Seminar of Algebra

Exotic triangulations and the octahedral axiom

Triangulated categories have become an important field of study in recent decades and are now a fundamental tool in areas such as algebraic geometry, algebraic topology, and representation theory. Most known triangulated categories are either algebraic or topological, and only a few ''exotic'' examples are known. A theorem by A. Heller provides a method to construct triangulated structures (initially without assuming the octahedral axiom) from certain natural isomorphisms. Using this theorem, we introduce a new exotic triangulated category and outline our attempts to prove the octahedral axiom, including an original characterization of it in terms of a secondary cohomology operation.

$\ell$-adic sheaves and D-modules: a motivic connection

The study of exponential sums over finite fields goes back to Gauss. These sums exhibit deep arithmetic structure and later became a central object in analytic number theory. From a modern perspective, exponential sums can be interpreted as trace functions associated with $\ell$-adic sheaves, allowing them to be studied via the action of Frobenius on $\ell$-adic cohomology. On the other hand, the theory of D-modules provides an algebraic framework for the study of linear differential equations, with deep connections to perverse sheaves through the Riemann–Hilbert correspondence. The aim of this short talk is to present some structural analogies between these two theories through classical examples.

Spherical-type Artin-Tits groups: generalizing curves in the n-punctured disc

In this talk we will explain the relevance of braid groups when trying to generalize topological tools to an algebraic context: spherical-type Artin-Tits groups. For that purpose, we will show which elements play a crucial role in this problem and how we can ''translate'' certain topological ideas into algebraic objects, focusing on the ribbon groupoid.