Optimal transport consists in sending a given source probability measure ρ to a given target probability measure μ in an optimal way with respect to a certain cost. Optimal transport has been widely used in many fields, including analysis, probability, statistics, geometry, and optimization. Under classical assumptions, there exists a unique optimal transport map from ρ to μ (Brenier's, McCann's theorems, etc.). In this talk based on a collaboration with Quentin Mérigot, we provide a quantitative answer to the following stability question, notably motivated by numerical analysis and statistics: if μ is perturbed, can the optimal transport map from ρ to μ change significantly? The answer depends on the properties of the source measure ρ. We will also explain some mechanisms leading to instability and present a few conjectures.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

  We present the construction of a new class of quantum vertex algebras associated with a normalized Yang R-matrix. They are obtained as Yangian deformations of certain S-commutative quantum vertex algebras and their S-locality takes the form of a single RTT-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. These results were obtained jointly with Slaven Kozic.
  If time allows, we will discuss a recent generalization of these results to the case of the type A trigonometric R-matrix. These results were obtained jointly with Marijana Butorac and Slaven Kozic.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behaviour of “typical” solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are in fact typical in the integrable case. In this talk I shall consider a family of NLS equations parametrized by a smooth convolution potential and prove that for “most” choices of the parameter there is a full measure set of Gevrey initial data that give rise to almost-periodic solutions whose hulls are invariant tori. As a consequence the elliptic fixed point at the origin turns out to be statistically stable in the sense of Lyapunov. This is a joint work with L.Biasco, G.Gentile and M.Procesi.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

Gibbs measures of nonlinear Schrödinger equations (NLS) are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.

In earlier joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein, we studied this problem for variants of the cubic NLS with defocusing (positive) interactions. The latter models physically correspond to pair interactions of bosons. In these works, the problem was studied in dimensions d=1,2,3.

In this talk, I will explain how one can obtain an analogous result for the 1D quintic NLS, which corresponds to three-body interactions of bosons. In this setting, we consider focusing interactions, due to which we need to add a truncation in the mass and rescaled particle number. Our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the 1D quintic NLS. This is joint work with Andrew Rout.

We are interested in (viscosity) solutions of Hamilton-Jacobi equations of the form $G( lambda u_lambda(x),x,D_x u_lambda) = cst $ where $u_lambda : M o mathbb{R}$ is a continuous function defined on a closed manifold and $G$ verifies convexity and growth conditions in the last variables. Such solutions carry invariant sets for the contact flow associated to $G$. The parameter $lambda>0$ is aimed to be sent to $0$. It has been known that when $G$ is increasing in the first variable, $u_lambda$ exists, is unique and the family converges as $lambda o 0$. We will explain that when this hypothesis is dropped, there can be non uniqueness of solutions $u_lambda$ at $lambda>0$ fixed. Moreover, there can be coexistence of converging families of solutions $(u_lambda)_lambda$ and diverging ones. (Collaboration with Davini, Ni and Yan)

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).

Tropical geometry studies a piecewise linear, combinatorial shadow of degenerations of algebraic varieties. In many cases, usual algebro-geometric objects such as divisors or line bundles on curves have tropical analogues that are closely tied to their classical counterparts. For instance, the theory of divisors and line bundles on metric graphs has been crucial in advances in Brill–Noether theory and the birational geometry of moduli spaces. In this talk, I will present an elementary theory of tropical principal bundles on metric graphs, generalizing the case of tropical line bundles to bundles with arbitrary reductive structure group. Our approach is based on tropical matrix groups arising from the root datum of the corresponding reductive group, and leads to an appealing geometric picture: tropical principal bundles can be presented as pushforwards of line bundles along covers equipped with symmetry data from the Weyl group. Building on Fratila's description of the moduli space of semistable principal bundles on an elliptic curve, we describe a tropicalization procedure for semistable principal bundles on a Tate curve. More precisely, the moduli space of semistable principal bundles on a Tate curve is isomorphic to a natural component of the tropical moduli space of principal bundles on its dual metric graph. This is based on ongoing work with Andreas Gross, Martin Ulirsch, and Dmitry Zakharov. <strong> Note: </strong> 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

Multisymplectic manifolds generalize symplectic manifolds by featuring a closed nondegenerate differential form of degree higher than 2. Such structures are natural candidates for a geometric formalization of classical field theories. In this context, Rogers (2010) showed that just as a symplectic manifold yields a Poisson algebra of functions, an n-plectic manifold yields an n-terms Lie infinity algebra of observables. The remarkable aspect of Rogers' construction is that it is essentially algebraic and relies only on the axioms of Cartan calculus, suggesting that this higher version of the "observable Poisson algebra" can be generalized beyond the realm of manifolds. In this talk, we propose such a generalization in the setting of Gerstenhaber algebras and Batalin–Vilkovisky (BV) modules, which provide an algebraic formulation of Cartan calculus of interests in the context of non-commutative geometry. This framework allows us to construct Lie infinity algebras of observables in a purely algebraic way, without reference to an underlying manifold. As an application, we turn to the problem of reducing multisymplectic observables in the presence of constraints or symmetries. Building on the work of Dippel, Esposito, and Waldmann, who introduced the notion of a "constraint triple" as a categorical package for coisotropic reduction, we adapt this formalism to our BV-module context and the associated Lie infinity algebras. This construction provides a conceptual framework for the algebraic reduction procedure of multisymplectic observables, as developed in our recent joint work with Casey Blacker (SIGMA 2024). The results presented here are part of a collaboration with Leonid Ryvkin, published in Differential Geometry and its Applications (2025).
  This talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  Which quivers have a path algebra that is PI? What can we say about their T-ideal? And what happens if we add relations? In this talk, I will address these questions.
  This is joint work with Elena Pascucci and Javier De Loera Chavez.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)