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)

Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita–Takesaki modular theory and have many applications to quantum field theory. In this talk, standard subspaces are considered as a subject of interest in their own right (independently of von Neumann algebras). A particular focus are inclusions of standard subspaces, which have similarities to subfactors, and several new methods for investigating the relative symplectic complement of an inclusion will be discussed. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.

Joint work with Ricardo Correa da Silva, see https://link.springer.com/article/10.1007/s00220-025-05458-4.

The dynamical formulation of optimal transport between two probability measures $\mu_0,\mu_1$ on a (sub)Riemannian manifold $M$, aims at minimizing the square integral of a Borel family of vector fields
$$
\int_0^1\int_M||v_t||^2dmu_t dt,
$$
where the narrowly continuous curve of probabilities $\mu_t$ and $v_t$ must respect the continuity equation. The equivalence between this Benamou-Brenier formulation and the Kantorovich formulation of optimal transport, is well known in Riemannian context, but still open in sub-Riemannian manifolds (in the SR case, $v_t$ is a family of {em horizontal} vector fields).
I will present some recent advancements in this problem and a joint work (with Giovanna Citti and Andrea Pinamonti), proving the equivalence under general regularity assumptions in the case of a sub-Riemannian manifold with no non-trivial abnormal geodesics. The key idea is the formulation of a relaxed version of the dynamical problem that hinges the other two versions, and allows to prove the equivalence of the Kantorovich formulation with the relaxed and the original Benamou-Brenier formulation.

NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

Weyl groups for the Cuntz algebras were introduced implicitly by Cuntz at the end of the seventies. However, they remained largely ignored probably because of computational difficulties until about 30 years later, when the breakthrough work by Conti and Szymanski allowed to determine explicitly a huge number of their elements by a sophisticated condition with a clear combinatorial flavour. In recent times, Brenti, Conti and Nenashev pushed the boundaries of the involved combinatorial structures, obtaining for instance the first enumerative results (at the moment, only for cycles). In this talk I will report on recent work joint with F. Brenti and G. Nenashev (in preparation) where, building on the combinatorial machinery developed over the last few years, we construct certain subgroups of outer automorphisms of O_n. In particular, we are able to describe in detail the 46 distinct finite groups of outer automorphisms of O_4 lying in the outer reduced Weyl group and maximal at level 2, which were first determined by Szymanski and collaborators by clever computer-assisted methods. The notion of bicompatible subgroup of the permutations of a square grid will play a role in the discussion.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n ge 3$ and $P_g$ be a conformally-covariant operator on $(M,g)$. We consider in this talk two problem at the crossroads of conformal geometry and spectral theory: 1) determining the extremal value that the renormalized eigenvalues of $P_g$ take as $g$ runs through a fixed conformal class and 2) determining whether these extremal values are attained at an extremal metric. Examples of such operators $P_g$ include the famous conformal Laplacian of the Yamabe problem, $P_g = Delta_g + c_n S_g$, but also its higher-order generalisations such as the GJMS operators of order $2k$ for any positive integer $k$.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We prove a conjecture of Cherednik describing the Betti numbers of compactified Jacobians of unibranch planar curves via superpolynomials of algebraic knots. The methods of the proof use the theory of orbital integrals and affine Springer theory. No prior knowledge about any of these will be assumed. 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

&nbsp; Given a simple finite Lie algebra over the complex numbers, we can consider two other Lie algebras attached to it: its Langlands dual Lie algebra and the affine algebra at the critical level. It is a theorem of the nineties, by Feigin and Frenkel, that the center of the completed enveloping algebra of the affine algebra at the critical level is canonically isomorphic to the algebra of functions on the space of Opers on the pointed disk for the Langlands dual Lie algebra. These objects are actually pointwise instances of a more general picture: the space of opers for example enhances to a space which lives over an arbitrary smooth curve that is equipped with a natural factorization structure. This structure is fundamental for the Geometric Langlands community: factorization patterns allow for local to global arguments. In this talk I will explain the construction of the objects mentioned above and elaborate on a joint work with Andrea Maffei in which we prove the factorizable version of the Feigin-Frenkel theorem. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

  In this talk, I will introduce restricted Poisson algebras in characteristic 2 and explore their connection with restricted Lie–Rinehart algebras. For the latter, a cohomology theory is deve-loped and abelian extensions are investigated. I will also construct a cohomology complex for restricted Poisson algebras in characteristic 2 that controls formal deformations. This complex is shown to be isomorphic to the cohomology complex of a suitable restricted Lie–Rinehart algebra. Several examples are provided to illustrate the constructions.
  This is a joint work with Sofiane Bouarroudj and Quentin Ehret.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)