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1500 questions
107
votes
36 answers
Interesting examples of vacuous / void entities
I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty set is a non-empty set):
"Perhaps as a result of…
Michael Hardy
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107
votes
10 answers
What is the oldest open problem in mathematics?
What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated.
Browsing Wikipedia list of open problems, it seems that the Goldbach conjecture (1742, every even integer greater than 2 is the sum of two…
coudy
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107
votes
8 answers
What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The Atiyah-Singer index theorem can be proven using heat…
Qiaochu Yuan
- 114,941
106
votes
2 answers
What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?
Atiyah states that this function…
Keshav Srinivasan
- 4,529
106
votes
26 answers
Fields of mathematics that were dormant for a long time until someone revitalized them
I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go).
Can people name examples of fields of mathematics that were once very active, then fell dormant for a while…
Andy Putman
- 43,430
106
votes
7 answers
What is the field with one element?
I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-dimensional, namely $\mathop{\text{Spec}}\mathbb…
Benjamin Antieau
- 4,982
106
votes
32 answers
Special rational numbers that appear as answers to natural questions
Motivation:
Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. Many more such numbers are described for example…
Dan Romik
- 2,480
106
votes
15 answers
Most striking applications of category theory?
What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
Joyal's Combinatorial Species
Grothendieck's Galois…
muad
- 1,402
106
votes
9 answers
solving $f(f(x))=g(x)$
This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a bounded interval. [EDIT: actually he can do rather…
Kevin Buzzard
- 40,559
106
votes
0 answers
A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't mention the characterization, then it's not an…
Martin Brandenburg
- 61,443
106
votes
10 answers
Analogues of P vs. NP in the history of mathematics
Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P from the NP-complete ones, and that this phenomenon…
Scott Aaronson
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105
votes
3 answers
Has the Lie group E8 really been detected experimentally?
A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum Criticality in an Ising Chain: Experimental Evidence…
Richard Borcherds
- 20,442
105
votes
5 answers
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
…
T. Amdeberhan
- 41,802
104
votes
10 answers
"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when they say this? The Kronecker-Weber theorem gives…
Jonah Sinick
- 6,942
104
votes
11 answers
What is the exterior derivative intuitively?
Actually I have several related questions, not worth opening different threads:
What is the exterior derivative intuitively? What is its geometric meaning?
A possible answer I know is, that it is dual to the boundary operator of singular homology.…
Jan Weidner
- 12,846