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

  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)

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)

Topological defects in quantum field theory have received considerable attention in the last few years as generalizations of the concept of symmetry. In the context of two dimensional conformal field theory, the properties of topological defects have been studied since the 90s, in particular in a series of works by Froehlich, Fuchs, Runkel and Schweigert. In this talk, I will discuss some applications of these ideas from physics to the theory of vertex operator (super-)algebras. In particular, I will describe some recent results about topological defects in the Frenkel-Lepowsky-Meurman Monstrous module, as well as in the Conway module, i.e. the holomorphic vertex operator superalgebra at central charge 12 with no weight 1/2 states. Finally, I will speculate about possible generalizations of the Moonshine conjectures.

This is partially based on ongoing joint work with Roberta Angius, Stefano Giaccari, and Sarah Harrison.

A famous theorem by Mukai (1988) provides a classification of all possible finite groups admitting a faithful action by symplectic automorphisms on some K3 surface. In 2011, in collaboration with Gaberdiel and Hohenegger, we proposed that a 'physics version' of Mukai theorem should hold for certain two dimensional conformal quantum field theories, called non-linear sigma models (NLSM) on K3, that describe the dynamics of a superstring moving in a K3 surface. In particular, we provided a classification of all possible groups of symmetries of NLSM on K3, that commute with the N=(4,4) algebra of superconformal transformations. This result was later re-interpreted by Huybrechts as a classification of the finite groups of autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. In the last few years, the concept of symmetry group in quantum field theory has been vastly generalized. In the context of two dimensional conformal field theories, these developments suggest that the idea of 'group of symmetries' should be replaced by 'fusion category of topological defects'. We discuss how our previous classification result could be extended to include fusion categories of topological defects in non-linear sigma models on K3. The geometric interpretation of these categories is still mysterious. This is based on joint work in collaboration with Roberta Angius and Stefano Giaccari.

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 talk discusses a framework to analyze certain model-based reinforcement learning algorithm. Roughly speaking, this approach consists in designing a model to deal with situations in which the system dynamics is not known and encodes the available information about the state dynamics that an agent has as a measure on the space of functions. In this framework, a natural question is if whether the optimal policies and the value functions converge, respectively, to an optimal policy and to the value function of the real, underlying optimal control problem as soon as more information on the environment is gathered by the agent. We provide a positive answer in the linear-quadratic case and discuss some results also in the control-affine nonlinear case.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006