The Gruenberg-Kegel graph of a group is defined as the graph whose vertices are the primes that appear as orders of elements of the group, and there is an edge between two primes p and q if and only if pq is the order of an element of the group. For solvable groups with small fields of characters, like rational groups or inverse semirstional groups, the set of vertices of this graph is known to be bounded (e.g., for rational solvable groups, only the primes 2,3 and 5 can appear).

This has been an open question for more than 70 years. I'll review what is known, including some recent developments.

------------------------------------------------

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

As a consequence of the Littlewood-Richardson  commuters coincidence and the Kumar-Torres branching model  via  Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey-Lenart conjecture on the  bijections between the Kwon and Sundaram branching models for  the pair (GL_2n(C), Sp_2n(C)) consisting of the general linear group GL_2n(C) and  the symplectic group Sp_2n(C).

In the 1950’s Davenport, Mirsky, Newman and Rado proved that if the integers are partitioned by a finite set of arithmetic progressions, then the largest difference must appear more than once. In other words, if g1, . . ., gn and a1 ≤ a2 ≤ . . . ≤ an are integers such that {gi+aiℤ}n i=1 is a partition of ℤ then an−1 = an. This confirmed a conjecture of Erdös and opened a broad area of research (see Covering systems of Paul Erdös. Past, present and future, Paul Erdös and his mathematics, I (Budapest, 1999), Bolyai Soc. Math. Stud., vol. 11, pp. 581-627. János Bolyai Math.

We investigate finite and profinite groups with the Magnus property, where a group is said to have the Magnus property if whenever two elements generate the same normal subgroup then the elements are conjugate or inverse-conjugate. This is joint work with Martino Garonzi and Pavel Zalesskii.

--------------------------------------------------------------------------------

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

We introduce the concept of a nonassociative (i.e. not necessarily associative) inverse semialgebra
over a field, the Lie version of which is inspired by the set of all partially defined derivations of
a nonassociative algebra, whereas the associative case is based on such examples as the set of all
partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a
scheme, the set of all regular functions defined on open subsets of an algebraic variety and the

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.

----------------------------------------------------

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.