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1500 questions
104
votes
6 answers
Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a sufficiently short formal proof could count as an…
gowers
- 28,729
103
votes
12 answers
Where are mathematics jobs advertised if not on mathjobs (e.g. in Europe and elsewhere)?
My impression is that in the US, there is a canonical place for finding math jobs, namely mathjobs.org. For those of us who live and apply for jobs elsewhere, life is more complicated, and searching for advertised academic mathematics jobs for…
Andreas Holmstrom
- 5,517
103
votes
5 answers
Independent evidence for the classification of topological 4-manifolds?
Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? The argument there is extraordinarily complicated…
Brendan Guilfoyle
- 2,297
103
votes
13 answers
How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
There is one strong argument against…
Frank Thorne
- 7,199
103
votes
10 answers
What is (co)homology, and how does a beginner gain intuition about it?
This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question would be the name of such a text. (At this…
Qiaochu Yuan
- 114,941
103
votes
15 answers
Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course multidimensional arrays are useful:
every programming…
Joseph O'Rourke
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- 933
103
votes
17 answers
Theorems that are essentially impossible to guess by empirical observation
There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold for all even integers up to $10^{18}$, but which…
BubbleZ
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103
votes
19 answers
When are two proofs of the same theorem really different proofs
Many well-known theorems have lots of "different" proofs. Often new proofs of a theorem arise surprisingly from other branches of mathematics than the theorem itself.
When are two proofs really the same proof? What I mean is this. Suppose two…
Martyguy
- 1,031
103
votes
23 answers
eBook readers for mathematics
For a while I have been eying stand-alone eBook readers that use "electronic ink" displays, the most popular ones seem to be the Amazon Kindle readers.
My criteria are as follows: It should be able to display pdf's and math formulas in them just…
Lars
- 4,400
102
votes
61 answers
Which mathematicians have influenced you the most?
There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems,…
Jose Brox
- 2,962
102
votes
8 answers
When should a result be made into a paper?
I recently posted a short (6 page) note on arXiv, and have more or less decided that I should not submit it to a journal. I could have tacked it onto the end of a previous paper, but I thought it would be somewhat incongruous -- it is an…
Andrew D. King
- 1,814
102
votes
19 answers
Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently published a paper proving that quantum Turing…
Peter Shor
- 6,272
102
votes
3 answers
Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric objects referred to in the title of this…
Sam Hopkins
- 22,785
102
votes
12 answers
What is entropy, really?
I first saw the term "entropy" in a chemistry course while studying thermodynamics.
During my graduate studies I encountered the term in many different areas of mathematics.
Can anyone explain why this term is used and what it means. What I am…
Mustafa Said
- 3,679
102
votes
4 answers
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville :
“The big experts in the field had
already tried to make this approach
work,” Granville said. “He’s not a
known expert, but he succeeded…
pageman
- 1,053