The classical Pieri rule is a multiplication formula for Schubert class and Chern classes of the tautological bundle. Combinatorially, it is given by adding a chain of boxes on partitions. In this talk, we will discuss its generalization to equivariant Motivic Chern classes and its dual basis Segre motivic classes. Our formula is in terms of ribbon Schubert operators, which is roughly speaking adding ribbons on partitions. As an application, we have found a little surprising relation between motivic Chern classes and Segre motivic classes, extending the relation between ideal sheaves and structure sheaves over Grassmannians.

  Pairs of cointeracting bialgebras appear recently in the literature of combinatorial Hopf algebras, with examples based on formal series, on trees (Calaque, Ebrahimi-Fard, Manchon and Bruned, Hairer, Zambotti), graphs (Manchon), posets... These objects have one product (a way to combine two elements in a single one) and two coproducts (the first one reflecting a way to decompose a single element into two parts, maybe into several ways, the second one reflecting a way to contract parts of an element in order to obtain a new one). All these structures are related by convenient compatibilities.
  We will give several results obtained on pairs of cointeracting bialgebras: actions on the group of characters, antipode, polynomial invariants... and we will give applications to a Hopf algebra of graphs, including the Fortuin and Kasteleyn's random cluster model, a variation of the Tutte polynomial.

This seminar aims to introduce the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the strong connection between crossing-symmetric twists and endomorphisms of standard subspaces. Crossing symmetry has many interesting connections, including T-twisted Araki-Woods algebras, q-Systems, and algebraic Fourier transforms.

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.