In this talk, we discuss Zariski topologies on groups and semigroups—topologies generated solely by their underlying algebraic structure. We show that in certain cases these topologies are, in a meaningful sense, canonical: they are the unique Polish topologies compatible with the given algebraic operations. We also present additional results illustrating an interplay between algebra and topology.
This is the third and last part of a series of 3 talks; the first part "Around Nyikos problem" has been on December 4 and the second part "Schur ultrafilters" has been on December 11.