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

A celebrated theorem by Martens states that the dimension of the locus of line bundles of fixed degree d with at least k sections on a smooth, projective, irreducible curve of genus g doesn't exceed d-2k+2. We establish similar results for various Brill-Noether loci and give some applications for higher rank Brill-Noether loci.

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

When minimizing a regularized functional - i.e., one of the form $H(u) = F(u) + alpha G(u)$, where $G$ is a regularization term and $alpha$ is the regularization parameter - one generally expects multiple minimizers to exist; one might furthermore expect the term $G$ to assume different values in correspondence of different minimizers. We show, however, that for most choices of the parameter $alpha$, all minimizers of the regularized functional share the same value of $G$. This holds without requiring any assumptions on the domain nor on the smoothness/convexity properties of the involved functionals. We also prove a stronger result concerning the invariance of the limit of $G$ along minimizing sequences. Moreover, we demonstrate how these findings extend to multi-regularized functionals and - when an underlying differentiable structure is present- to critical points.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a quantum operator, playing the role of an renormalizable effective density matrix and to describe the matter fields topologically, according to a Dirac-Kähler formalism, as the direct sum of a 0-form, a 1-form and a 2-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the G-field, which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remain second order in the metric and in the G-field. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the G-field could have for dark matter.

  Let G be a connected reductive algebraic group over an algebraically closed field k. Lusztig (1984) partitioned G into subvarieties which play a fundamental role in the study of representation theory, the Jordan classes. An analogue partition of the Lie algebra Lie(G) into subvarieties, called decomposition classes, dates back to Borho-Kraft (1979). When k = C the study of geometric properties (e.g., smoothness) of a point g in the closure of a Jordan class J in G can be reduced to the study of the geometry of an element x in the closure of the union of finitely many decomposition classes in Lie( M), where M is a connected reductive subgroup of G depending on g.
  The talk aims at introducing such objects and at generalizing this reduction procedure to the case char(k) > 0.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)