26 October 2018, Colloquium on Mathematical Logic, Denis Saveliev

Abstract:

In the first part of my talk, I'll give a brief overview of Hindman's finite sums theorem, a famous result in algebraic Ramsey theory, including some its prehistory and a few of its further generalizations. I'll outline a modern technique used in proving these and close results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups.

The second, main part of my talk will contain an application of a generalization of Hindman's theorem to the topologizability problem in algebra. First I'll remind the problem and related classical results concerning groups and rings as well as some newer results. Then I'll define a wide class of universal algebras called polyrings; instances of such algebras include Abelian groups, rings, modules, algebras over a ring, differential rings, and other classical structures. Generalizing earlier results, I'll show that the Zariski topologies of all infinite polyrings are nondiscrete. Actually, I'll prove the following, much stronger fact:

Main Theorem: If $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$.

In a certain sense, the theorem shows that Zariski topologies of polyrings, although generally even non-Hausdorff, admit a reasonable notion of topological dimension. My proof of this theorem essentially uses a stronger, multidimensional version of Hindman's finite sums theorem established by Bergelson and Hindman. As a corollary, I'll prove a result on the Hausdorff topologizability of polyrings. In conclusion, if time will permit, I'll briefly discuss some related problems.

For more information, see http://www.staff.science.uu.nl/~ooste110/seminar.html or contact Benno van den Berg at bennovdberg at gmail.com.