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1500 questions
52
votes
5 answers
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer examination it does not even seem well-defined. The…
Dick Palais
- 15,150
52
votes
22 answers
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do in class or give as a project. In particular,…
Joe Johnson
- 812
52
votes
3 answers
Is the "Napkin conjecture" open? (origami)
The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure with perimeter $P'$.
Napkin conjecture: You…
Jérôme JEAN-CHARLES
- 1,286
52
votes
11 answers
What is an important mathematical question?
$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste. I am having people ask me why study certain…
ArB
- 688
52
votes
2 answers
Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The group theoretic explanation for this that I know…
Eugene Stern
- 541
52
votes
5 answers
Metamathematics of buts
Something I learned (probably in middle school) that always bothered me is that the truth value of "and" and "but" are basically the same. If you were going to assign a truth-functional interpretation of "but" in first-order logic, it would be the…
arsmath
- 6,720
52
votes
15 answers
What is the Implicit Function Theorem good for?
What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are:
the result that the solution space of a non-degenerate system of equations naturally has the structure of a smooth manifold;
the…
jlk
- 3,254
52
votes
5 answers
Why do bees create hexagonal cells ? (Mathematical reasons)
Question 0 Are there any mathematical phenomena which are related to the form of honeycomb cells?
Question 1 Maybe hexagonal lattices satisfy certain optimality condition(s) which are related to it?
The reason to ask - some considerations with the…
Alexander Chervov
- 23,944
52
votes
1 answer
Mathematics of imaging the black hole
The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .
It has been claimed that state-of-the-art imaging algorithms were an enabler…
Piyush Grover
- 2,261
52
votes
3 answers
Mathematical research in North Korea -- reference request
Question: Where can one find information on which areas of mathematics
are represented at which of the more than 20 universities in the
Democratic People's Republic of Korea (DPRK), and on which mathematicians
are working there?
The DPRK is a…
Stefan Kohl
- 19,498
- 21
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- 136
52
votes
1 answer
Atiyah's May 2018 paper on the 6-sphere
A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he apparently published a follow-up which fleshes out the…
Paul Siegel
- 28,772
52
votes
4 answers
Do there exist chess positions that require exponentially many moves to reach?
By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an $n\times n$ board, but I will not dwell on this point…
Timothy Chow
- 78,129
52
votes
3 answers
What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?
Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent preprint, Schoen and Yau show how the usual…
Steve McCormick
- 573
52
votes
2 answers
Silver's approach to the inconsistency of $\mathrm{ZFC}$
As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency of measurable cardinals and $\mathrm{ZFC}$:
As…
Rahman. M
- 2,341
52
votes
9 answers
Undergraduate ODE textbook following Rota
I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do yourself a favor and follow the link I…
Jeff H
- 1,412