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General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences
June 9, 2008 14:00, St. Petersburg, POMI, room 311 (27 Fontanka)
 


Unification theorems in algebraic geometry over algebraic structures

V. N. Remeslennikov

Omsk

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Abstract: Quite often relations between sets of elements of a fixed algebraic structure $\mathcal A$ can be described in terms of equations over $\mathcal A$. In the classical case, when $\mathcal A$ is a field, the area of mathematics where such relations are studied is known under the name of algebraic geometry. It is natural to use the same name in the general case. Algebraic geometry over arbitrary algebraic structures is a new area of research in modern algebra, nevertheless, there are already several breakthrough particular results here, as well as, interesting developments of a general theory. There are general results which hold in the algebraic geometries over arbitrary algebraic structures, we refer to them as the universal algebraic geometry. Research in this area started with a series of papers by Plotkin, Baumslag, Kharlampovich, Myasnikov and Remeslennikov.
In the talk three approaches to the main task of the algebraic geometry over algebraic structures will be considered (algebraic, geometric and logical approach). Two main theorems (“Unification theorems”, which coordinate different approaches) will be stated.

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