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1500 questions
68
votes
7 answers
When is one 'ready' to make original contributions to mathematics?
This is quite a philosophical, soft question which can be moved if necessary.
So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an enjoyable and rewarding experience. However, I have…
Hollis Williams
- 4,958
68
votes
6 answers
The logic of Buddha: a formal approach
Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features of such a non-standard logic in his…
68
votes
1 answer
Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime factorizations:
$$
\begin{split}
n_1 &= 3,\\
n_2 &= 2^2 + 5 =…
Max Alekseyev
- 30,425
68
votes
3 answers
Derived algebraic geometry: how to reach research level math?
I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.
My goal is to study derived algebraic geometry, where derived schemes are built out of simplicial commutative…
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68
votes
2 answers
Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in $\mathbb{R}^2$ which contains the image…
Abcd
- 629
68
votes
5 answers
Mathematics of path integral: state of the art
I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I…
Qfwfq
- 22,715
68
votes
3 answers
What is the status of the Gauss Circle Problem?
For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) \sim \pi r^2$ as $r \rightarrow \infty$. The Gauss…
Pete L. Clark
- 64,763
68
votes
12 answers
Algebraic topology beyond the basics: any texts bridging the gap?
Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts - both geometric ones like Allen Hatcher's and…
68
votes
7 answers
Can anyone give me a good example of two interestingly different ordinary cohomology theories?
An answer to the following question would clarify my understanding of what a cohomology theory is. I know it's something that satisfies the Eilenberg-Steenrod axioms, and I know that those axioms allow you to work out quite a lot. But what sort of…
gowers
- 28,729
68
votes
49 answers
Which mathematical ideas have done most to change history?
I'm planning a course for the general public with the general theme of "Mathematical ideas that have changed history" and I would welcome people's opinions on this topic. What do you think have been the most influential mathematical ideas in terms…
JCollins
- 111
68
votes
3 answers
Yitang Zhang's 2007 preprint on Landau–Siegel zeros
The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint On the Landau-Siegel zeros conjecture? If this result is correct, then (in my opinion) it is even bigger news for…
GH from MO
- 98,751
68
votes
19 answers
Mathematicians whose works were criticized by contemporaries but became widely accepted later
Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's theory of transfinite numbers was originally…
Nilotpal Kanti Sinha
- 4,638
68
votes
20 answers
Fun applications of representations of finite groups
Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not know much about what is a repersentation (say…
Dmitri Panov
- 28,757
- 4
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67
votes
4 answers
Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first published in 1991 (the geometric quantization…
John Pardon
- 18,326
67
votes
11 answers
What is significant about the half-sum of positive roots?
I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra (say over $\mathbb{C}$) and $\mathfrak{h} \subset…
Justin Campbell
- 3,535