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user 766
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.
9
votes
What is the most useful non-existing object of your field?
Non-trivial approximate subrings of ${\bf R}$ or of ${\bf F}_p$.
The existence of such objects is ruled out by a number of "sum-product theorems", a typical one of which asserts that given a subset $ …
9
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Zagier's one-sentence proof of Fermat's two square theorem (discussed previously on MathOverflow here) seems to qualify.
30
votes
Using slides in math classroom
Slides can, in principle, enhance a lecture, but there is one important difference between slides and blackboard that definitely needs to be kept in mind, and that is that slides are much more transie …
14
votes
How do you present a non-existence theorem?
One important class of "non-existence theorems", which includes Liouville's theorem as a model example, are the various rigidity theorems throughout mathematics that tend to have the general flavour o …
26
votes
Oddities of evenness
The hairy ball theorem is only valid for even-dimensional spheres (or odd-dimensional ambient Euclidean spaces).
Similarly, the strong Huygens principle is only valid in odd-dimensional physical space …
26
votes
Examples of conjectures that were widely believed to be true but later proved false
I believe that Fefferman's disproof in 1971 of the $L^p$ boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner a …
44
votes
Mathematicians who were late learners?-list
According to this Notices article, Raoul Bott was undistinguished in high school, but displayed impressive talent once he reached graduate school (though his thesis was actually in electrical engineer …
27
votes
Most intricate and most beautiful structures in mathematics
The Littlewood-Richardson coefficients. (Or, if one wants a single object, as per the rules of the game: the representation ring of $S_n$ or $GL_n({\bf C})$. Or the ring of symmetric functions. Or …
13
votes
Accepted
Are reduced residue systems relative primorials an active area of research? If not, why not?
The Chinese remainder theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of the finite fields ${\bf Z}/q {\bf Z}$, where $q$ ranges over the pr …
- 114k
7
votes
Accepted
"Universal" differential identities
Let's work in two dimensions for notational simplicity. We claim that there is no non-trivial polynomial identity of the form
$$ P( f, f_x, f_y, f_{xx}, f_{xy}, \dots ) = 0$$
relating some finite num …
- 114k
17
votes
Where is number theory used in the rest of mathematics?
Number theory naturally arises when analysing nonlinear partial differential equations on the torus, basically one wants to understand the extent to which nonlinear resonances between frequencies can …
20
votes
Where is number theory used in the rest of mathematics?
One of the first conditional proofs of the undecidability of Hilbert's tenth problem, by Davis and Putnam in 1959, assumed the existence of arbitrarily long arithmetic progressions of primes, as well …
73
votes
Still Difficult After All These Years
Difficulty is not additive, and measuring the difficulty of proving a single result is not a good measure of the difficulty of understanding the body of work in a given field as a whole.
Suppose for …
58
votes
Why are characters so well-behaved?
The trace is about the strongest general way we have to linearly project a non-abelian situation (matrices) to an abelian situation (scalars): tr(AB)=tr(BA). By using the trace, the representation th …
- 114k
19
votes
When do we study maps into an object or from the object to another object?
In combinatorics, one considers both maps out of a space $X$ (colourings) and maps into a space $X$ (tuples). But there is one key difference between the two: if $X$ has $n$ elements, then the number …