Posting some theorems, and occasionally other stuff. By @bahran_cihan
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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.
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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.
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This might also stem from them having only 16 references which is absurd for a paper of this length
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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…
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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.
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[Brehm–Kühnel 1987]: For fixed d ∈ℤ⁺, every combinatorial d-manifold with strictly less than
3⌈d/2⌉ + 3
vertices is a d-sphere.
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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.
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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.
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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.
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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.
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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.
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[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.
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Given a finite group G, write P(G) for the commuting probability of G, that is, the ratio of commuting ordered pairs in G x G.
[Browning 2023]: Under the reverse ordering of rational numbers in [0,1], the set
{P(G) : G finite}
is well-ordered with order type ω^ω.
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42 = (-80538738812075974)³ + (80435758145817515)³ + (12602123297335631)³
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[Antonyan–Dobrowolski 2015]. For a locally compact Hausdorff group G with a closed subgroup H, TFAE:
(1) G/H is locally contractible.
(2) G/H is a manifold.
[Antonyan 2011]: For a compact Hausdorff group G with a closed subgroup H, TFAE:
(1) The coset space G/H is contractible.
(2) H = G.
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ALT From the introduction of https://arxiv.org/abs/1902.04011v1
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ALT Implications involved in Lam's treatment (specifically Theorem 4.10 in his noncommutative rings book) of the Hopkins–Levitzki theorem.
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