In this talk I will describe the normal stable surfaces with K²=2p_g-3 whose only non canonical singularity is a cyclic quotient singularity of type 1 4k  (1,2k-1) and the corresponding locus 𝔇 inside the KSBA moduli space of stable surfaces. The main result is the following: for p_g≥ 15,   (1) a general point of any irreducible component of 𝔇 corresponds to a surface with a singularity of type 1 4k  (1,1),   (2) the closure of 𝔇 is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K²=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of 𝔇. Moreover 𝔇 has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4. This is joint work with Rita Pardini.

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 the PS-12 split introduced by Powell and Sabin in 1977 we present an optimal symmetric 4 point quadrature rule and a collection of weighted rules. These are useful for an efficient formation of the linear system arising in Galerkin discretization on this split. We use the S-spline version of simplex splines introduced by Cohen, L., Riesenfeld in 2013, and a global basis based on the theory of minimal determining sets adapted to S-splines on the PS-12 split. This is joint work with Salah Eddargani, Carla Manni, and Hendrik Speleers. The talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

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

Fourier analysis is a powerful tool in analysis. In the setting of abelian schemes, Fourier-Mukai transformation and the weight decomposition play a similar role. For degenerate abelian fibrations, the relative group structure disappears and understanding the intersection theory leads to many interesting questions, such as the P=W conjecture, χ-independence phenomenon, and multiplicative splitting of the perverse filtration for the Beauville-Mukai system. In this talk, I will connect Fourier transform between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, I will compute the pushforward of monomials of divisor classes on compactified Jacobians via the twisted double ramification formula. Along the way, we will encounter instances of χ-independence and the multiplicativity of perverse filtration for compactified Jacobians. This is a joint work with Samouil Molcho and Aaron Pixton.

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

This talk presents recent results on the generic properties of conjugate points in viscosity solutions to first-order Hamilton–Jacobi equations. In this context, we introduce a quantitative version of the transversality theorem and apply it to estimate the total number of shock curves in weak entropy solutions to scalar conservation laws. Additionally, we establish sharp quantitative bounds on the critical sets of smooth functions and provide an explicit upper bound for the (d−1)-dimensional Hausdorff measure of the zero set of nontrivial multivariable polynomials.
N.B.This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

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.