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1500 questions
122
votes
4 answers
What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\times ({B\Sigma_\infty})_+$, where $\Sigma_\infty$ is…
Daniel Litt
- 22,187
122
votes
12 answers
Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
While there are a few notable examples of this (for…
Dylan Wilson
- 13,183
122
votes
7 answers
Topology and the 2016 Nobel Prize in Physics
I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In particular, the prize was awarded "for theoretical…
Mark Grant
- 35,004
122
votes
35 answers
Rediscovery of lost mathematics
Archimedes (ca. 287-212BC) described what are now known as the 13
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
reconstructed.
(Image from…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
122
votes
25 answers
"Mathematics talk" for five year olds
I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the members of the community share their experience…
121
votes
9 answers
Breakthroughs in mathematics in 2021
This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important breakthroughs in 2021 in different mathematical…
Johnny Cage
- 1,543
121
votes
18 answers
How do you decide whether a question in abstract algebra is worth studying?
Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me…
Alex B.
- 12,817
121
votes
33 answers
Mathematicians who were late learners?-list
It is well-known that many great mathematicians were prodigies.
Were there any great mathematicians who started off later in life?
Abel
- 41
121
votes
17 answers
Pressure to defend the relevance of one's area of mathematics
I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “internal perspective,” I mean a constant…
Monroe Eskew
- 18,133
121
votes
16 answers
How do you keep your research notes organized?
One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the other hand, this means that I end up with several…
Gabe Cunningham
- 1,861
- 6
- 26
- 31
121
votes
12 answers
How to solve $f(f(x)) = \cos(x)$?
I found the following equation on some web page I cannot remember, and found it interesting:
$$f(f(x))=\cos(x)$$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an iterative equation except for trial…
user4503
- 1,551
120
votes
41 answers
What are some very important papers published in non-top journals?
There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly different. In the course of a discussion I am having, the…
gowers
- 28,729
119
votes
33 answers
Examples of theorems misapplied to non-mathematical contexts
For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments based on (what they take to be) Godel's theorem,…
JSE
- 19,081
119
votes
38 answers
Noteworthy, but not so famous conjectures resolved recent years
Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.
Question…
Alexander Chervov
- 23,944
118
votes
15 answers
Sum of 'the first k' binomial coefficients for fixed $N$
I am interested in the function $$f(N,k)=\sum_{i=0}^{k} {N \choose i}$$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other notable properties? Any literature references?
In…
mathy
- 1,258