*Abstract*: A thin Lie algebra is a Lie algebra , graded over the positive integers, with its first homogeneous component of dimension two and generating , and such that each nonzero ideal of lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. If is the only diamond, then is a graded Lie algebra of maximal class.

We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is one less than the dimension of their largest metabelian quotient.

]]>Recall that “Kourovka Notebook” is a famous collection of open problems in group theory proposed by hundreds of mathematicians from all over the world, published every 2-4 years since 1965.

As an additional feature of the Kourovka Notebook, a new series of online seminars “Kourovka Forum” has been launched recently: http://mca.nsu.ru/kourovkaforum/ . Previous talks, with slides and recorded lectures, can be found at http://mca.nsu.ru/kourovkaforum/scheduleandabstracts/.

This Regional Meeting was followed by a 2-day **workshop on Profinite Groups and Related Aspects**. The speakers for the workshop included Gunnar Traustason (University of Bath), Anastasia Hadjievangelou (University of Bath), Nadia Mazza (Lancaster University), John Wilson (University of Cambridge and University of Leipzig), Pavel Shumyatsky (University of Brasilia), Colin Reid (University of Newcastle, Australia), Alejandra Garrido (Autonomous University of Madrid), Henry Bradford (University of Cambridge), and Rachel Camina (University of Cambridge). For this workshop more than 40 people participated.

Below are some photos from the event.

]]>Title: On the Modular Isomorphism Problem

Abstract: The Isomorphism Problem for group rings asks for which groups *G* and *H* the group rings of these groups over a given commutative ring *R* are isomorphic as rings. Less formally speaking, it asks how much the linear representations of *G* over *R* know about the structure of the group *G*. Several concrete formulations of the problem were studied, but the only classical formulation which remains open today and was explicitly formulated by R. Brauer is the *Modular Isomorphism Problem*: Does the group ring of a finite *p-*group *G* over a field of characteristic *p* determine *G* up to isomorphism?

Though in contrast to other cases the Modular Isomorphism Problem in its strongest form studies a finite object, allows algorithmic approaches and many ideas have been developed, progress has been slow and for any fundamental classes of *p*-groups the problem remains open. I will review some history of the general Isomorphism Problem, present techniques used in the study of the modular version and give some recent results.

This is joint work with Tobias Moede and Mima Stanojkovski.

]]>Title: Automorphism groups, elliptic curves, and the PORC conjecture

Abstract:

In 1960, Graham Higman formulated his famous PORC conjecture in relation to the function f(p,n) counting the isomorphism classes of groups of order p^n . By means of explicit formulas, the PORC conjecture has been verified for n < 8. Despite that, it is still open and has in recent years been questioned. I will discuss (generalizations of) an example of du Sautoy and Vaughan-Lee (2012), together with a conceptualization of the phenomena they observe. Hidden heroes of this story turn out to be Hessian matrices and torsion points of elliptic curves. This is joint work with Christopher Voll.