See some recommendations from me and my colleagues at the University of Maryland for your summer reading list, including a rollercoaster biography of a remarkable Polish logician and mathematician Alfred Tarski, "the man who defined truth"!
Eight faculty and staff members share their favorite books—both old and new.
In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups! Previously only known in the amenable case, by "Definably amenable NIP groups", Chernikov, Simon, J. Amer. Math. Soc. 2018
We also provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup.
In this paper we obtain a remarkable counterpart of the aforementioned classical theory for locally compact topological groups, and classify idempotent generically stable measures in abelian groups as (unique) translation invariant measures on type-definable subgroups.
This led to many applications, e.g. resolution of Pillay’s conjecture for compact o-minimal groups, or Hrushovski’s work on approximate subgroups. And brought to light the study of invariant measures on definable subsets of the group, and the methods of topological dynamics.
More recently, many of the ideas of stable group theory were extended to the class of NIP groups, which contains both stable groups and groups definable in o-minimal structures or over the p-adics.
The class of stable groups is at the core of model theory, and the corresponding theory was developed in the 1970s-1980s borrowing many ideas from the study of algebraic groups over algebraically closed fields.
We are interested in a counterpart of this in the definable category. In the same way as algebraic or Lie groups are important in algebraic or differential geometry, the understanding of groups definable in first-order structure is important for model theory and its applications.
Classical work by Wendel, Rudin, Cohen (before inventing forcing) and others classifies idempotent Borel measures on locally compact abelian groups, showing that they are precisely the Haar measures of compact subgroups.
Very excited about this new preprint with Kyle Gannon and Krzysztof Krupinski! "Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture" arxiv.org/abs/2406.00912#ModelTheory#math
We study idempotent measures and the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is...