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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.
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
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 …
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 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
15
votes
Good papers/books/essays about the thought process behind mathematical research
The introduction to Wiles's famous paper on Fermat's Last Theorem (from the Annals in
the mid 1990s) gives an unusually detailed account of the process by which Wiles developed the arguments of the pa …
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
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 …
10
votes
The current status of the Birch & Swinnerton-Dyer Conjecture
Reading Olivier's comment reminded me that it is always possible to verify rigorously, for a given elliptic curve $E$ over $\mathbb Q$ for which the rank of $E({\mathbb Q})$ is at most $3$, that one h …
- 57.6k
13
votes
Co-Objects are better
Dear Martin,
As Harry points out in his comments, in certain settings (e.g. moduli spaces) an object is characterized by maps in. In others (e.g. the free abelian group on one generator), the charac …
2
votes
what notions are "geometric" (characterized by geometric fibers)?
If you have a flat, or even better a smooth morphism, then the intuition is that there is some sort of continuity along the fibres. It is then not unreasonable to make definitions by looking at fibre …
- 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
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
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 …
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