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)

Our search for a quantum theory of gravity is aided by a unique and perplexing feature of the classical theory: General Relativity already knows" about its own quantum states (the entropy of a black hole), and about those of all matter (via the covariant entropy bound). The results we are able to extract from classical gravity are inherently non-perturbative and increasingly sophisticated. Recent breakthroughs include a derivation of the entropy of Hawking radiation, a computation of the exact integer number of states of some black holes, and the construction of gravitational holograms in our universe using techniques from single-shot quantum communication protocols.

We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3^[2]-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. Interestingly enough, these constructions are almost never infinitesimally rigid, and more precisely we show how to get (infinitely many) 20 and 40 dimensional families. This is a joint work with Claudio Onorati. Time permitting, I will also present a work in progress with Alessandro D'Andrea and Claudio Onorati on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of GL(n).

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 analyzing the survival threshold for a species in population dynamics, one is led to consider the principal eigenvalue of some indefinite weighted problems in a bounded domain. The minimization of such eigenvalue, associated with either Dirichlet or Neumann boundary conditions, translates into a shape optimization problem. We perform the analysis of the singular limit of this problem, in case of arbitrarily small favorable region. We show that, in this regime, the favorable region is connected, and it concentrates at points depending on the boundary conditions. Moreover, we investigate the interplay between the location of the favorable region and its shape. Joint works with Lorenzo Ferreri, Dario Mazzoleni and Benedetta Pellacci.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Each angle formed by two rays with integer slopes has two basic integer invariants: its height and its width. An angle is called Vitruvian (after a Roman architect Vitruvius advocating proportions between height and width) if its height divides its length. A Vitruvian polygon is a polygon, such that all of its angles are Vitruvian. Vitruvian polygons form a distinguished class of polygons in Tropical Planimetry. After a breakthrough idea of Galkin and Usnich from 2010, Vitruvian triangles (studied, under a different guise, by Hacking and Prokhorov, buiding up on an earlier work of Manetti to obtain the complete classification of toric degenerations of the plane) started to play a prominent role also in Symplectic Geometry. In the talk, I review some of these applications, as well as a new symplectic application, involving use of Vitruvian quadrilaterals (work in progress, joint with Richard Hind and Felix Schlenk).

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

  Moduli spaces of representations associated to quivers are algebraic varieties encoding the continuous parameters of linear algebraic classification problems. In recent years their topological and geometric properties have been explored to investigate wild quiver classification problems.
  The goal of this talk is the construction of the Nakajima variety of a quiver as one of these moduli spaces. Starting with basic definitions from the representation theory of quivers, fundamental concepts like doubling and framing are introduced. Via geometric invariant theory Nakajima varieties can be defined. We will discuss their properties focusing on the example of the quiver consisting of one vertex and no arrows. Finally, further relations to recent research interests are explained.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)