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1500 questions
62
votes
7 answers
Euler-Maclaurin formula and Riemann-Roch
Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = 1- e^{-D}$ or, by taking the inverses,
$$…
VA.
- 12,929
62
votes
1 answer
Normalizers in symmetric groups
Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$.
Background: If true, this would for instance give…
Peter Mueller
- 20,764
61
votes
7 answers
Are higher categories useful?
Of course, personally, I think the answer is a big Yes!
However once, a while ago, while giving a talk about higher category theory, I was asked a question about whether higher category theory was useful outside of the realm of higher category…
Chris Schommer-Pries
- 27,144
61
votes
3 answers
What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good resources for this problem that gives some…
Johan Öinert
- 1,035
61
votes
11 answers
Non-commutative algebraic geometry
Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which parts don't?
Edit: I also welcome any…
Kevin H. Lin
- 20,738
61
votes
1 answer
Double affine Hecke algebras and mainstream mathematics
This is something of a followup to the question "Kapranov's analogies", where a connection between Cherednik's double affine Hecke algebras (DAHA's) and Geometric Langlands program was mentioned.
I am interested to hear about known connections of…
David Jordan
- 6,053
61
votes
4 answers
Hirzebruch's motivation of the Todd class
In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, the key relation $f(x)$ must satisfy is that…
Dan Kneezel
- 1,405
- 16
- 13
61
votes
3 answers
Why do filtered colimits commute with finite limits?
It's not hard to show that this is true in the category Set, and proofs have been written down in many places. But all the ones I know are a bit fiddly.
Question 1: is there a soft proof of this fact?
For example, here's a soft proof of the fact…
Steve Lack
- 3,081
61
votes
41 answers
Most intricate and most beautiful structures in mathematics
In the December 2010 issue of Scientific American, an article "A Geometric Theory of
Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most
intricate structure known to mathematics, the exceptional Lie group E8."…
Richard Stanley
- 49,238
61
votes
6 answers
Discovered Phd topic has already been worked on
I am a second-year French Ph.D. student and two days ago I found out the topic I had been working on has already been studied, and the result I wanted to prove is basically already known. Unfortunately, neither I nor the supervisor were aware of…
M.S.L.
- 561
61
votes
9 answers
Arguments against large cardinals
I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally regular, is unprovable in ZFC. Nevertheless large…
user8996
- 765
61
votes
7 answers
Is there a measure zero set which isn't meagre?
A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set is meagre. The Cantor set is nowhere dense, so it's…
Anton Geraschenko
- 23,718
61
votes
9 answers
There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? If not, is it because:
It is not as interesting…
Ola Sande
- 625
61
votes
11 answers
Why certain diophantine equations are interesting (and others are not) ?
It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mathematical generalizations thereof, others come…
Qfwfq
- 22,715
61
votes
9 answers
Techniques for debugging proofs
After writing many proofs, most of which contained errors in their initial form, I have developed some simple techniques for "debugging" my proofs. Of course, a good way to detect errors in proofs is to send them to a colleague for review. But even…
Erel Segal-Halevi
- 3,585