I will show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of $mathbbm{Q}$-ample line bundles on higher models. As already known by previous works, I will then explain how this implies a solution to the Yau-Tian-Donaldson conjecture, connecting the $K$-stability of a smooth polarized projective variety $(X,L)$ to the existence of constant scalar curvature Kähler metrics in $c_1(L)$.
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

Perturbation theory is the construction of local quasi-integrals through the composition of a series of canonical or near-identity transformations, called perturbation steps. It has been, historically, the fundamental analytical method for the study of nearly integrable Hamiltonian systems, as it allows the study of the local behavior in an otherwise complex dynamical setting, as well as the construction of local analytical solutions, and the study of the effective stability of the trajectories in the associated phase space. Naturally, the construction of such quasi-integrals is a non-convergent process , with the rate of non-convergence heavily depending from the domain that is considered. In celestial mechanics, the domains of fast divergence are called resonances. In their vicinity, stable and unstable manifolds are formed, splitting the phase space into rotation regions, foliated with KAM rotational tori, libration regions, which contain secondary tori, and chaotic domains, where the tori have broken down. In the non resonant domain, quasi-integrals allow the study of the secular dynamics of celestial bodies, their classification into groups of common origin, and the study of their stability . In contrast, resonant regions (Dogkas & Guido 2026) restrict the applicability domain of such methods. Nevertheless, effective stability estimates can be found in their vicinity (Celletti, Dogkas, et al. 2026), when an optimized selection of the associated parameters is considered, while quasi-integrals can still be defined in the libration regions, deep inside the resonant domain (Dogkas & Vartolomei 2026). N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

  In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard d-dimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle π : E ⟶ B , we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles π₁ and π₂ as input, we can output a "bracket bundle" [π₁, π₂]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's class of [π₁, π₂] with those of π₁ and π₂ . The main input of the proof is a generalization of the Fulton-MacPherson configuration spaces.
This is joint work with Robin Koytcheff and Sander Kupers.
    N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

&nbsp; The Khovanov-Lauda-Rouquier (KLR) algebras associated to quivers have been used to categorify positive halves of quantum enveloping algebras of Kac-Moody algebras. Suitable quotients of these algebras, the cyclotomic KLR algebras, categorize highest weight integrable re-presentations, and also lead to interesting knot and link invariants. In this talk, we will present some work which goes towards an analogous theory in which the quiver gets replaced by a smooth projective curve (the case of the projective line is already interesting). <br> This is joint work with Fang Yang. <br> &nbsp; &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 prove a weaker version of the transcendental base-point-freeness conjecture using techniques from the relative minimal model program (MMP). As an application, we establish a lower bound for the diameter in the case of finite-time singularities of the Kähler-Ricci flow.

Randomized methods can be applied to speed up the computation of the singular value decomposition of a large matrix of low rank. They also can be used to accelerate the convergence of Krylov subspace methods for the solution of certain linear systems of equations. This talk discusses several approaches to apply randomized methods to the solution of large-scale linear systems of equations that arise when solving linear inverse problems. These kind of problems appear when one seeks to determine the cause of an observed effect. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV.

In this talk, I will give some recent results on the Hartree system with Hardy-Littlewood-Sobolev critical exponent. Under some technical conditions on potentials, we investigate the existence and multiplicity of standing waves by using variational method combined with Brouwer degree theory and Ljusternik-Schnirelmann theory.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We investigate several possible notions of "canonical'' metrics that naturally arise in Hermitian non-Kähler geometry. In particular, we study an analogue of the Yamabe problem in the non-Kähler setting, concerning the existence of Hermitian metrics with constant scalar curvature with respect to the Chern connection. We also develop a moment map interpretation of the Chern scalar curvature in the locally conformally Kähler setting. Another tool for highlighting ''canonical structures'' is the Chern–Ricci flow. The long-time behavior of its solutions is expected to reflect the underlying complex structure, and we present some evidence of this in the case of compact complex surfaces. This talk is based on joint work with Simone Calamai, Mauricio Corrêa, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti, and Oluwagbenga Joshua Windare.

Point and line arrangements in the plane arise in various contexts, including topology (Zariski pairs), algebra (freeness), and combinatorics. Notably, Hirzebruch utilized these configurations to construct specific ball-quotient surfaces. In this talk, I will introduce operators acting on these arrangements and their corresponding parameter spaces. We shall see how certain elliptic modular surfaces and modular curves can be recovered as parameter spaces of point arrangements under the action of these operators.
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 question of whether a Hamiltonian system is typically integrable or chaotic is a central topic in dynamical systems, which traces back to the pioneering works of Poincaré in Celestial Mechanics. A satisfactory picture of the typical dynamics of such systems did not emerge until the 1970s, when Markus and Meyer established that a generic (in the Baire category sense) Hamiltonian system on a compact symplectic manifold is neither integrable nor ergodic. On the contrary, the case of natural Hamiltonian systems is much less studied, in spite of its central relevance in mathematical physics. Specifically, a natural Hamiltonian corresponds to the situation in which the symplectic manifold is the cotangent bundle of a manifold M , and the Hamiltonian is given by the sum of a fixed kinetic energy term and a potential field V in C^{infty}(M ; R). It is known that a generic potential field on a compact manifold is non-ergodic. Moreover, near the potential maximum, the system may exhibit positive topological entropy under (non-generic) suitable conditions. Nevertheless, the fundamental question of whether motion at low energy levels is typically integrable or chaotic remains open to date. This difficulty arises because standard transversality methods are no longer applicable, raising the conjecture of whether classical results on generic non-integrability extend to the setting of potential fields. In this talk we shall show that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential V on T^2 exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. To put this result in perspective, the existence of hyperbolic periodic orbits is the natural starting point to establish the presence of chaos in dynamical systems.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).