Some theorems

Posting some theorems, and occasionally other stuff.
By @bahran_cihan

[Some theorems @CihanPostsThms]

NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.

[Some theorems @CihanPostsThms]

@ElliotGlazer $10k scale experimentation with Astra is not essential for modern math research.

[Some theorems @CihanPostsThms]

@tilmanbayer @nihilunbounded Bullying requires power, which an anon twitter account does not have over math professors in Caltech.

[sloth12 @slothanalyst]

@plain_simon This might also stem from them having only 16 references which is absurd for a paper of this length

[Some theorems @CihanPostsThms]

I asked Claude to generate a word cloud image based on the first names appearing in the list here (weighted by number of papers):
https://t.co/X4JW1uuZVB

[Some theorems @CihanPostsThms]

@littmath @ultima_shifl @SebastienBubeck @Jacob_Tsimerman I find the "human capital" term rather problematic. It objectifies/commodifies human beings. Reminds me of this icky quote

[Some theorems @CihanPostsThms]

The recent LLM news and achievements in math are of course impossible to ignore. However, I take every "omg what now" think piece penned by people (including mathematicians) getting paid by the involved companies with a grain of salt.

[Some theorems @CihanPostsThms]

@abacustimes6 Thanks!

[Some theorems @CihanPostsThms]

Up to oriented diffeomorphism,

- There is a unique smooth structure on the 61-sphere.

- There are exactly 24 smooth structures on the 62-sphere.

- There are exactly 142211872163171481167115958878208 smooth structures on the 63-sphere.

[Some theorems @CihanPostsThms]

[Brehm–Kühnel 1987]: For fixed d ∈ℤ⁺, every combinatorial d-manifold with strictly less than
3⌈d/2⌉ + 3
vertices is a d-sphere.

[Some theorems @CihanPostsThms]

Up to oriented diffeomorphism,

- There is a unique smooth structure on the 61-sphere.

- There are exactly 24 smooth structures on the 62-sphere.

- There are exactly 142211872163171481167115958878208 smooth structures on the 63-sphere.

[Some theorems @CihanPostsThms]

The "look and say sequence" {L[n]} goes 1, 11, 21, 1211, 111221, 312211, 13112221, ...

[Conway 1986]: Writing d[n] for the number of digits in L[n], the limit of the ratio d[n+1]/d[n] exists, and is an algebraic integer of degree 71.

[Some theorems @CihanPostsThms]

The quadratic integer ring ℤ[√14] is a Euclidean domain, but 𝐍𝐎𝐓 with respect to the natural norm
N(a + b√14) := a² - 14b².

The first unconditional proof of the Euclidean-ness of ℤ[√14] was by Harper in 2004.

[Some theorems @CihanPostsThms]

The maximum number of colors necessary to properly color a map drawn on a closed connected surface with Euler characteristic χ is
⌊½(7 + sqrt(49-24χ))⌋,
with a single exception: for the Klein bottle it is 6 (not 7).

The 2-sphere case is the Four Color Theorem.

[Some theorems @CihanPostsThms]

@LukePaulPL In some sense, this is an "acceptable surprise" since it is essentially bootstrapped from the set bijection ℕ² ≅ ℕ.

More disturbing are the existence of rings R with R² ≇ R but R³ ≅ R !!

[Some theorems @CihanPostsThms]

Given a ring R with the invariant basis number (IBN) property, every fin-gen stably free (left) R-module M has a well-defined rank r(M): whenever
R^a ⊕ M ≅ R^b
we have b – a = r(M).

[Cohn 1966]. There exists a ring R with IBN and a stably free R-module M such that
r(M) < 0.

[Some theorems @CihanPostsThms]

@geoffreyirving It seems that the only known examples of this phenomenon have p=5 and P≅C₅.

[Some theorems @CihanPostsThms]

[Conder 2017]. There is a finite group G with a Sylow 5-subgroup P and g∈G–P such that every element of the coset Pg has order 5.

[Some theorems @CihanPostsThms]

A finitely presented residually finite group has a decidable word problem.

[Some theorems @CihanPostsThms]

The AI-bro "Plan accordingly" discourse is insufferable.