Extension Monads: Some Structure Theorems

Abstract: To some extent, this presentation follows the seminar titled "Non-standard Methods and Universal Algebra" by Matt Insall presented back in February. In this presentation, we take algebraic structures---a pair consisting of (1) a set of elements to compute upon and (2) a set of computational instructions, called algebras heretofore---into the nonstandard setting. This process can both enlarge and endense said algebra, often causing new elements to appear both "far away" from and "in between" the elements of the original algebra, while preserving all of the same logical properties of the original.

The additional elements are often clustered around directly mapped elements from the standard (original) setting. In the case of these algebras, we define a third structure, the extension monad, of the elements that are very close to the original algebra. This extension monad is, as a set, in between the standard and nonstandard version of the algebra and while not logically identical, captures the local behavior of the computational instructions, known as operations.

Bio: Danielle Bowerman studied mathematics at Evangel University and earned a bachelor of science in 2019. She then went on to join a mathematics PhD program as a Chancellor's Distinguished Fellow at the Missouri University of Science and Technology and expects her PhD in the fall of 2025. She taught a few sections of College Algebra at some point.

Moderator: This talk will be moderated by Matt Insall. Matt Insall is a mathematician, father of four (two sons and two daughters), and grandfather of four boys and two girls. He’s written poems and a song, and performs occasionally at open mic events. Matt retired in September 2024 from his position in the faculty of the Missouri University of Science and Technology Department of Mathematics and Statistics after 35 years in that position.

Computer scienceMathematics

Audience: researchers in the topic

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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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