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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.
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 …
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 …
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 …
57
votes
How seriously should a graduate student take teaching evaluations?
This is more of a comment than an answer, but I think it bears saying, based on some of the comments at the top. Anyone who is now a math grad student was not a typical member of an undergrad math cl …
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
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 …
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
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
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 …
27
votes
Accepted
The current status of the Birch & Swinnerton-Dyer Conjecture
The parity conjecture is known, i.e. it is known that if the order of vanishing of the $L$-function is even/odd, then the corank of $p$-Selmer is of the same parity (and I think this is known for ever …
- 57.6k
25
votes
Are there situations when regarding isomorphic objects as identical leads to mistakes?
A slightly more general answer: if the objects in question have no non-trivial automorphisms (i.e. non-identity isomorphisms from itself to itself) then no danger will come from treating isomorphisms …
- 57.6k
22
votes
Why is the concept of topos a "metamorphosis" of the concept of space?
If you have a space, you can consider the category of sheaves of sets on the space; the latter
is a topos (the archetypal example thereof). Since sheaves are (a) very flexible; and (b) highly attuned …
- 57.6k
21
votes
Would Euler's proofs get published in a modern math Journal, especially considering his trea...
My own belief is that contemporary discounting of the validity of Euler's arguments is misplaced. Euler wrote arguments to the standards of his day. If he were writing now, he would write arguments …
- 57.6k
19
votes
Reference letters for graduate school after a couple years in the industry
It may not suit your goals, but one approach is to enroll in a masters program before entering a doctoral program. This could help you get back into the groove of academic life,
and also give you a c …
17
votes
why haven't certain well-researched classes of mathematical object been framed by category t...
It might be worth noting that the problems of computing Feynmann integrals in quantum field theory is one that is traditionally phrased as one of analysis, but is now studied by pure mathematicians us …
- 57.6k