Over the complex numbers, generic vanishing theory is useful for studying the geometry of irregular varieties. A standard application of this theory is that if X is a smooth, complex projective variety of maximal Albanese dimension (i.e. dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is non-negative. This relies on vanishing theorems of analytic nature.

In this talk, we will show that the same statement holds in positive characteristic, assuming further that the Frobenius morphism acts bijectively on the cohomology (such a hypothesis tends to be true for "most" varieties). If we also assume that our variety is not of general type, then we also show that its Euler characteristic is in fact zero, and that the image of the Albanese morphism is fibered by abelian varieties (which are well-known statements over the complex numbers).

The proof relies on a positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, together with a Witt vector version of the Grauert-Riemenschneider vanishing theorem.

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 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.

We prove two types of inequalities for a foliation of general type on a smooth projective surface, the slope inequality and Noether inequality, both of which provide lower bounds on the volume vol(F). In order to define the slope, we first introduce three birational non-negative invariants c_1^2(F), c_2(F) and chi(F) for any foliation F, called the Chern numbers. If the foliation F is not of general type, the first Chern number c_1^2(F)=0, and c_2(F)=chi(F)=0 except when F is induced by a non-isotrivial fibration of genus g=1. If F is of general type, we obtain a slope inequality when F is algebraically integrable, which gives a lower bound on vol(sF) by chi(F). On the other hand, we also prove three sharp Noether type inequalities for a foliation of general type, which provides a lower bound on vol(F) by the geometric genus p_g(F). As applications, we also give partial solutions to the Poincaré and Painlevé problems using these two inequalities. This is a joint work with Professor S.L. Tan.

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