Given a dense open immersion U -> S, some smooth and proper families X_U -> U do not extend to smooth proper families over S. More often (but still not always), there is a "best smooth extension", the Neron model. I will talk about how to construct Neron models for families of smooth curves and their Jacobians. Neron models are not compatible with base change, so there are no "moduli spaces of Neron models", but we will see that they relate to some logarithmic moduli functors, and that their base change behaviour can be understood tropically (in terms of combinatorics of dual graphs).Given a dense open immersion U -> S, some smooth and proper families X_U -> U do not extend to smooth proper families over S. More often (but still not always), there is a "best smooth extension", the Neron model. I will talk about how to construct Neron models for families of smooth curves and their Jacobians. Neron models are not compatible with base change, so there are no "moduli spaces of Neron models", but we will see that they relate to some logarithmic moduli functors, and that their base change behaviour can be understood tropically (in terms of combinatorics of dual graphs).

In this talk we will introduce Ulrich sheaves on projective algebraic varieties. The corresponding notion for modules over rings originated from the work of Ulrich, but only after a paper by Eisenbud and Schreyer its geometric side received many attention due to the connection with determinantal and Pfaffian representations of (Chow forms of) varieties. The main questions are existence of such sheaves and, if so, their minimal rank. Positive answer to the first question is known for curves, surfaces (up to change of polarization), Veronese varieties, Segre varieties and complete intersections. Outside those examples, very few is known in dimension at least 3; therefore we will focus on Fano 3-folds.

We consider the Microcanonical Variational Principle for the vortex system in a bounded domain. In particular, we are interested in the thermodynamic properties of the system in domains of second kind, i.e. for which the equivalence of ensembles does not hold. For connected domains close to the union of disconnected disks (dumbbell domains), we show that the system may exhibit first-order phase transitions, while the entropy is convex for large energy (joint work with Dario Benedetto and Emanuele Caglioti).

Note: This talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV.

  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