BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Matt Insall (Missouri University of Science and Technology) DTSTART:20250228T163000Z DTEND:20250228T173000Z DTSTAMP:20260422T171812Z UID:CompMath/11 DESCRIPTION:Title: Non-standard Methods and Universal Algebra\nby Matt Insall (Missouri University of Science and Technology) as part of Relatorium seminar\n\n\n Abstract\nThe subject of Universal Algebra (aka General Algebra) has at it s core a class of structures called “Algebras”. An algebra can be tho ught of as a universe of discourse for a computer or calculator that “ru ns” deterministic code\, because calculations in an algebra use function s (in the mathematical sense\, no input has multiple outputs). Algebras ca n be organized into various types or “kinds”\, according to a formal l anguage that uses only universally quantified equations. The algebras form a kind of “semantic playground” for mathematicians\, and the equation s can be thought of as a “syntactic grammar” for establishing “rules of games” for any given section of the playground. At a higher level\, one may study the relations and interactions between the rules (syntax) an d the algebras (semantics)\, and classes of algebras with certain desirabl e properties. Some of the games use finite algebras as their section of t he playground but others use infinite algebras. The study of infinite alg ebras can in some nice ways benefit from knowledge about finite algebras. The goal of this discussion will be to see how Nonstandard Methods can be used to create new concepts in universal algebra\, mostly by “playing” in two (generic) “playground sections”. One section is loosely calle d “the standard world”\, and the other is “the nonstandard world”. Finite algebras we consider in the standard world are also finite algebr as in the nonstandard world\, but some of the algebras viewed as “finite ” in the nonstandard world are extensions of infinite algebras in the st andard world\, so that from the standard perspective\, some “nonstandard ly finite” or “hyperfinite” algebras are infinite. Results known abo ut finite algebras in the standard world carry over to results about hyper finite algebras\, and can be used then to draw conclusions about the stand ard algebras they extend. For example\, \n\nA standard algebra is locally finite (each of its finitely generated subalgebras is finite) if and only if it has a hyperfinite extension. \n\nOther properties of algebras (other than the property of being finite) in the standard world have nonstandard analogues as well\, and we can use this general framework to “create” new properties of algebras in the standard world. Hopefully\, with audien ce participation\, we will be able to create such a property that is new ( to us\, at least). \n\nReferences:\n\nHurd\, Albert E.\; Loeb\, Peter A. An introduction to nonstandard real analysis. Pure and Applied Mathematics \, 118. Academic Press\, Inc.\, Orlando\, FL\, 1985.\n\nStanley N. Burris\ , Stanley N.\; H.P. Sankappanavar\, H. P. A Course in Universal Algebra\, https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html\n\nInsall\, M. (1 991)\, Nonstandard Methods and Finiteness Conditions in Algebra. Mathemati cal Logic Quarterly\, 37: 525-532. https://doi.org/10.1002/malq.1991037330 3\n\n--------------------------------------------------------------------- ---------------------\n\nThe talk will be moderated by Irfan Alam. Irfan i s a mathematician\, abstract artist\, and aspiring philosopher. He current ly works as a postdoctoral fellow in the department of Computer and Mathem atical Sciences at University of Toronto Scarborough. Irfan’s research b ackground is in the area of nonstandard analysis\, especially in its appli cations to other areas of mathematics such as probability theory and topol ogy.\n\nMatt Insall studied chemical engineering (BS\, 1986) and mathemati cs (BS\, 1985) at University of Houston\, doing a MS (1987) and PhD (1989) with Professor Klaus Kaiser\, also at University of Houston. He taught ma thematics in Rolla\, Missouri\, at Missouri University of Science and Tech nology (S&T) from 1989 to 2024 and has now retired. He and his wife have f our children\, two step grandsons\, two grandsons\, and two granddaughters . Dr. Insall’s research career has included solo and collaborative proje cts in mathematics and its applications\, mainly with colleagues in variou s departments at S&T. He is currently on a courtesy appointment in the S& T mathematics and statistics department\, finishing work with PhD students . In retirement he is continuing to learn mathematics and its applications \, following some current events\, continuing some writing projects in mat hematics\, and enjoying meeting new people over coffee or online\, to disc uss mathematics and education and science.\n LOCATION:https://researchseminars.org/talk/CompMath/11/ END:VEVENT END:VCALENDAR