Posting some theorems, and occasionally other stuff. By @bahran_cihan
Joined July 2022
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Bullying requires power, which an anon twitter account does not have over math professors in Caltech.
ALT An r/mathematics comment by kieransquared1: Only 16 citations, half of which aren't even from this century. There's only one citation from this *decade*. They don't cite *any* of works on blowup for 3D Euler, like Elgindi, Chen-Hou, Cordoba-Martinez-Zoroa, etc. At best, this is terrible citation practice and at worst it's blatant plagiarism.
I asked Claude to generate a word cloud image based on the first names appearing in the list here (weighted by number of papers):
annals.math.princeton.edu/au…
I find the "human capital" term rather problematic. It objectifies/commodifies human beings. Reminds me of this icky quote
ALT A photo of Sam Altman with the quote "We see a future where intelligence is a utilitu, like electricity or water, and people buy it from us on a meter" — Sam Altman, Open AI CEO.
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 retweeted
[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 retweeted
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 retweeted
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 retweeted
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 retweeted
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.
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 !!
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.
[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.