A reduced word for a permutation of the symmetric group is its own commutation class if it has no commutation moves available. In this talk we provide an upper bound for the number of one-element commutation classes of a permutation. Using this upper bound, we prove a conjecture that relates the number of reduced words with the number of commutation classes of a permutation.

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There will be coffee and cake after the seminar in the common room

The concept of a rank function on an additive category with cokernels, or more generally, on one admitting finite weak cokernel resolutions, generalises the classical notion of a Sylvester rank function on a ring, which itself extends familiar invariants such as the dimension of a vector space or the rank of a matrix. Analogous notions have also been developed for triangulated categories and for (d+2)-angulated categories.

In the 1960s, Auslander and Bridger introduced the concept of G-dimension for finitely generated modules over a Noetherian ring. In 1995, Enochs and Jenda extended this concept to modules that are not necessarily finitely generated and defined what are now known as Gorenstein projective and injective modules, along with their homological dimensions. However, their investigation was limited to specific classes of rings.

The conjugacy problem is, alongside the word and the isomorphism problems, one of the three classical algorithmic problems in group theory introduced by Max Dehn in 1911. It asks whether it is decidable if two given elements of a group are conjugate. Since its introduction, the problem has been extensively studied from algebraic, asymptotic, topological, and language-theoretical perspectives.

In this talk, we consider Novikov algebras with derivation and algebras obtained from its dual operad. It turns out that the obtained dual operad has a connection with bicommutative algebras. The motivation for this work comes from the white and black Manin product of the Novikov operad with itself.

This talk provides a brief overview of a Lie bracket on affine spaces, known as a Lie affgebra, introduced by K. Grabowska, J. Grabowski, and P. Urbański in 2003. Lie affgebras have been studied in several works by Tomasz Brzeziński and his collaborators. In particular, Lie affgebra structures on various classes of affine spaces of matrices have been investigated. It has been shown that any Lie affgebra is isomorphic to a Lie algebra equipped with an additional element and a specific generalized derivation.

The goal of this talk is to state and motivate Serre's modularity conjectures on Galois representations. The talk includes applications and, if time permits, some generalizations to real quadratic fields.

There will be coffee and cake after the seminar in the common room

We will describe how arithmetic properties of some orders in quaternion algebras can be used solve Diophantine problems about representations of integers by some quaternary quadratic forms. Some open problems will be described along the way.

There will be coffee and cake after the seminar in the common room

In commutative algebra, a major subject of investigation is the study of ideals in polynomial rings that are defined by graphs; the goal is to understand how the properties of the ideals relate to the properties of the associated graphs. One of the several ideals studied in this context is the Eulerian ideal of a graph G. This ideal is generated by binomials which identify Eulerian subgraphs of G with an even number of edges.

Partial representation theory is a relatively recent area of research, originating in the study of partial dynamical systems and C∗-algebras generated by partial isometries. This talk offers a brief tour of the field, highlighting some of its key ideas and recent developments: the globalisation problem and a new approach to it, the connections with inverse semigroups, the partial representation theory of finite groups and the isomorphism problem for partial group algebras, and – time permitting – some recent progress towards understanding partial (co)modules over Hopf algebras.