&nbsp; T.B.A. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

&nbsp; T.B.A. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

&nbsp; T.B.A. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

&nbsp; T.B.A. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

Weyl groups for the Cuntz algebras were introduced implicitly by Cuntz at the end of the seventies. However, they remained largely ignored probably because of computational difficulties until about 30 years later, when the breakthrough work by Conti and Szymanski allowed to determine explicitly a huge number of their elements by a sophisticated condition with a clear combinatorial flavour. In recent times, Brenti, Conti and Nenashev pushed the boundaries of the involved combinatorial structures, obtaining for instance the first enumerative results (at the moment, only for cycles). In this talk I will report on recent work joint with F. Brenti and G. Nenashev (in preparation) where, building on the combinatorial machinery developed over the last few years, we construct certain subgroups of outer automorphisms of O_n. In particular, we are able to describe in detail the 46 distinct finite groups of outer automorphisms of O_4 lying in the outer reduced Weyl group and maximal at level 2, which were first determined by Szymanski and collaborators by clever computer-assisted methods. The notion of bicompatible subgroup of the permutations of a square grid will play a role in the discussion.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n ge 3$ and $P_g$ be a conformally-covariant operator on $(M,g)$. We consider in this talk two problem at the crossroads of conformal geometry and spectral theory: 1) determining the extremal value that the renormalized eigenvalues of $P_g$ take as $g$ runs through a fixed conformal class and 2) determining whether these extremal values are attained at an extremal metric. Examples of such operators $P_g$ include the famous conformal Laplacian of the Yamabe problem, $P_g = Delta_g + c_n S_g$, but also its higher-order generalisations such as the GJMS operators of order $2k$ for any positive integer $k$.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

&nbsp; Given a simple finite Lie algebra over the complex numbers, we can consider two other Lie algebras attached to it: its Langlands dual Lie algebra and the affine algebra at the critical level. It is a theorem of the nineties, by Feigin and Frenkel, that the center of the completed enveloping algebra of the affine algebra at the critical level is canonically isomorphic to the algebra of functions on the space of Opers on the pointed disk for the Langlands dual Lie algebra. These objects are actually pointwise instances of a more general picture: the space of opers for example enhances to a space which lives over an arbitrary smooth curve that is equipped with a natural factorization structure. This structure is fundamental for the Geometric Langlands community: factorization patterns allow for local to global arguments. In this talk I will explain the construction of the objects mentioned above and elaborate on a joint work with Andrea Maffei in which we prove the factorizable version of the Feigin-Frenkel theorem. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

  In this talk, I will introduce restricted Poisson algebras in characteristic 2 and explore their connection with restricted Lie–Rinehart algebras. For the latter, a cohomology theory is deve-loped and abelian extensions are investigated. I will also construct a cohomology complex for restricted Poisson algebras in characteristic 2 that controls formal deformations. This complex is shown to be isomorphic to the cohomology complex of a suitable restricted Lie–Rinehart algebra. Several examples are provided to illustrate the constructions.
  This is a joint work with Sofiane Bouarroudj and Quentin Ehret.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Cubic hypersurfaces over the rational numbers often contain infinitely many rational points. In this situation, the asymptotic behavior of the number of rational points of bounded height is predicted by conjectures of Manin and Peyre. After reviewing previous results, we discuss the chordal cubic fourfold, which is the secant variety of the Veronese surface. Since it is isomorphic to the symmetric square of the projective plane, a result of W. M. Schmidt for quadratic points on the projective plane can be applied. We prove that this is compatible with the conjectures of Manin and Peyre once a thin subset with exceptionally many rational points is excluded from the count. 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 introduce a family of functionals approximating the conformally invariant Dirichlet n-energy of maps between two Riemannian manifolds (M^n,g) and (N,h), which admit critical points. Along the approximation process, these critical points may incur a bubbling phenomenon, due to the conformal invariance of the limit Dirichlet n-energy. We prove an energy identity result for this approximation, ensuring that no energy gets lost along the formation of bubbles, under a Struwe type entropy bound assumption. We then show that min-max problems for the n-energy are always solved by a "bubble tree" of n-harmonic maps. This is a joint work with T. Lamm.<br> <b>NB</b>:<i>This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006</i>