Algebra and Topology


L'attivita` 2007/08 del Seminario Dottorato iniziera` mercoledi` prossimo con un seminario in stile "colloquium", indirizzato ad un largo pubblico di matematici.

Mercoledi` 26 settembre 2007 alle ore 11:30, in Aula 1C/150
Dan SEGAL (professor at Oxford - All Souls College) - "Algebra and Topology"

Actually the subject begins with number theory. In the 1930s
Wolfgang Krull extended the Fundamental Theorem of Galois Theory from finite Galois extensions to infinite Galois extensions. In order to obtain a bijective correspondence between intermediate fields and subgroups of the Galois group, Krull realized that it is necessary to consider the
latter as a topological group: each field corresponds to a closed subgroup and conversely. The topology is defined by taking as neighbourhoods of the identity the Galois groups of the big field over (larger and larger) finite sub-extensions of the small field. In this way, the Galois group appears as the inverse limit of a system of finite
(Galois) groups. A group that is the inverse limit of an inverse system of finite groups is called a profinite group. It is in a natural way a compact, totally disconnected topological group (inheriting these properties from the
finite groups considered as finite discrete spaces). An infinite abstract group may have many different structures as a profinite group (i.e. different topologies) (or of course none). But it was discovered by J-P. Serre in the 1970s that for certain kinds of profinite group, the topology is uniquely determined by the underlying group. These are the so-called finitely generated pro-$p$ groups. Serre wondered whether the same might be true for finitely generated profinite groups in general;
after about 30 years of partial results by several mathematicians, we have recently shown that the answer is "yes". In fact, what the proof does is to show that many closed subgroups can be constructed in a purely algebraic way. In the talk I will try to sketch some of the mathematics involved in the proof, and mention other related results and open problems.

Rif. int. C. Marastoni, T. Vargiolu

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