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1500 questions
55
votes
7 answers
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have…
Amitesh Datta
- 1,983
55
votes
3 answers
Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?
For a finite group G, let |G| denote the order of G and write $D(G) = \sum_{N \triangleleft G} |N|$, the sum of the orders of the normal subgroups. I would like to call G "perfect" if D(G) = 2|G|, since then the cyclic group of order n is perfect…
Tom Leinster
- 27,167
55
votes
14 answers
Does any research mathematics involve solving functional equations?
This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those given in this handout. While I can see the…
Qiaochu Yuan
- 114,941
55
votes
2 answers
What is the state of our ignorance about the normality of pi?
Famously, it is not known whether $\pi$ is a normal number. Indeed, there are far weaker statements that are not known, such as the statement that there are infinitely many 7s in the decimal expansion of $\pi$. I'd like to have some idea of where…
gowers
- 28,729
55
votes
7 answers
On referee-author communications
Every time I referee a paper, I dream of a system which would allow me to ask the author a question without troubling the editors. It would save time for everyone involved, most importanly the referee; in my case it would probably half the…
Igor Belegradek
- 28,360
55
votes
8 answers
Applications of Grothendieck-Riemann-Roch?
I am currently trying to learn a bit about Grothendieck-Riemann-Roch...
To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \to Y$ where $Y$ is not a point. I already I know of…
Kevin H. Lin
- 20,738
55
votes
19 answers
Memorizing theorems
I always have trouble memorizing theorems. Does anybody have any good tips?
Kim Greene
- 3,583
- 10
- 42
- 41
55
votes
9 answers
Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. After a closer look at his proof I found that,…
Hvjurthuk
- 573
55
votes
6 answers
(How) is category theory actually useful in actual physics?
An answer to a recent question motivated the following question:
(how) is category theory actually
useful in actual physics?
By "actual physics" I mean to refer to areas where the underlying theoretical principle has solid if not conclusive…
Steve Huntsman
- 15,258
55
votes
8 answers
Is there a Whitney Embedding Theorem for non-smooth manifolds?
For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can every topological manifold embed continuously into…
Jake
- 795
55
votes
0 answers
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic progression (i.e., such that for any two distinct…
Sebastien Palcoux
- 25,922
55
votes
5 answers
How do we know that Fermat wrote his famous note in 1637?
It is widely stated that Fermat wrote his famous note on sums of powers ("Fermat's last theorem") in, or around, 1637. How do we know the date, if the note was only discovered after his death, in 1665?
My interest in this stems from the fact that if…
Angelo
- 26,786
55
votes
3 answers
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers, but what is known about the higher homotopy groups?
Jonathan Wise
- 7,754
55
votes
3 answers
Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and only if the links can be related by a sequence of…
algori
- 23,231
55
votes
6 answers
Is it possible to partition $\mathbb R^3$ into unit circles?
Is it possible to partition $\mathbb R^3$ into unit circles?
Zarathustra
- 1,404