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Potentially dense sets in abelian groups

Dimitri Shakhmatov

Department of Mathematics of Faculty of Science of Ehime University (Matsuyama, Japan)

Abstract
We consider 65 years old problem of Markov: Given a subset S of a group G, can we find a Hausdorff group topology on G in which the set T becomes dense? A related problem: Given a subset S of an abelian group G, can one find a homomorphism (or even better, a monomorphism) f: G --> T^k from G to some power T^k of the circle group T such that the image f(S) of S becomes dense in T^k?

Bio sketch

Prof. Dimitri Shakhmatov was born in 1961 in Minsk (former Soviet Union, now Belorussia), graduated from Faculty of Mechanics and Mathematics of Moscow State University in 1983, and received PhD in mathematics from Moscow State University in 1987. Professor Shakhmatov worked as Assistant Professor and Research fellow at the Faculty of Mechanics and Mathematics of Moscow State University from 1986 till 1991. In 1991-1992 he was Distinguished Visiting Associate Professor at Miami University (Oxford, Ohio, U.S.A.), in 1992-1993 he was Visiting Associate Professor at University of Toronto (Toronto, Ontario, Canada). Since 1993 till present he is a Professor at the Department of Mathematics of Faculty of Science of Ehime University (Matsuyama, Japan). He authored 80 publications in set-theoretic and general topology and in topological algebra, especially topological groups.