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Results tagged with soft-question
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user 2874
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.
88
votes
How to escape the inclination to be a universalist or: How to learn to stop worrying and do ...
I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from te …
- 4,713
28
votes
Why do so many textbooks have so much technical detail and so little enlightenment?
I believe that normal subgroups were first defined in the context of Galois theory (in particular, normal field extensions), by Galois. If one wants to abstract the situation slightly and see what ki …
- 5,670
28
votes
Accepted
Consequences of the Langlands program
There are many, many consequences of the general Langlands program (which I'll interpret to mean both functoriality for automorphic forms and reciprocity between Galois representations and automorphic …
- 57.6k
173
votes
Mathematical habits of thought and action which would be of use to non-mathematicians
In my experience, mathematicians will frequently argue (in general, not just in mathematics) by passing to an extreme case at the beginning. Non-mathematicians (again in my experience) sometimes ob …
- 57.6k
39
votes
Accepted
how to use arxiv?
My comments above formulated as an answer:
People typically post a preprint on the arxiv at the same time that they post it on their own homepage, with the goal of disseminating their work to their c …
- 57.6k
11
votes
Accepted
Characterization of algebraic points on Shimura varieties?
If you haven't, you should first think about these questions just for modular curves, which are the simplest Shimura varieties. Then there are only finitely many $N$ for which the modular curve of le …
- 57.6k
97
votes
Accepted
What notions are used but not clearly defined in modern mathematics?
One of the most important contemporary mathematical concepts without a rigorous definition is
quantum field theory (and related concepts, such as Feynman path integrals).
Note: As noted in the com …
- 57.6k
16
votes
Should there be a specified standard knowledge of mathematicians?
Many (most? all?) North American graduate programs have some form of qualifying exam (which goes by different names at different institutions) whose goal is to establish a baseline knowledge of the ki …
- 57.6k
8
votes
Can breadth hurt a job candidate?
Just to reiterate what has been mentioned in some of the other answers: the standard way that complications in an application (e.g. explaining TCS work/publication criteria/other issues to a hiring co …
- 57.6k
1
vote
Accepted
Removing a hypersurface when applying the Representation theorem to prove Positivstellensatz...
The real point of $g = 0$ are the empty set, but $g = 0$ is still a non-trivial hypersurface over $\mathbb R$, just with no real points. (The fact that is has no real points is the reason why the rea …
- 57.6k
6
votes
Varieties as an introduction to algebraic geometry / How do professional algebraic geometers...
I stand by my answer to the question that you linked. In particular, I think that the distinction between "classical" and "modern" algebraic geometry is a little artificial, and I don't think that a …
CommunityBot
- 1
8
votes
Is Galois theory necessary (in a basic graduate algebra course)?
First, my perspective: at my institution, we teach two streams of undergrad algebra, a standard stream, and an honours-type stream. Both cover some Galois theory, certainly with more being done in th …
- 57.6k
53
votes
How many people fully understand the proof of Fermat's Last Theorem?
Dear Michael,
The methods introduced by Wiles, and by Taylor and Wiles, in the two papers that proved FLT, as well as the methods introduced by Ribet in his earlier paper reducing FLT to Shimura--Ta …
- 57.6k
33
votes
Tools for the Langlands Program?
This answer deals with the classical Langlands program (if you like, the Langlands program
for number fields).
There are (at least) two aspects to this program:
(a) functoriality: this is Langlands …
- 57.6k
17
votes
Why are proofs so valuable, although we do not know that our axiom system is consistent?
To address the issue of Fermat's Last Theorem: the reasoning behind Fermat's Last Theorem,
while elaborate, in the end rests on basic intuition about the integers. (I'm not sure that
it is actually p …
- 57.6k