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1500 questions
50
votes
8 answers
Problems known to be in both NP and coNP, but not known to be in P
One such problem I know is integer factorization.
What are other interesting cases?
falagar
- 2,761
50
votes
15 answers
Strengthening the induction hypothesis
Suppose you are trying to prove result $X$ by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result $X'$ (that you don't really care about) by induction. This has a chance of succeeding because you have more…
Tony Huynh
- 31,500
50
votes
8 answers
Examples of Mathematical Papers that Contain a Kind of Research Report
What are examples of well received mathematical papers in which the author provides detail on how a surprising solution to a problem has been found.
I am especially looking for papers that also document the dead ends of investigation, i.e. ideas…
Manfred Weis
- 12,594
- 4
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- 71
50
votes
5 answers
Motivated account of the prime number theorem and related topics
Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often see assertions of things like "the prime number…
Sarah
- 482
50
votes
23 answers
Great mathematical figures and/or diagrams?
Most math papers have few figures, if any, although sometimes a well-chosen figure can be a tremendous help in understanding mathematical concepts. Does anyone have any examples of notable uses of figures in mathematical writing and/or texts that…
Elisha Peterson
- 590
50
votes
5 answers
The unification of Mathematics via Topos Theory
In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is this an exaggeration, and if not is this a new idea or has…
Roy Maclean
- 1,140
50
votes
5 answers
What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")?
While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into the notion of dual Coxeter number but am…
Jim Humphreys
- 52,369
50
votes
0 answers
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in particular on the forgetful map from topological complex…
David C
- 9,792
50
votes
16 answers
Equality vs. isomorphism vs. specific isomorphism
This question prompted a reformulation:
What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?
I believe this to be a serious question because it actually is oftentimes a good idea casually to…
Minhyong Kim
- 13,471
50
votes
6 answers
Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?
Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that
$$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$
for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the same as the roots of $f^{(n)}(x)$ and, by repeated…
David E Speyer
- 150,821
50
votes
4 answers
What is a symplectic form intuitively?
Hi,
to completely describe a classical mechanical system, you need to do three things:
-Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system.
-Specify a hamilton function $H:X\rightarrow…
Jan Weidner
- 12,846
50
votes
48 answers
Describe a topic in one sentence.
When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good high-level view (sometimes!) What I'm looking for are…
Gabe Cunningham
- 1,861
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- 26
- 31
50
votes
4 answers
What algorithm in algebraic geometry should I work on implementing?
This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I want to learn how to program from them, so we were…
Steven Gubkin
- 11,945
50
votes
7 answers
Why are local systems and representations of the fundamental group equivalent
My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally isomorphic to a constant sheaf. Does anyone know of a…
bavajee
- 1,147
50
votes
3 answers
How did "normal" come to mean "perpendicular"?
How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to see was that it isn't any stranger than "right"…
Michał Masny
- 1,425