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1500 questions
138
votes
17 answers
What makes four dimensions special?
Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds special?
What makes four-dimensional Lorentzian…
Cristi Stoica
- 4,304
137
votes
7 answers
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after reflection and some research,
I find little support…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
137
votes
2 answers
Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros
Very recently, Yitang Zhang just gave a (virtual) talk about his work on Landau-Siegel zeros at Shandong University on the 5th of November's morning in China. He will also give a talk on 8th November at Peking University.
The 111-page preprint now…
Blanco
- 1,503
- 2
- 9
- 12
137
votes
28 answers
Which mathematical definitions should be formalised in Lean?
The question.
Which mathematical objects would you like to see formally defined in the Lean Theorem Prover?
Examples.
In the current stable version of the Lean Theorem Prover, topological groups have been done, schemes have been done, Noetherian…
Kevin Buzzard
- 40,559
136
votes
9 answers
Is there an underlying explanation for the magical powers of the Schwarzian derivative?
Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} \Big(\frac{f''}{f'}\Big)^2$
Here is a somewhat more conceptual definition, which justifies the…
Paul Siegel
- 28,772
136
votes
14 answers
Careers advice for Ph.D.s without current postdocs or university jobs
Hi,
I'm sure I'm not the only Ph.D. mathematician on MO in serious need of career advice. I'm sure there will be other readers in similar situations, who will find any good advice very helpful. Can anyone suggest anything? Honest, serious answers…
Zen Harper
- 1,930
136
votes
15 answers
Statistics for mathematicians
I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of statistics.
Most texts I've come across are…
Nokker
- 31
135
votes
26 answers
What are some famous rejections of correct mathematics?
Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner
Theorem: There are only a finite
number of imaginary quadratic fields
that have unique factorization. They
are $\sqrt{d}$ for $d \in…
Steve Huntsman
- 15,258
135
votes
43 answers
What are the most attractive Turing undecidable problems in mathematics?
What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on the Wikipedia page.
Standard community wiki…
Joel David Hamkins
- 224,022
134
votes
69 answers
Mathematical "urban legends"
When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee, and noted that the class of topological spaces…
Igor Rivin
- 95,560
133
votes
5 answers
Does the inverse function theorem hold for everywhere differentiable maps?
(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each point $x_0 \in {\bf R}^n$, the derivative…
Terry Tao
- 108,865
- 31
- 432
- 517
133
votes
6 answers
what mistakes did the Italian algebraic geometers actually make?
It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations (comm alg etc) were only just being laid, and…
Kevin Buzzard
- 40,559
132
votes
3 answers
When is the tensor product of two fields a field?
Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by some particular case of the following…
Georges Elencwajg
- 46,833
131
votes
22 answers
Books that teach other subjects, written for a mathematician
Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read?
Or say I want to learn about Einstein's theory of relativity, but I don't even know much basic physics. What sources should I read?
I am…
Josh
- 501
129
votes
74 answers
Most helpful math resources on the web
What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let the votes decide which are the best!
vonjd
- 5,875