We construct braided Hopf algebroids in a braided monoidal category over a field k and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids associated to braided Hopf Galois extensions.
  We introduce braided Lie-Rinehart algebras, which generalize the definition in a paper by Aschieri, Landi, and Pagani, and study their universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the modules of a triangular Hopf algebra. We then focus on the case of braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the H-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras.
  Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the k-order universal enveloping algebra of the braided right invariant vector fields and the k-th jet space of a braided Hopf algebroid, and furthermore a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid, under a finiteness condition.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

  An algebra A is called PI if there is some polynomial in non-commuting variables that vanishes under every evaluation on elements of A. Simple examples of PI algebras include finite-dimensional algebras and commutative algebras. In general, it is a hard problem to recognize when an algebra is PI. In this talk, I will describe a simple characterization of the quivers with a PI path algebra. I will then explain how to describe the full T-ideal of their polynomial identities in terms of a preorder incidence algebra that is naturally associated with the quiver.
  This is based on joint work with A. Berele, G. Cerulli Irelli, and E. Pascucci.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

Orthogonal Shimura varieties arise naturally as moduli spaces in algebraic geometry, one example being the moduli space of quasi-polarized K3 surfaces. Their geometry is closely related to the theory of automorphic forms. I will discuss two instances of this relationship. First, the obstruction space for extending a line bundle on an orthogonal Shimura variety to its Baily-Borel compactification can be identified with a certain space of theta series for a symplectic group. By showing that these theta series span the space of cusp forms we conclude that the Picard group of this Baily-Borel compactification has rank one and is spanned by a multiple of the hodge line bundle if the underlying lattice splits two hyperbolic planes globally and three locally. In the second part, I will discuss a result in which Eisenstein series for orthogonal groups are used to study the birational geometry of orthogonal Shimura varieties. We construct pluricanonical forms from Eisenstein series, yielding new results on when such varieties are of general type.
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

In this talk we will focus for N\geq 2 on fractional W^(1-1/N,N) maps from an (N−1)-dimensional sphere into itself as traces of maps with finite energies modelled on the N-Dirichlet integral. We consider energy minimization among maps with trace of prescribed topological degree. Our goal is to prove existence and quantitative stability for minimizers in homotopy classes. Due to the lack of compact Sobolev embeddings and conformal invariance, minimizing sequences may fail to converge strongly, leading to a concentration-compactness alternative and bubbling phenomena. When they occur, these concentration effects are investigated using PDEs tools like blow-up analysis and measure-theoretic tools such as Cartesian currents. Finally, we discuss genericity results under perturbations of the energy functional and we present some open problems. This is a joint ongoing work with Xavier Lamy (Toulouse).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

The classical many body problem with constant masses is non-integrable, therefore, to investigate the dynamics of planetary systems, perturbation theory based on the exact solution of the two-body problem is usually applied. However, in the case of variable masses, the two-body problem is generally non-integrable as well. Consequently, to apply perturbation theory, another integrable model problem is used as the unperturbed motion, the solutions of which are close to the solution of the two-body problem with variable masses (in particular, when the mass changes according to Meshchersky's first law, the solutions coincide exactly). In this talk, perturbation theory is used to derive the differential equations determining the orbital elements of the relative motion of one body around another. Here, the models of perturbed aperiodic motion along a conic section and along a quasi-conic section are considered and compared. It is shown that the model based on aperiodic motion along a quasi-conic section is more suitable for generalization to the many body problem with variable masses. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

  There is a beautiful "positroid" stratification of the Grassmannian, studied by Lusztig, Postnkov, Rietsch, Williams, K-Lam-Speyer, and many others (with strata most naturally indexed by juggling patterns!). In [KLS] we gave a formula representing the class of a positroid variety, which [Fan-Guo-Su-Xiong '25] recently expressed as a sum over "periodic pipe dreams". In this talk I'll explain all that, and introduce "matrix positroid varieties", which are what the FGSX formula is really for. Time permitting, I'll explain the necessary upgrades for Chern-Schwartz-MacPherson classes. This work is joint with Paul Zinn-Justin.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  In this talk, I will report on work in progress, joint with B. Elias, H. Ko, and N. Libe-dinsky, which aims at giving a diagrammatic presentation of the category of singular Soergel bimodules.
  Soergel bimodules categorify the Hecke algebra of a Coxeter group and play a central role in modern representation theory. Their singular counterparts categorify the Hecke algebroid, which is governed by the rich combinatorics of double cosets in Coxeter groups. I will describe this category and explain how its diagrammatic presentation will allow us to make explicit computations and to work in great generality, for example over Z.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV 2023/2027 (CUP E83C23000330006)

  Around 1985 three groups of jugglers came up with the same system for recording, studying, and creating juggling patterns. I'll explain and demonstrate this system, and explain 21st-century connections to probability and algebraic geometry.
  Note:   this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV 2023/2027 (CUP E83C23000330006)

  A permutation   a₁ , a₂ , ... , a_n   of   1 , 2 , ... , n   is alternating if   a₁ > a₂ < a₃ > a₄ < ...   The number E_n of alternating permutations of   1 , 2 , ... , n   is called an Euler number. We will survey the theory of alternating permutations and Euler numbers, beginning with the famous formula of Désiré André:   ∑_n E_n xⁿ/n! = sec(x) + tan(x) .   Connections will be given to such topics as convex polytopes, tridiagonal matrices, probability theory, and the representation theory of the symmetric group. We will explain how the enumeration of alternating permutations that are also fixed-point-free involutions is related to an asymptotic result in one of Ramanujan's notebooks.
  Note: this talk is part of the activity of the MUR Excellence Department Project CUP MatMod@TOV 2023-27 (CUP E83C23000330006)

Does the heterogeneous Fisher-KPP equation admit a unique stationary solution? We will present a positive result which is valid up to dimension 6. Then we will see that some surprises may occur in higher dimension. These results shed a light on the question of long-time convergence of solutions of the reaction-diffusion equation. This is a joint work with O. Tough.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006