Non-standard Methods and Universal Algebra

Abstract: The subject of Universal Algebra (aka General Algebra) has at its core a class of structures called “Algebras”. An algebra can be thought of as a universe of discourse for a computer or calculator that “runs” deterministic code, because calculations in an algebra use functions (in the mathematical sense, no input has multiple outputs). Algebras can be organized into various types or “kinds”, according to a formal language that uses only universally quantified equations. The algebras form a kind of “semantic playground” for mathematicians, and the equations 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) and the algebras (semantics), and classes of algebras with certain desirable properties. Some of the games use finite algebras as their section of the playground but others use infinite algebras. The study of infinite algebras 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 called “the standard world”, and the other is “the nonstandard world”. Finite algebras we consider in the standard world are also finite algebras in the nonstandard world, but some of the algebras viewed as “finite” in the nonstandard world are extensions of infinite algebras in the standard world, so that from the standard perspective, some “nonstandardly finite” or “hyperfinite” algebras are infinite. Results known about finite algebras in the standard world carry over to results about hyperfinite algebras, and can be used then to draw conclusions about the standard algebras they extend. For example,

A standard algebra is locally finite (each of its finitely generated subalgebras is finite) if and only if it has a hyperfinite extension.

Other 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 audience participation, we will be able to create such a property that is new (to us, at least).

References:

Hurd, Albert E.; Loeb, Peter A. An introduction to nonstandard real analysis. Pure and Applied Mathematics, 118. Academic Press, Inc., Orlando, FL, 1985.

Stanley N. Burris, Stanley N.; H.P. Sankappanavar, H. P. A Course in Universal Algebra, www.math.uwaterloo.ca/~snburris/htdocs/ualg.html

Insall, M. (1991), Nonstandard Methods and Finiteness Conditions in Algebra. Mathematical Logic Quarterly, 37: 525-532. doi.org/10.1002/malq.19910373303

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The talk will be moderated by Irfan Alam. Irfan is a mathematician, abstract artist, and aspiring philosopher. He currently works as a postdoctoral fellow in the department of Computer and Mathematical Sciences at University of Toronto Scarborough. Irfan’s research background is in the area of nonstandard analysis, especially in its applications to other areas of mathematics such as probability theory and topology.

Computer scienceMathematics

Audience: advanced learners

Comments: Matt Insall studied chemical engineering (BS, 1986) and mathematics (BS, 1985) at University of Houston, doing a MS (1987) and PhD (1989) with Professor Klaus Kaiser, also at University of Houston. He taught mathematics in Rolla, Missouri, at Missouri University of Science and Technology (S&T) from 1989 to 2024 and has now retired. He and his wife have four children, two step grandsons, two grandsons, and two granddaughters. Dr. Insall’s research career has included solo and collaborative projects in mathematics and its applications, mainly with colleagues in various 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 mathematics, and enjoying meeting new people over coffee or online, to discuss mathematics and education and science.

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Relatorium seminar

Series comments: The name "Relatorium" combines "relator" with the Latin root "-ium," meaning "a place for activities" (as in "auditorium" or "gymnasium"). This seminar series is a platform to relate ideas, interact with math, and connect with each other.

In this series, we explore math beyond what we usually hear in standard talks. These sessions fall somewhere between a technical talk and a podcast: moderately formal, yet conversational. The philosophy behind the series is that math is best learned by active participation rather than passive listening. Our aim is to “engage and involve,” inviting everyone to think actively with the speaker. The concepts are accessible, exploratory, and intended to spark questions and discussions.

The idea of relatability has strong ties to compassion — creating space for shared understanding and exploration - which is the spirit of this seminar! This is a pilot project, so we’re here to improvise, learn, and evolve as we go!

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