We prove the Shafarevich conjecture for a large class of irregular varieties. Our proof relies on the Lawrence-Venkatesh method as used by Lawrence-Sawin, together with the big monodromy criterion from our previous work with Javanpeykar, Lehn and Maculan. This is joint work in progress with Marco Maculan (IMJ Paris).

  We will talk about minimal monomial lifting of cluster algebras. That is sort of a homogenisation technique, whose goal is to identify a cluster algebra structure on some schemes "suitable for lifting", compatibly with a base cluster algebra structure on a given subscheme. We will see how to apply this technique to study some branching problems, in representation theory of complex reductive groups and, time permitting, we will discuss some possible development as the construction of polyhedral models for multiplicities.

  Quasi-invariants are special polynomials associated with a finite reflection group W and a multiplicity function. They appeared in 1990 in the study of Calogero-Moser integrable systems by Chalykh and Veselov, in which case they are the highest symbols of differential operators which form a large commutative ring. Similarly to all the polynomials, quasi-invariants form a free module over invariant polynomials of rank |W|, and they have other good properties. Quasi-invariants form representations of spherical Cherednik algebras as was established by Berest, Etingof and Ginzburg in 2003, which gives a way to establish the freeness property. I am going to explain a more recent application of quasi-invariants to the theory of free multi-arrangements of hyperplanes. In this case one is interested in the module of logarithmic vector fields which is known to be free over polynomials for some arrangements including Coxeter ones. Quasi-invariants can be used to construct elements of this module, and they also lead to new free multi-arrangements in the case of complex reflection groups.
  The talk is based on a joint work with T. Abe, N. Enomoto and M. Yoshinaga.

I will explore some recent results based on the interaction of operator space theory and quantum nonlocality. In particular I will emphasize the connection between large violations of Bell inequalities and certain norms in Banach and operator space categories. Finally, using Grothendieck's inequality, I will derive some interesting consequences for the parallel repetition problem in the context of XOR games.

If X is a projective variety cut out byquadrics, the p^th Syzygy scheme Syz_p(X) is the scheme cut out by quadrics involved in a p^th syzygy of X, and it turns out to capture refined geometrical properties of X in its embedding. We report on joint work in progress with M. Aprodu and E. Sernesi, concerning the second Syzygy scheme Syz_2(C) of a smooth curve in case C is embedded either by the canonical line bundle or by a non-special line bundle L, aiming at a classification of all (C,L) such that Syz_2(C) strictly contains

  A Yetter-Drinfeld module (=YD-module) over a bialgebra H is at the same time module and comodule over H satisfying a compatibility condition. It is well-known that the category of YD-modules (over a finite-dimensional Hopf algebra H) is equivalent to the center of the monoidal category of H-(co)modules as well as the category of modules over Drinfeld doubles of H. Canaepeel, Militaru, and Zhu introduced generalized YD-modules. More precisely, they consider two bialgebras H, K, together with a bicomodule algebra C and bimodule coalgebra over them. A generalized YD-module in their sense, is simultaneously an A-module and a C-comodule with a compatibility condition. Under a finiteness condition, they showed that these modules are exactly modules over a suitable constructed smash product build out of A and C. The aim of this talk is to show how the category of the generalized YD-modules can be obtained as a relative center of the category A-modules, viewed as a bimodule category over categories of H-modules and K-modules. Moreover, we also show how other variations of YD-modules, such as anti-YD-modules, arise as a particular case.
  This talk is based on ongoing work with Joost Vercruysse.

  Many well-known examples of "noncommutative spaces", like Woronowicz' quantum SU(2) or Vaksman-Soibelman odd-dimensional quantum spheres, can be described by C*-algebras associated to directed graphs. More generally, many compact quantum groups and quantum homogeneous spaces, can be described by convolution C*-algebras of "nice" groupoids. C*-algebras associated to combinatorial data (graphs, diagrams, groupoids) allow efficient models to attack key open problems in noncommutative geometry.
  The aim of this talk is to present some basic ideas of noncommutative geometry, using graph and groupoid C*-algebras as examples of "noncommutative spaces that one can see".

In the first part of this talk I will present the Solvability Complexity Index — a theory developed with several coauthors over the last decade to help us classify the complexity of everyday computations. This theory serves as a bridge between classifications of degrees of complexity in theoretical computer science, and applications in numerical analysis. In the second part of the talk, I will discuss recent results where these ideas were applied to the computation of resonances, providing the first algorithms that work in any dimension.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

The construction of feedback-like control fields for kinetic models in phase space are discussed. The purpose of these controls is to drive an initial density of particles in the phase space to reach a desired cyclic trajectory on the phase space and follow it in a stable way. For this purpose, an ensemble optimal control problem governed by the kinetic model is formulated in a way that is amenable to a Monte Carlo approach. The proposed formulation allows to define a one-shot solution procedure consisting in a backward solve of an augmented adjoint kinetic model. Results of numerical experiments demonstrate the effectiveness of the proposed control strategy.

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the stationary Dirac equation in presence of a (possibly non-Hermitian) matrix-valued perturbation V such that |V(x)| goes as |x|^(-e) at infinity, for -infty < e < 1. Also, we show that our results are sharp for n=2,3, providing explicit examples of solutions that have the prescribed decay, in presence of a potential with the related behaviour at infinity. In this sense, our work is a result of unique continuation from infinity for the Dirac operator.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006