The latest edition (the 19th) of the Kourovka Notebook has just been released. It now has its own website, https://kourovka-notebook.org/. The Kourovka Notebook…]]>

The latest edition (the 19th) of the Kourovka Notebook has just been released. It now has its own website, https://kourovka-notebook.org/.

The Kourovka Notebook has been going for more than 50 years, longer than my life as a mathematician. It is a problem book for group theory and related areas, and now contains somewhere around a thousand problems, 111 of them new in this edition. The first edition appeared in 1965, and the KN did a remarkable job of encouraging communication across the Iron Curtain in those tense times. There are about 20 problems from that first edition still unsolved, and the first two to be attributed (rather than just labelled “Well known problem”) are by Mal’cev and Magnus. The last two questions ask whether the identical relations of a polycyclic group, or of a matrix group in characteristic zero, have a finite basis. This was a time of great…

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“Algebraic extensions of uniform algebras, and more general commutative Banach algebras, are well documented and generally well understood. However, more general classes of uniform algebra extensions have received relatively little attention in the literature. In fact, some important examples from uniform algebra theory appear as extensions. In this talk, we discuss some of the important examples of uniform algebra extensions, and how the action of groups play an important role in the theory.”

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The new edition (now also on a dedicated web-site) is sponsored by School of Mathematics and Physics of the University of Lincoln and Sobolev Institute of Mathematics, Novosibirsk.

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*Abstract*: The upper rank of a group is the supremum of the Prüfer ranks of its finite quotient groups. The upper *p*-rank, for a prime *p*, is defined analogously, using the Sylow *p*-subgroups of finite quotients. I’ll discuss to what extent the upper rank of a finitely generated group is controlled by its upper *p*-ranks.

*Abstract: *Thompson’s group *V* is probably one of the best known examples of a finitely presented infinite simple group. The presentation originally given by Thompson in his notes has remained the best in terms of fewest generators and relations until very recently. I will present recent work with Collin Bleak that gives new presentations for *V* including one involving fewer relations than Thompson’s presentation. There are a number of other infinite simple groups that arise as generalisations of Thompson’s group *V*. One set of examples are the groups *nV* (for positive integers *n*) introduced by Brin and that act upon n-dimensional Cantor space. I will present ongoing work that produces presentations for the groups *nV* and that draws attention to the interaction between the baker’s maps on *n*-dimensional Cantor space and transpositions of basic open sets.

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The speaker was Dr Jason Lotay (University College London), who spoke on Adventures in the 7th Dimension. The lecture was followed by drinks and nibbles.

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