algebra final tmrw 😤 watch me cook

#algebra #abstractalgebra #grouptheory

Fundamentals Of Hypercomplex Numbers | UCLA Extension

https://boffosocko.com/2025/12/03/fundamentals-of-hypercomplex-numbers-ucla-extension/

The determinant of transvections. — New blog post on Freedom Math Dance

A transvection in a K-vector space V is a linear map T(f,v) of the form x↦x+f(x)v, where f is a linear form and v is a vector such that f(v)=0. It is known that such a linear map is invertible, with inverse given by f and −v. More precisely, one has T(f,0)=id and T(f,v+w)=T(f,v)∘T(f,w). In finite dimension, these maps have determinant 1 and it is known that they generate the special linear group SL(V), the group of linear automorphisms of determinant 1.

When I started formalizing in Lean the theory of the special linear group, the question raised itself of the appropriate generality for such results. In particular, what happens when one replaces the field K with a ring R and the K-vector space V with an R-module?

https://freedommathdance.blogspot.com/2025/11/the-determinant-of-transvections.html

#math #LinearAlgebra #AbstractAlgebra

Numperphile - “Lord of the Commutative Rings”

This was the sort of stuff I loved in upper-level undergraduate mathematics.

https://www.youtube.com/watch?v=1oqqpqaDgfI

#Numberphile #math #maths #abstractAlgebra

A geometric link: Convexity may bridge human & machine intelligence
https://phys.org/news/2025-07-geometric-link-convexity-bridge-human.html

On convex decision regions in deep network representations
https://www.nature.com/articles/s41467-025-60809-y

Convexity (algebraic geometry)
https://en.wikipedia.org/wiki/Convexity_(algebraic_geometry)

#ML #RepresentationLearning #convexity #AbstractAlgebra #DeepLearning #intelligence #NetworkTheory

A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra \(A\) into its "basic parts". Formally, we say that \(A\) is a subdirect product of \(A_1\), \(A_2\), ..., \(A_n\) when \(A\) is a subalgebra of the product
\[
A_1\times A_2\times\cdots\times A_n
\]
and for each index \(1\le i\le n\) we have for the projection \(\pi_i\) that \(\pi_i(A)=A_i\). In other words, a subdirect product "uses each component completely", but may be smaller than the full product.

A trivial circumstance is that \(\pi_i:A\to A_i\) is an isomorphism for some \(i\). The remaining components would then be superfluous. If an algebra \(A\) has the property than any way of representing it as a subdirect product is trivial in this sense, we say that \(A\) is "subdirectly irreducible".

Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic \(p\)-groups \(\mathbb{Z}_{p^n}\) and the Prüfer groups \(\mathbb{Z}_{p^\infty}\).

In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (https://math.chapman.edu/~jipsen/posets/si_lattices92.html) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?

We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", https://arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

#UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra

I've been on a longer hiatus from livestreaming than I originally intended, but you can see me give a seminar talk this evening at the The New York City Category Theory Seminar:

https://www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html

I'll be talking about the invariant theory part of my thesis (https://arxiv.org/abs/2402.18063) at 7PM, New York time. I'll discuss how I found that every (positive) property of finite structures can be checked by counting small* substructures.

*Terms and conditions may apply. Small is constrained by the logical complexity of a property and may not conform to mundane notions of smallness in bad cases.

#CategoryTheory #combinatorics #logic #Bourbaki #algebra #AbstractAlgebra