Numerous orbits exist in the solar system or in astrodynamics with very peculiar motions. Their common feature is that they consist of two moons or satellites around a much heavier central attractor with almost equal semi-major axes, this is called a co-orbital motion. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results in this setting have been obtained even in the restricted three-body problem. Actually, the nearly equal semi major axes of the moons implies also nearly equal orbital periods (or 1:1 mean motion resonance), and this last point prevent the application of the usual Hamiltonian perturbation theory for the three body problem.
Adapting the idea of Arnold to a resonant case, hence by an applcation of KAM theory to the planar planetary three-body problem, we provide a rigorous proof of existence of a large measure set of Lagrangian invariant tori supporting quasi-periodic co-orbital motions, hence stable over infinite times.
(Joint work with L. Biasco, L. Chierchia, A. Pousse and P. Robutel)
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

&nbsp; &nbsp; In this talk, we first review the band bases of the skein algebras (equivalently, the quantum cluster algebras) associated with unpunctured surfaces. We then show that these bases coincide with the common triangular bases, that is, the Kazhdan--Lusztig--type bases of these quantum cluster algebras. In our approach to this result, we discover a phenomenon in which the null loops involved are arranged in configurations resembling beads on a necklace. <br> &nbsp; This is a joint work with Chao Shen. <br> &nbsp; <em><small><small> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </small></small></em>

Given an alphabet, which can be infinite, we consider subshifts defined by a set of forbidden words as a combinatorial object and define algebras associated with them. When the alphabet is finite, these include Carlsen's C*-algebras associated with subshifts. In this talk, I will explain how one naturally obtains a partial action from a subshift and how to use it to define the subshift algebras. As with graphs, even though we start with a combinatorial object, certain topological spaces arise from these algebras, one of them being the Ott-Tomforde-Willis (OTW) subshift. I will show how these C*-algebras are related to the conjugacy of Ott-Tomforde-Willis subshifts. At the end, I will briefly talk about the K-theory of the subshift C*-algebras. (Joint work with G. Boava, D. Gonçalves and D. van Wyk.)

Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this talk, I will explain how to make combinatorial methods functorial, and show that collapsing an outsplitted graph to the original graph and transforming a graph to a shifted graph can be implemented by admissible graph homomorphisms and admissible path homomorphisms, respectively. To include the inverses of such isomorphisms, I will introduce a new category of graphs where morphisms are given as regular homomorphisms of graph inverse semigroups. This new category admits a covariant functor to the category of C*-algebras and *-homomorphisms which extends the known covariant functor from the category of graphs and admissible path homomorphisms. I will demonstrate the new framework by applying it to various graphs that yield matrix algebras, Cuntz algebras, and matrices over Cuntz algebras. (Based on joint work with G. G. de Castro, M. Lowiel, E. Pacheco and M. Tobolski.)

I'll discuss commuting varieties and a new upper bound for the density of pairs of commuting n x n matrices with integer entries. Our approach uses Fourier analysis and reduction modulo a suitably chosen prime, together with a result about the flatness of the commutator Lie bracket, which we also solve. This is joint work with Will Sawin and Victor Wang. <em> Note: </em> 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

The Frenkel-Kontorova model is a standard model in condensed matter physics describing particles having nearest-neighbor spring-like interactions. Mathematical analysis of this model leads to studying standard-like twist maps. In the 80s, Suris found a remarkable family of potentials for this model with integrable dynamics. In some sense, this is similar to the role that ellipses play in planar billiards. In the talk, we will highlight this connection via the action-angle coordinates of the two systems. Then we will also show that an integrable perturbation of a Suris potential has to be a Suris potential itself. This is in the spirit of local results proven for the Birkhoff conjecture in billiards. The proof relies heavily on Fourier analysis, as well as the construction of a suitable basis for L2, which captures the dynamics of the system. Joint work with Corentin Fierobe.

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

Dynamical Galois groups are profinite groups that are constructed via iterations of rational functions. They are intimately connected with, and in a way they are a wild generalization of, Galois representations on Tate modules of elliptic curves. The problem of determining which rational functions yield abelian dynamical Galois groups has attracted quite some attention in recent years; in this talk I will survey on what is known about this problem, and explain how to solve certain instances of it over number fields and over function fields. This is based on joint works with P. Ingram, A. Ostafe, C. Pagano and U. Zannier. <em> Note: </em> 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

  Differentiable stacks are a class of singular spaces in differential geometry including orbifolds, leaf spaces of foliations and orbit spaces of Lie group actions. One possible definition is: a differentiable stack is a Morita equivalence class of Lie groupoids. It follows from this definition that geometry on differentiable stacks is more or less the same as Morita invariant geometry of Lie groupoids. Following this principle, several different geometries on differentiable stacks have been introduced and studied recently, including vector fields, differential forms, symplectic and Poisson structures, with several applications in Poisson geometry and mathematical physics.
  In this talk, I will first review Lie groupoids, Morita equivalence and differentiable stacks. In the second part of the talk, based on joint work with A. Maglio and A. Tortorella, I will briefly discuss contact structures on differentiable stacks.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  Tensor categories (and their module categories) take center stage in many interactions between algebra and low-dimensional topology inspired by physics. If we are lucky enough, a useful tensor category can be described neatly as the representation category of a suitable Hopf algebra, and any further embellishments of the situation can be described in terms of that Hopf algebra; in fact one can argue that this is exactly what Hopf algebras are for. We visit a recent installment of such a connection. Laugwitz-Walton-Yakimov introduce the "reflective center" of a module category over a braided tensor category (setting aside for this abstract any technical requirements of course). The notion of a braided tensor category is closely related to (representations of) the Artin braid group (of type A), and in the same fashion the braided module category constructed by Laugwitz-Walton-Yakimov is related to the Artin braid group of type B which is to a Weyl group of type B what the Artin braid group of type A is to the symmetric group. If the braided tensor category is the module category of a Hopf algebra H, and the module category is described by an algebra with an H-action, Laugwitz-Walton-Yakimov construct a new algebra with H-action describing the "reflective center". This "reflective algebra" is parallel in a sense to the famous Drinfeld double construction producing, from any Hopf algebra, a new Hopf algebra whose module category is braided. After explaining what all this is about, we humbly redo the reflective algebra construction of Laugwitz-Walton-Yakimov in a more conceptual (and slightly selfish) way, making full use of Majid's "transmuted" Hopf algebra and a peculiar and often overlooked structure the latter enjoys.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

I will present a randomised algorithm to compute the local zeta function of a fixed smooth, projective surface over the rationals, at any large prime p of good reduction. The runtime of the algorithm is polynomial in log p, answering a question of Couveignes and Edixhoven. The main ingredient is to explicitly compute cocycles associated to a Lefschetz pencil on the surface. This is based on joint work with Nitin Saxena. 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