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Results tagged with soft-question
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user 25510
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
106
votes
Nontrivially fillable gaps in published proofs of major theorems
In 1970, I. N. Baker published a proof of a basic result in holomorphic dynamics:
a transcendental entire function cannot have more than one completely invariant domain.
A completely invariant d …
94
votes
Mistakes in mathematics, false illusions about conjectures
The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other t …
79
votes
Mathematical conjectures on which applications depend
An important specific conjecture is that you cannot factor large integers fast. Many security systems for Internet and other transactions, depend on this.
73
votes
Good "casual" advanced math books
What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.
But still you may try these books:
Michio Kuga, Galois' dream, …
67
votes
Lost soul: loneliness in pursing math. Advice needed.
One famous mathematician answered to all such questions from young people: No, you should not do math.
You will do it successfully only in the case that you cannot do anything else.
In which case you …
62
votes
Accepted
Is spherical trigonometry a dead research area?
It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geo …
- 91.8k
61
votes
Is St. Petersburg a good place for the 2022 Int. Congress of Mathematicians
The real question is not about the Congress. There will be no ICM in Petersburg. Maximum what they can stage is a Russian-Chinese conference. Which decent Russians and Chinese will not attend anyway.
…
57
votes
Do you know important theorems that remain unknown?
Here is one little-known and one completely unknown result.
The little known result is the Mean Motion Theorem. This says that for all real numbers $\lambda_j$
and all complex numbers $a_j$ the foll …
56
votes
How can a mathematician handle the pressure to discover something new?
I can share my own experience: similar considerations always disturbed me too.
To deal with this, I always considered mathematics research as a hobby. From the very beginning I thought that my main jo …
51
votes
Accepted
Most intriguing mathematical epigraphs
The most interesting epigraphs I have seen in mathematical books are in:
Bender and Orszag, Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory. …
50
votes
Is it possible to have a research career while checking the proof of every theorem that you ...
Possible or not, this should be a goal:-) Let me put it slightly differently: you should understand every result that you use. First of all, a theorem that you use
can be wrong. So whenever you rely y …
48
votes
What are examples of (collections of) papers which "close" a field?
Index theorem of Atiyah and Singer closed a substantial field of research in the 1960s. I knew people who were working in this field, and had to switch the field of their research
completely.
A more …
47
votes
What areas of pure mathematics research are best for a post-PhD transition to industry?
Probability. A strong background in probability will permit to qualify for jobs in
statistics and financial math. See AMS Notices where a lot of statistics about the
employment of recent PhD is publi …
47
votes
When is one 'ready' to make original contributions to mathematics?
Mathematics is not learned by reading books. One becomes a research mathematician by solving problems. Most people need an adviser to recommend a good problem. Then you start thinking and reading what …
47
votes
Trichotomies in mathematics
Almost everything in mathematics, indeed, can be "hyperbolic", "parabolic" or "elliptic".
Like PDE's, Riemann surfaces, or manifolds of higher dimension, fractional-linear transformations, fixed point …