In General Relativity, Euclidean 3-space is replaced by a 3-dimensional Riemannian manifold arising as a space-like hypersurface in a Lorentzian space-time. Energy conditions on the space-time matter lead to curvature restrictions on the 3-manifold such as non-negativity of the scalar curvature. In this context it is important to find geometric structures resembling classical physical concepts such as mass and momentum both locally and globally to describe physically interesting phenomena like gravitational collapse in a way that is independent of coordinates. The lecture discusses new geometric concepts for the mass and diameter of a finite region of a 3-manifold that aim in this direction.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

The talk aims to discuss the existence and regularity problem for spacelike hypersurfaces in Lorentz-Minkowski space whose mean curvature is a prescribed measure. The same equation also appears in Born-Infeld’s theory of electrostatics, according to which the unknown describes the electric potential generated by a given charge. Even though the problem is formally the Euler-Lagrange equation of a nice convex functional, the lack of smoothness when the graph becomes lightlike in Lorentz-Minkowski space may prevent the unique variational minimizer to solve the equation. A chief difficulty comes from the possible presence of “light segments” in the graph of the solution, a fact that we will describe in detail. Various open problems and research directions will be discussed. The talk is based on joint works with J. Byeon, N. Ikoma, A. Malchiodi and L. Maniscalco.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

In this presentation, we will analyze the various domains in which the ergodic mean field game (MFG) system arises. Specifically, we will explore how weak KAM theory can be used to study this system and derive results regarding long-time behavior or the approximation of Nash equilibria. Finally, we will introduce a quasi-stationary system—a model in which, at each moment, agents optimize their expected future cost under the assumption that their environment remains static.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Irreversibility theorems for the renormalization group for quantum field theories in space-time dimensions d=2,3,4 have been shown to follow from strong subadditivity of entropy and Lorentz invariance. A companion set of irreversibility theorems were conjectured to hold for defects in conformal field theories. In this case the scales of the theory come from the defect Hamiltonian. The simplest case of a one-dimensional defect, called g-theorem, can be proved using monotonicity of relative entropy. We will show how to extend this proof to defect dimensions 2,3,4 using the QNEC property (quantum null energy condition) of the relative entropy. We will briefly comment on the connection of this proof with strong subadditivity for holographic theories.

The duality between algebraic structures and geometric spaces is of paramount importance in mathematics and physics, because provides a dictionary to describe manifolds and variaties in a purely algebraic fashion. In his seminal paper, Gelfand showed that a topological space can be functorially reconstructed from its Banach algebra of continuous functions. Conversely, the Gelfand spectrum of the algebra of continuous functions is homeomorphic to the underlying topological space. The goal of this talk is to construct a sufficiently robust notion of spectrum for general rings that allows one to implement a noncommutative analog of Gelfand duality. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others.

This is a joint project with Federico Bambozzi and Matteo Capoferri.

We consider the Dirichlet eigenvalues of the fractional Laplacian related to a smooth bounded domain. We will prove that there exists an arbitrarily small perturbation of the original domain for which all Dirichlet eigenvalues of the fractional Laplacian are simple. Also, the same result of simplicity of eigenvalues holds for a generic perturbation of the coefficients of the eigenvalue equation. Finally we study the set of perturbations which preserve the multiplicity of eigenvalues.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

In this talk we discuss semi-homogeneous vector bundles on abelian varieties and show that, from a cohomological point of view, they play the role that fractional polarizations should play. In the first part of the seminar we discuss some Mukai's results regarding the existence and structure of semi-homogeneous bundles. In the second part we revisit the theory of cohomological rank functions giving a vector bundle interpretation of them. As an application, we show how this perspective allows us to bound certain numerical invariants that measure the positivity of polarizations.

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

  For a Riemann surface X and a complex reductive group G, G-character varieties are moduli spaces parametrizing G-local systems on X. When G=GL_n, the cohomology of these character varieties have been deeply studied and under the so-called genericity assumptions, their cohomology admits an almost full description, due to Hausel, Letellier, Rodriguez-Villegas, and Mellit. An interesting aspect is that the geometry of these varieties is related to the representation theory of the finite group GL_n(F_q). We expect in general that G-character varieties should be related to Ĝ(F_q)-representation theory, where Ĝ(F_q) is the Langlands dual. In the beginning of the talk, I will recall the results concerning GL_n. Then, I will explain how to generalize some of these results when G=PGL₂ . In particular, we will see how to relate PGL₂-character varieties and the representation theory of SL₂(Fq).
  This is joint work with Emmanuel Letellier.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Haag duality is a simple property of algebras attached to regions in QFT that expresses a form of completeness of the theory. Violations of Haag duality are due to "non-local operators". These may be charged with respect to global symmetries. When this happens for a continuous symmetry there is an obstruction for the validity of Noether's theorem. This is behind all known examples when the Noether current is absent, including the ones covered by Weinberg-Witten theorem. An abstract classification of the simplest possibilities is divided into two classes. In the first one there are non compact sectors, which leads to free models. The other possibility, allowing interacting models, corresponds to the ABJ anomaly. This interpretation unifies the features of the anomaly --- anomaly matching, anomaly quantization, non-existence of the Noether current, and validity of Goldstone theorem --- from a symmetry based perspective.

We discuss the emergence of logarithmic Sobolev inequalities from energy/entropy inequalities and then derive from them the existence and uniqueness of the ground state of Hamiltonians as well the spectral gap. The method is an infinitesimal extension of the one introduced by Len Gross in case the ground state is a probability or a trace and is based on the monotonicity of the relative entropy.