Let k be a field, G be a reductive group over k, B be a Borel subgroup of G and C be a smooth curve over k. The B-orbit stratification of the flag variety G/B induces a natural stratify-cation on the moduli space of maps from C to G/B . The open stratum is strictly related to the so-called Zastava spaces.
  In this talk, we give an overview of the main properties of these spaces. If time permits, we also present some consequences on the moduli space of G-bundles.

  We define a specialization map for quiver Grassmannians of Dynkin type and prove that it is surjective in type A. This generalizes a beautiful theorem of Lanini and Strickland concerning the cohomology of degenerate flag varieties.
  This is a joint ongoing work with Francesco Esposito, Ghislain Fourier and Fang Xin.

We analyze the problem of controlling a multi-agent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with a SDE constraint, we introduce a rigorous limit process towards an infinite dimensional optimal control problem constrained by the coupling of a system of ODE for the leaders with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the Gamma-limit of the cost functionals for the finite dimensional problems. Joint work with Giacomo Ascione (SSM Napoli) and Francesco Solombrino (Napoli Federico II).
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Strata of differentials have been studied from several perspectives, from dynamics to geometry and topology. Algebraic geometers need compactifications. One, which is a posteriori logarithmic, was introduced by Bainbridge--Chen--Gendron--Grushevsky--Möller in terms of multiscale (collections of) differentials (on reducible nodal curves). This natural compactification is unfortunately not irreducible, but they were able to single out the main component in terms of a condition on global residues; their proof rests on transcendental methods. I will mention a conjecturally equivalent condition which is purely algebraic. In joint work with Sebastian Bozlee, we give a proof of concept of this conjecture in the case of hyperelliptic differentials. The technical core of our work is a flexible tool for constructing hyperelliptic Gorenstein contractions of reducible nodal curves, given a hyperelliptic tropical differential, i.e. a suitable piecewise-linear function, on the dual graph.

A few years ago, the idea of formalising modern research level mathematics seemed completely out of reach. Since then, more and more examples have appeared. I'll go through several examples (some related to the mathematics of Scholze, Tao and Gowers), and talk about how the process is evolving, enabling multiple people to collaborate in the formalisation of modern research in real time. Computer-verified proofs: 48 hours in Rome

In this talk I will describe what goes on behind the scenes of Lean. I will explain the logic of Lean, called dependent type theory, what Lean tactics are and explain why we can trust proofs that are checked by Lean. Computer-verified proofs: 48 hours in Rome

This presentation will explore the pivotal role of the Lean in enhancing first-year undergraduates' understanding of mathematical proofs. I will share insights from my experiences and initial educational research on teaching a large first-year undergraduate cohort with Lean, focusing on how this tool can significantly impact student perception of proofs. Additionally, I will address the challenges encountered in teaching with Lean and the implications for learning and comprehension. Computer-verified proofs: 48 hours in Rome

In this talk I will illustrate how certain programs, of which Lean is an example, permit to interact with a computer about the logical soundness of mathematical arguments. I will go through the details of well-known proofs trying to understand the feedback provided by the computer and will try to share the fun involved in the process. Computer-verified proofs: 48 hours in Rome

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. The phenomenon of internal waves in a fluid occurs in many situations and has been intensively studied by physicists. One of the first experiments, which became paradigmatic, was done in a container shaped like a rectangular trapezoid (with some thickness). For a class of tables including rectangular trapezoids, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). If time permits, we will also discuss the prominent case where the table is an actual trapezoid, studying the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnold tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero. Joint work with C. Bonanno and G. Cristadoro.

In this talk, we are going to discuss a generalization to weighted blowups of the classical Castelnuovo' contraction theorem. Moreover, we will show as a corollary that the moduli stack of n-pointed stable curves of genus 1 is a weighted blowup. This is a joint work with Arena, Di Lorenzo, Inchiostro, Mathur, Obinna.