, another integrable model problem is used as the unperturbed motion, the solutions of which are close to the solution of the two-body problem with variable masses (in particular, when the mass changes according to Meshchersky's first law, the solutions coincide exactly). In this talk, perturbation theory is used to derive the differential equations determining the orbital elements of the relative motion of one body around another. Here, the models of perturbed aperiodic motion along a conic section and along a quasi-conic section are considered and compared. It is shown that the model based on aperiodic motion along a quasi-conic section is more suitable for generalization to the many body problem with variable masses. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).