BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Thomas Kucera (University of Manitoba) DTSTART:20220412T190000Z DTEND:20220412T200000Z DTSTAMP:20260423T004134Z UID:PALS/33 DESCRIPTION:Title: Sa turated free algebras and almost indiscernible theories: an overview\n by Thomas Kucera (University of Manitoba) as part of PALS Panglobal Algebr a and Logic Seminar\n\n\nAbstract\nThis is work motivated by questions at the intersection of algebra and model theory\, and using advanced techniqu es of model theory.\nBaldwin and Shelah (Algebra Universalis\, 1983) studi ed saturated free algebras. Pillay and Sklinos (Bull. Symb. Logic 2015)\, following the lead of this paper\, studied "almost indiscernible theories" \, taking the opportunity to refine the statements of the major results an d improve the proofs. We extend these results to large infinite contexts\, both in the size of the language and the kinds of tuples allowed in a "ba sis"\; and return to examples and applications in algebra\, in particular in the theory of modules.\nThe theory develops by noting various analogies . The model-theoretic concept 'indiscernible sequence' generalizes 'linear ly independent set' in a vector space\, 'free (generating) set' of an alge bra\, 'algebraic independence' in an algebraically closed field\, and simi lar concepts. 'Saturated model' generalizes concepts such as 'injective en velope of a module'\, 'algebraic closure of a field'\, and similar constru ctions. A complete first-order theory is "almost indiscernible" if it has a (sufficiently large) saturated model which lies in the algebraic closure of an indiscernible set (of sequences). Requiring that a saturated model be generated by an indiscernible set imposes strong structural constraints \, but nonetheless there are natural motivating examples.\nI start with so me history and motivation from algebra\, then I will give an overview of t he main model theoretic concepts and techniques\, motivating them as much as possible by examples from algebra. I'll state the new technical structu ral results for almost indiscernible theories in our more general context\ , with no more than informal 'hand-waving' about the proof techniques. The n I will present some consequences for free algebras and for theories of m odules\, including structure theorems and some examples. I conclude with a list of open questions.\nThis is joint work with Anand Pillay.\n LOCATION:https://researchseminars.org/talk/PALS/33/ END:VEVENT END:VCALENDAR