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1500 questions
146
votes
10 answers
What non-categorical applications are there of homotopical algebra?
(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More and more lately I have been unexpectedly running…
mme
- 9,388
145
votes
4 answers
What are "perfectoid spaces"?
This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit more info can be found in Peter Scholze's seminar…
Thomas Riepe
- 10,731
145
votes
15 answers
Is a free alternative to MathSciNet possible?
How could a free (i.e. free content) alternative for MathSciNet and Zentralblatt be created?
Comments
Some mathematicians have stopped writing reviews for MathSciNet because they feel their output should be freely available. (The Pricing for…
Anton Petrunin
- 43,739
144
votes
21 answers
How does one justify funding for mathematics research?
G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be beautiful, but unlike music, visual art or literature,…
anon
143
votes
24 answers
Occurrences of (co)homology in other disciplines and/or nature
I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion,…
Noah Giansiracusa
- 1,871
143
votes
20 answers
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed in the literature since the primary interest of…
Mark Grant
- 35,004
142
votes
6 answers
Gaussian prime spirals
Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer,
moving initially $\pm$ in the horizontal
or vertical directions. When it hits a Gaussian prime, it turns left $90^\circ$.
For example, starting at $12 - 7 i$, moving…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
142
votes
7 answers
Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?
Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the integrand is manifestly positive. This formula is…
Noam D. Elkies
- 77,218
141
votes
14 answers
Why study Lie algebras?
I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics and physics. I visited a course on Lie groups, and…
Olivier Bégassat
- 2,663
141
votes
12 answers
Solutions to the Continuum Hypothesis
Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was the continuum hypothesis born?
Background
The…
Gil Kalai
- 24,218
141
votes
0 answers
Grothendieck -sad news
Sorry for that this is not a real question. But I thought people would like to know.
Alexandre Grothendieck died today: http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614
DamienC
- 8,113
141
votes
4 answers
If $2^x $and $3^x$ are integers, must $x$ be as well?
I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
If $n^x$ is an integer for every $n \in \mathbb{N}$ then $x$ must be an…
Alon Amit
- 6,414
139
votes
17 answers
Why is differentiating mechanics and integration art?
It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/u-substitution/often tricky).
There are all kinds of…
vonjd
- 5,875
139
votes
59 answers
Jokes in the sense of Littlewood: examples?
First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following
passage in A Mathematician's Miscellany by J.E. Littlewood
(Methuen 1953, p. 79):
I remembered the Euler formula $\sum…
John Stillwell
- 12,258
138
votes
0 answers
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $G_{\mathbb{Q}}$ is the absolute Galois group and…
AFK
- 7,387