Some like it abstract
Geometry & Topology in Rome

Francesco D'Andrea
(U Napoli Federico II)

On morphisms of Hopf-Galois extensions
The ring structure of the K-theory of a compact space can be reconstructed from the knowledge of compact principal bundles over the space. The K-theory of a noncommutative C^*-algebra has no obvious ring structure, but one can use Hopf-Galois extensions to construct a replacement of the K-theory ring. I will present some results about morphisms of Hopf-Galois extensions that are a preparation for the study of the "multiplicative K-theory" of a C^*-algebra. Joint work with T. Maszczyk.

Fabio Gavarini
(U Roma Tor Vergata)

Multiparameter quantum supergroups, deformations, and specializations
From a joint work with G. A. García and M. Paolini, I will present a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in 1994. For these objects we consider: (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; (2) their semiclassical limits, which are multiparameter Lie superbialgebras; (3) the deformations by twist and by 2-cocycle (of toral type) of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations; (4) we compare toral deformations at the quantum and the semiclassical level, thus proving that "(toral) deformation commutes with quantization".

Emma Lepri
(U Glasgow)

Prorepresentability of noncommutative deformation functors
Schlessinger's Theorem, which gives necessary and sufficient conditions for a commutative deformation functor to be prorepresentable, does not generalise easily to noncommutative deformations because of the appearance of inner automorphisms in the category of Artinian rings. This talk is based on a joint work in progress with Eugenio Landi, where we try to overcome this difficulty using bicategories.

Hsuan-Yi Liao
(U Tsing Hua Taiwan)

Brown functors of directed graphs
In this talk, we will consider Brown functors of directed graphs — that is, contravariant functors from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom. Our main theorem states that such a functor is representable. In addition, I will show that the first path cohomology of directed graphs provides a nontrivial example of a Brown functor. This talk is based on joint work with Zachary McGuirk, Dang Khoa Nguyen, and Byungdo Park.

Antonio Miti
(U Roma La Sapienza)

A canonical morphism between twisted and untwisted higher Courant Lie infinity algebras
In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" (arXiv:1003.1004), Marco Zambon constructs a Lie infinity algebra associated with any higher standard Courant algebroid (also known as a Vinogradov algebroid), and exhibits an explicit Lie infinity morphism from the Lie algebra associated with a standard Lie algebroid twisted by a closed 2-form to the Lie 2-algebra of the standard Courant algebroid. He poses the question of whether analogous canonical morphisms exist in higher degrees — namely, for any standard higher Courant algebroids twisted by closed (n+1)-forms. In this talk, we present a general framework that naturally yields such canonical Lie infinity morphisms for arbitrary n, clarifying the geometric and homotopical structures underlying these constructions. Time permitting, we also discuss how this framework accommodates the canonical morphism between the observable Lie infinity algebra of a pre-n-plectic manifold and the higher Courant algebra we described in "Observables on multisymplectic manifolds and higher Courant algebroids" (arXiv:2209.05836). Joint work with Domenico Fiorenza.

Thomas Weber
(Charles U Prague)

Quantization of infinitesimal braidings
There is a well-known correspondence between braided monoidal categories and quasitriangular bialgebras: the representation category of a quasitriangular bialgebra is braided monoidal and, conversely, for every 'nice enough' braided monoidal category, one obtains a quasitriangular bialgebra via Tannaka-Krein reconstruction. In this talk, we consider first-order deformations of braided monoidal categories, meaning we introduce infinitesimal braidings, and discuss the emerging quantization problem. This is done parallelly to the algebraic picture, where infinitesimal R-matrices and so-called pre-Cartier bialgebras appear. It turns out that the quantization problem admits a solution in terms of Drinfeld associators, resulting in quasitriangular quasi-bialgebras. As an explicit example, the deformation of the pointed E(n) Hopf algebras is discussed. This is based on a collaboration with Chiara Esposito, Jonas Schnitzer, and Andrea Rivezzi.