Artem Chernikov<p>Very excited about this new preprint, with Kyle Gannon and Krzysztof Krupinski!</p><p>&quot;Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski&#39;s conjecture&quot;<br /><a href="https://arxiv.org/abs/2406.00912" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/2406.00912</span><span class="invisible"></span></a></p><p>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.<br />We are interested in a counterpart of this phenomenon in the definable category. In the same way as e.g. algebraic or Lie groups are important in algebraic or differential geometry, the understanding of groups definable in a given first-order structure (or in certain classes of first-order structures) is important for model theory and its applications. 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. 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. This led to multiple applications, e.g. a resolution of Pillay’s conjecture for compact o-minimal groups, or Hrushovski’s work on approximate subgroups. This brought to light the importance of the study of invariant measures on definable subsets of the group, as well as the methods of topological dynamics. In particular, deep connections with tame dynamical systems as studied by Glasner, Megrelishvili and others have emerged. </p><p><a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="tag">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/ModelTheory" class="mention hashtag" rel="tag">#<span>ModelTheory</span></a> <a href="https://mathstodon.xyz/tags/TopologicalDynamics" class="mention hashtag" rel="tag">#<span>TopologicalDynamics</span></a></p>