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1500 questions
86
votes
7 answers
How many orders of infinity are there?
Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions. Let's say that one growth function $F$ dominates…
Terry Tao
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- 517
86
votes
27 answers
Which popular games have been studied mathematically?
I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with this, so I'm writing to ask for some help. I…
David White
- 29,779
86
votes
38 answers
Books about history of recent mathematics
I draw on this question to ask something that has always been a pet peeve of mine. It is very easy to find books about the history of mathematics, much less so if one wants books about the recent (say > 1850) one.
Of course I know that this is…
Andrea Ferretti
- 14,454
- 13
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- 111
86
votes
33 answers
Parodies of abstruse mathematical writing
Perhaps under the influence of a recent question
on perverse sheaves,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
that I can no longer remember in detail nor find…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
86
votes
11 answers
Is there a complex structure on the 6-sphere?
I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a number of published proofs that are not taken…
Deane Yang
- 26,941
86
votes
2 answers
History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$
Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is equivalent to
$$
\theta =…
Francois Ziegler
- 29,500
86
votes
1 answer
Are there non-scalar endomorphisms of the functor $V\mapsto V^{**}/V$?
Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but got no answer.
EDIT (Apr 15'14). Here is a…
Pierre-Yves Gaillard
- 4,336
86
votes
16 answers
Teaching homology via everyday examples
What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory?
To be more precise, I am teaching a short course on homology, from chapter two of Hatcher's book. Before diving into the…
Sam Nead
- 26,191
85
votes
11 answers
What is Quantization ?
I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we quantize a function?, a set?, a theorem?, a…
85
votes
4 answers
Etale cohomology -- Why study it?
I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using the machinery of etale cohomology. I know a…
Joel Dodge
- 2,779
85
votes
7 answers
How many mathematicians are there?
Although we are not so numerous as other respected professionals, like for example lawyers, I wonder if we could come up with a reasonable estimate of our population.
Needless to say, the question more or less amounts to the definition…
Georges Elencwajg
- 46,833
85
votes
9 answers
Demystifying the Caratheodory approach to measurability
Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all sets $A$ satisfying $m^* (S)=m^* (S\cap A)+m^*…
Michael Greinecker
- 12,555
85
votes
4 answers
The enigmatic complexity of number theory
One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, relations and problems--some of which can be…
David G. Stork
- 2,518
85
votes
12 answers
Why is the gradient normal?
This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I have read. Quoting Wikipedia, the theorem is:
The…
Kim Greene
- 3,583
- 10
- 42
- 41
85
votes
6 answers
Are there any serious investigations of whether "mathematicians do their best work when they're young"?
There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious (preferably academic rather than journalistic) research…
