We start by reviewing the classical spectral theory of the Dirichlet-Laplacian, on a general open set. It is well-known that the spectrum may fail to be purely discrete, in this generality. We then turn our attention to a nonlinear variant of this problem, by considering the case of the $p-$Laplacian with Dirichlet homogeneous conditions. More precisely, we analyze the minmax levels of the constrained $p-$Dirichlet integral: we show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ''at infinity'', it actually defines an eigenvalue. We also prove a quantitative exponential fall-off at infinity for the relevant eigenfunctions: this can be seen as a generalization of v{S}nol-Simon--type estimates to the nonlinear case. Some of the results presented have been obtained in collaboration with Luca Briani (TUM Monaco) and Francesca Prinari (Pisa). <br> <b>NB</b>: <i> This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006 </i>

È ben noto grazie a un lavoro di Ein del 1988 che le intersezioni complete molto generali di multigrado $(d_1,...,d_c)$ nello spazio proiettivo di dimensione $n$ non contengono curve razionali non appena $d_1+...+d_c geq 2n-c-1 $. Questo risultato è stato reso ottimale ed esteso nel caso delle ipersuperfici grazie a un metodo introdotto da Voisin che ha ispirato lavori ulteriori di Clemens, Ran e miei. Nonostante lavori più recenti di Coskun, Riedl e Yang sempre nel caso delle ipersuperfici, il risultato di Ein è rimasto l’unico disponibile per intersezioni complete di codimensione arbitraria. Nel seminario parlerò di un lavoro in collaborazione con Francesco Bastianelli in cui estendiamo il lavoro di Ein, mescolando l’approccio di Voisin con quello di Coskun, Riedl e Yang.
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

  The argument shift method is a simple but efficient way to produce a commutative subalgebra in a Poisson algebra, in particular in the symmetric algebra of a Lie group. Its main ingredient is an operator on the Poisson algebra that generates the elements of the commutative subalgebra by acting on its center. In my talk I will describe an analogous operator on the universal enveloping algebra of gl(d), and show why it satisfies a similar property.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

&nbsp; The full symmetric Toda system is a Hamilton system on the space of real symmetric matrices, given by the Lax equation <em>L'</em> = [<em>M</em>(<em>L</em>),<em>L</em>] , where <em>M</em>(<em>L</em>) is the na&iuml;ve antisymmetrisation of the symmetric matrix <em>L</em> (deleting the diagonal and inverting the sign of the lower-triangular half). This system turns out to be Liouville-integrable and even super-integrable. In my talk I will prove that it satisfies one more integrability criterion, the Lie-Bianchi criterion, based on the study of its symmetries. <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>

We introduce the concept of horospherical billiard in the universal covering of a compact surface without focal points and prove some rigidity results assuming the existence of some geometric first integrals.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

Exponential sums over finite fields are essential ingredients in the solution of many arithmetic problems. Their study often relies on algebraic geometry, and especially on Deligne's Riemann Hypothesis over finite fields, which reveals deep structural features of these sums. In turn, one can exploit these to construct some remarkable combinatorial objects. The talk will provide a survey of these various aspects. (Joint works with A. Forey, J. Fresán and Y. Wigderson) <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

Recently the learning problems took the center stage in area of theoretical computer science. An amazing and beautiful thing is that they are harmonic analysis problems at heart. The lecture concerns with some natural and elementary question of learning theory and the approach to learning via harmonic analysis. Suppose you wish to find a N by N matrix by asking this matrix question that it honestly answers. For example you can ask question ''What is your (1,1) element?'' Obviously you will need $N^2$ many questions like that. But if one knows some information on Fourier side one can ask only log log N questions if they are carefully randomly chosen. Of course one pays the price: first of all one would find the matrix only with high confidence (high probability bigger than $1-delta$), secondly with the error $epsilon$. Such learning is known as PAC learning, PAC stands for 'probably approximately correct'. The origins of the problem are in theoretical computer science, but the methods are pure harmonic analysis and probability. The main ingredient is dimension free Bernstein—Remez inequality.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

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.)