[Turkmath:9482] MSGSÜ Seminar - Dikran Dikranjan - 23.01.2014 Thursday, 16:00

[Kıvanç Ersoy]

Dear all,

This week Prof. Dikran Dikranjan will give a talk about
"Zariski Topology of an Abstract Groups" on  23.01.2014 Thursday, at 16:00
in Mimar Sinan University, Department of Mathematics.

The abstract is below. All interested are cordially welcome.

Prof. Dikranjan's visit is supported by TÜBİTAK 2221 Fellowship of Visiting
Scientists and Scientists on Sabbatical Program and Istanbul Center of
Mathematical Sciences. We thank TÜBİTAK and IMBM for support.

Best regards,

Kıvanç Ersoy

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*Zariski Topology of an Abstract Group*

This topology was implicitly introduced by A. Markov in 1944
and later explicitly introduced by R. Bryant, under the name
verbal topology. In the last ten years a wealth of new papers
appeared, where this topology is given the name Zariski topology
for its striking similarity with the Zariski topology studied in
algebraic geometry. Actually, one of the main direction of
research on this topic (pursued by Baumslag, Myasnikov and
Remeslennikov) has as principal objective the development of a
counterpart of Algebraic Geometry in abstract groups.

This cycle of lectures will be dedicated to another direction,
namely the one undertaken by Markov himself towards the solution
of one of his problems: the existence of non-discrete Hausdroff
group topologies on the infinite groups. To this end one can make
use of another topology, introduced again implicitly by Markov.
This topology was explicitly introduced by Dikranjan and
Shakhmatov under the name Markov topology. In these terms,
Markov's problem can be formulated as follows: is the Markov
topology of an infinite group always non-discrete. It is easy to
see that the Markov topology is finer than the Zariski topology.
Markov showed that they coincide for countable groups and asked if
this is always the case. The first infinite group with discrete
Markov topology was built by Shelah in 1980 under the assumption
of the Continuum Hypothesis. Shortly afterwards Ol'shankij gave an
example of a countable group with discrete Zariski topology.
Finally, some attention will be dedicated to a recently relevant
progress in this line obtained by Ol'shankij and his school, in
building groups with preassigned properties of the Zariski
topology.
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