This Week I Learned: October 09, 2026

[u/inherentlyawesome]

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

[u/grungegoth]

that the power rule for integration of xⁿ can be applied to the case where n = -1 if we use a limit and some other trickery to get the answer which is ln(x)

[u/Plastic-Rope5516]

I learned the definition of a Cauchy sequence and how they can be used to construct the real numbers
I also learned the formal definition of a linear mapping (that I had encountered before but not learnt formally)

[u/Plastic-Rope5516 → u/Plastic-Rope5516]

I am using Analysis I and Linear Algebra Done Right to study

[u/jellyv2000]

This week I learned more about Relational Algebra and functional dependencies which as a Math and CS student, is very cool

[u/Werwolvee]

I learned about weak and strong induction, induction being an axiom in Peano Arithmetic, and it being a theorem in ZFC, I learned about group isomorphisms as well. I'm not too experienced at math and have only been really studying it for about 6-8 months so I learn a lot of stuff pretty fast as its not too complicated.

[u/kiantheboss]

Integration theorems: bounded convergence theorem, monotone convergence theorem, etc

[u/KnownAd9773]

Not exactly "learned", but that's an interesting fact. Found that the OEIS is calling for nominations for the sequence which will become A400000 (till 16 October): https://oeis.org/wiki/Nominations_for_Sequence_A400000

Sent them a nominee (it is not a sequence proposed by me, but I found it interesting).

[u/hooverprover]

As an alternative to using logarithm tables to multiply numbers, a table of Mantel's Theorem quotients can be used to solve multiplications with just one addition and two subtractions: ab = 0.25((a+b)²⁾ - 0.25((a-b)²⁾

[u/BeyondTheAxioms]

Green correspondence Theorem! I am very surprised that there is little to none discussion on the web, with returned search results mainly involving ‘Green’s Theorem’ in calculus or ‘Correspondence Theorem’ in group theory.

[u/cereal_chick → u/BeyondTheAxioms]

What is the Green correspondence theorem?

[u/BeyondTheAxioms → u/cereal_chick]

Let k be a field of characteristic p where p| |G|. Let G be a finite group.

In representation theory, we care about irreducible representation, because it's like the single piece of LEGO, or the atom of representation. In this setting, however, something tricky happened: you have LEGOs that aren't the usual single pieces, but it just couldn't be pulled apart. In proper terminology, the LEGOs that can't be pulled apart are said to be indecomposable.

We might want to take a step back from understanding irreducible to understanding indecomposable, but indecomposable is surprisingly hard to classify. So the philosophy in representation theory is to try reducing this problem, even a little bit. This is where Green correspondence theorem came in, in which it reduces the problem from group G to its subgroup, by introducing a one-to-one correspondence.

--- the technical part ---

Let H be a subgroup of G. Then we say a kG-module U is relative H-projective if U is a direct summand of Ind^G_H(Res^G_H U), where Ind and Res are the induction and restriction functor.

For indecomposable kG-module U, a vertex vtx(U) of U is defined to be the smallest subgroup of G such that U is relatively projective to.

Let Q be a p-subgroup of G, and let L = N_G(Q), i.e. the normalizer of Q in G.

Green correspondence theorem says that there is a one-to-one correspondence between "the isomorphism classes of indecomposable kG-modules with vertex Q" and "the isomorphism classes of indecomposable kL-modules with vertex Q".

And as mentioned, this reduces the problem of understanding indecomposable from a group G to a subgroup L, and you have many 'structure' inside this subgroup L. Also, there is a much more generalised version of the presented statement, but it takes decade to build up to. I think I'll stop here.

[u/Kyle--Butler → u/[deleted]]

TIL ! How is this proved ?
EDIT : found a good exposition here (in french).

[u/Amaldevhari → u/[deleted]]

Makes me wonder how much of unknown “unknowns” are out there. A lot of mathematics is based on intuitions extended to an abstract idea. Many mathematical subjects arose to solve practical problems (like Newton discovering calculus to study motion, Fourier analysis for heat conduction, etc. ). Some others arose by questioning mathematical assumptions or directly exploring abstract idea, like Non-Euclidean geometry (what if Euclid’s parallel postulate doesn’t hold?) , set-theory (investigating infinity), etc.

However, the common theme is that, even these abstract topics began with questions we are able to formulate in the first place. This makes me wonder if there are entire branches of mathematics that we haven’t discovered, not because they’re too difficult to understand or prove, but because we haven’t even developed the intuition or conceptual framework needed to imagine them. We probably don't even possess the conceptual tools to formulate such questions.

Maybe there are infinitely many such mathematical subjects hiding in plain sight, but our limitations as humans makes us blind to them.

[u/jphamlore → u/Amaldevhari]

Stephen Wolfram's research is basically an example of new mathematics that might be out there to discover? But being so far out there, it's hard to decide what's actually there.