I made my first post to the #AI section of the arXiv this week! You can find the preprint "Discrete neural nets and polymorphic learning" at https://arxiv.org/abs/2308.00677.

In this paper a learning algorithm based on polymorphisms of finite structures is described. This provides a systematic way to choose activation functions for neural nets whose neurons can only act on a fixed finite set of values. These polymorphisms preserve any specified constraints imposed by the learning task in question.

This paper is the result of a 2021 REU at the University of #Rochester. I am working with a great group of students on a follow-up project right now, so videos of talks and a sequel preprint should be out soon!

#NeuralNets #MachineLearning #math #combinatorics #UniversalAlgebra #ComputerScience #teaching

After posting an answer on this MathOverflow question https://mathoverflow.net/questions/450930/is-there-an-identity-between-the-associative-identity-and-the-constant-identity , I wonder if it might be a suitable graduate research project to see if current generation #ProofAssistant / #MachineLearning / #AI tools can be used to determine the logical relationship between various universal equational laws that could be satisfied by a single binary operation + on a set (i.e., by a magma). For instance, in the answer to this related question https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?noredirect=1&lq=1 it was shown (by a slightly intricate argument) that the law (𝑥+𝑥)+𝑦=𝑦+𝑥 implies the commutative law 𝑥+𝑦=𝑦+𝑥, but not conversely, while I showed that the law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑤 is strictly intermediate between the triple constant law 𝑥+(𝑦+𝑧)=(𝑤+𝑢)+𝑣 and the associative law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑧. It seems that this is a restrictive enough fragment of #mathematics (or even of #UniversalAlgebra) that automated tools should function rather well, without being so trivial as to be completely solvable by brute force.

The new version of my paper with Semin Yoo, "Orientable triangulable manifolds are essentially quasigroups" is now available on arXiv! You can find the preprint at https://arxiv.org/abs/2110.05660 and you can find some videos of me talking about it on my YouTube channel (https://www.youtube.com/channel/UCT0qXiThOxzbCO36U-iXNTQ).

In addition to new images which illustrate our constructions we also have filled a gap in the proof of the main theorem. In order to show that all orientable triangulable manifolds could be created from an \(n\)-ary quasigroup by our construction, we needed to make an appropriate \(n\)-quasigroup for each manifold. What we actually did in the original paper was give a presentation of such an algebraic structure, which is not quite enough to prove the desired result. This new version contains an explicit description of such an \(n\)-quasigroup.

You can look forward to hearing more from me on connections between #quasigroups and #topology in the future!

#UniversalAlgebra #combinatorics #AlgebraicTopology

@RobertJackson58585858 @johncarlosbaez Seeing the isomorphism theorems was part of what got me interested in #UniversalAlgebra. Every time someone would say "and then everything follows just as it did in the version for groups" I would think to myself that these all must be consequences of the same theorem. Sure enough, they are.

The LOOPS'23 conference on #loops and #quasigroups just concluded. There were lots of great talks, great Polish food, and great people. This palace in Będlewo, Poland was a neat venue at which to study #AbstractAlgebra!

#conferences #UniversalAlgebra #math #combinatorics

The name for the algebraic structure consisting of a set \(A\) equipped with a binary operation \(f\colon A^2\to A\) which is not assumed to be commutative, associative, etc. has an interesting history. As far as I can tell, this sort of algebra was originally known as a "groupoid", and you can find recent literature using the term in that way. However, Nicolas Bourbaki called a set with a single binary operation a "magma" in his _Éléments de mathématique_ and this name came to be used when there might be confusion with the newer topological use of "groupoid" to mean a category whose morphisms are all invertible. A third contender is "binar". I don't know the history of this one but it seems reasonable to me.

I prefer "magma" myself. I don't know what a "lava" or a "volcano" should be nor have I had the audacity to try to name something like this in the literature. One reason I prefer "magma" is that I can talk about a set \(A\) equipped with a single \(n\)-ary operation \(f\colon A^n\to A\) as an "\(n\)-ary magma" (or just "\(n\)-magma" or even "magma").

#algebra #UniversalAlgebra #topology #bourbaki #terminology

Since I have a fair amount of work from before this venture into social media, I'll share some older things I did sometimes.

My first single-author paper was about multiplayer versions of rock-paper-scissors. You can find a copy on arXiv (https://arxiv.org/abs/1903.07252) which is pretty similar to the version published in the journal Algebra Universalis.

I had the idea for this paper when I was stranded in #Yosemite national park for a month in 2017 after I finished my bachelor's degree. I wanted to explain to my non-mathy friends that I was into #UniversalAlgebra and this is what I came up with. There are a lot of connections with tournaments from #GraphTheory. You can find videos of me talking about this on YouTube too.

#math #combinatorics #algebra #AbstractAlgebra

Hey everyone!

My name is Jesse, and I am an undergraduate student of #Mathematics and #ComputerScience. I really enjoy #Algebra the most out of the #math disciplines I've studied. I mainly self-taught myself a good portion of topics from a first year semester of #AbstractAlgebra (#Groups, #Rings, #Fields, and some basics of #GaloisTheory ), I plan to take a #RingTheory course next semester, and I am also trying to learn #UniversalAlgebra currently on the side. Glad to be here!

#introduction