T.B.A.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  T.B.A.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Curvature-driven geometric flows provide a natural framework for studying the evolution of curves and hypersurfaces. Starting from classical examples such as the curve shortening flow and the mean curvature flow, we will discuss how additional constraints lead to modified evolution equations designed to preserve geometric quantities such as length or enclosed area. We will then explore the role played by the geometry of the ambient manifold, considering examples in Hadamard manifolds and on the sphere. Through selected recent results, we will illustrate how ambient curvature and geometric constraints influence the evolution and qualitative behaviour of these flows, highlighting some of the analytical challenges that arise in their study. Some of the results presented are based on two collaborations: one with Esther Cabezas-Rivas and another with Esther Cabezas-Rivas and Shujing Pan.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Determining the algebraic rank of an elliptic curve is, in general, a hard problem. For families of elliptic curves, we have a fairly detailed conjectural understanding of the rank's behavior on average coming from Silverman's conjecture, Goldfield's conjecture, and the minimalist philosophy. Recent progress in this direction has relied on a much more tractable invariant: the 2-Selmer rank. For a polynomial $P\in \mathbb{Z}[t]$ of arbitrary degree, we consider elliptic surfaces of the form $y^2=x^3-P(t)^2x$. We show that, for $100 \%$ of polynomials $P$, the average 2-Selmer rank in such elliptic surface is $3$. As a direct consequence, we obtain information about the distribution of algebraic ranks elliptic surfaces of this form.
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

  Isotropic random fields on the sphere, whose law is invariant under rotations, arise na-turally in cosmology, geophysics and, more recently, in the infinite-width limit of deep neural net-works. Their structure is entirely governed by the action of SO(d+1) on L2(Sd). This talk presents a self-contained account of this connection, from the structure theory of compact Lie groups to the spectral representation of isotropic fields.
  These results show how the probabilistic structure of spherical random fields is dictated by repre-sentation theory.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  The aim of this talk is to provide a computation of the K-theory of the category of matroids with respect to the covering family of Tutte coverings. In particular, we show that this is equivalent to the K-theory spectrum of the category of graphic matroids on looped forests, with the covering family generated by isomorphisms.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

By the Liouville-Arnold theorem, for a completely integrable Hamiltonian system with finitely many degrees of freedom, the compact connected components of regular joint level sets of the integrals of motion are invariant Lagrangian tori and induce local foliations of the phase space. The vanishing of the symplectic form on each leaf allows for the construction of action-angle coordinates adapted to the foliation, which linearize the dynamics. Although there are several results concerning the integrability of infinite-dimensional Hamiltonian systems, none of them provides a decomposition of the phase space as rigid as the one given by the Liouville-Arnold theorem. In this talk, we shall see that this is largely due to the choice of topology on the phase space. I will present a robustness result for foliations by non-compact Lagrangian tori in $\ell^\infty$-based Banach spaces, together with the construction of analytic action-angle coordinates for a class of Hamiltonian PDEs on the circle, with the KdV hierarchy as a notable case, working in the space of analytic Fourier series with the topology induced by the weighted sup-norm on the coefficients. This is part of a joint work with J. E. Massetti e M. Procesi.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

In this talk we present the state of an ongoing project concerning the construction of local nets of real subspaces on causal homogeneous spaces M = G/H by starting with a unitary representation of G. Here G/H can be a causal symmetric spaces (non-compactly or compactly causal) or a causal flag manifold. In Physics, the case of timeoriented Lorentzian spaces, these three types correspond to de Sitter space, anti-de Sitter space and the causal compactification of Minkowski space, or some covering of such a space.
For a given unitary representation, we characterize in purely representation theoretic terms: a) the existence of a corresponding net on M with the Bisognano-Wichmann and the Reeh-Schlieder property. b) locality properties of the nets, including in particular spacelike and timelike duality on Minkowski space. This is joint work with V. Morinelli and G. Olafsson

We construct compactifications of M_{g,n} whose boundaries parametrize certain curves with A-type singularities. These singularities mildly generalize nodes and naturally appear in the birational study of the nodal stable compactification of M_{g,n}. We realize these compactifications as good moduli spaces of non-Deligne-Mumford algebraic stacks by verifying the Theta- and S-completeness criteria of Alper, Halpern-Leistner, and Heinloth. Along the way, we establish general results (such as generic S-completeness) and analyze isotrivial degenerations of curves with A-type singularities. Many such compactifications are proper but not projective, and for n=0 and r=5, they fail to be schemes. These are examples of modular compactifications of M_{g,n} in arbitrary genus that cannot be recovered by GIT techniques. This is joint work with Ludvig Modin and Michele Pernice.
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

Scalar conservation laws $\partial_t u + \operatorname{div} f(u) = 0$ are nonlinear first-order PDEs that model the transport of a conserved quantity $u$, with a flux $f$ that depends on $u$. A prototype, in one space dimension, is Burgers' equation, where $f(u) = u^2/2$. Even for smooth initial data, their solutions develop discontinuities (shocks), so one works with weak solutions, which may jump across a codimension-one set, the jump set. However, weak solutions are not unique. Uniqueness is restored by selecting entropy solutions, for which every convex function of the solution (an entropy) can only be dissipated. While the nonlinearity of the flux creates shocks, it also has a regularizing effect. In this talk, we discuss the fine regularity it provides outside the jump set. In several space dimensions, we show that entropy solutions are continuous there, assuming only that the flux is nonlinear in a qualitative sense. We then consider the broader class of weak solutions with finite entropy production, which can also create entropy, as long as the entropy production is a finite measure. For these, continuity fails already in one space dimension. For every nonlinear flux, we construct such solutions with zero initial datum whose essential infimum and supremum on every open set are two fixed states, so they are essentially continuous at no point. For Burgers' equation, we also construct a solution with finite entropy production that has a large set of non-Lebesgue points outside the jump set.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006