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1500 questions
74
votes
14 answers
How to write popular mathematics well?
Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead of giving up in the face of a Very Hard Problem,…
74
votes
51 answers
An example of a beautiful proof that would be accessible at the high school level?
The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about beauty in the teaching of school mathematics.
I'm…
Manya
- 339
74
votes
29 answers
(Preferably rare) Audio/Video recordings of famous mathematicians?
Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not sure if it is really Hilbert's voice in the link,…
Koundinya Vajjha
- 829
74
votes
11 answers
Why hasn't mereology succeeded as an alternative to set theory?
I have recently run into this Wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set theory, which is founded on the idea of set…
godelian
- 5,682
74
votes
9 answers
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things…
Aaron Mazel-Gee
- 6,029
74
votes
15 answers
$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential
The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://scottaaronson.com/blog/?p=263
The growth rate of the function…
Gil Kalai
- 24,218
74
votes
9 answers
Motivating the Laplace transform definition
In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, for $t \geq 0$ is defined by $\int_{0}^{\infty}…
elaichi
- 883
74
votes
17 answers
Facts from algebraic geometry that are useful to non-algebraic geometers
A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic information about Galois theory and algebraic…
Charles Staats
- 7,218
74
votes
29 answers
Proofs where higher dimension or cardinality actually enabled much simpler proof?
I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or cardinality than the actual theorem.
Specific…
Claus
- 6,777
74
votes
7 answers
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion.
Background
I think I understand the basic idea…
Theo Johnson-Freyd
- 52,873
74
votes
8 answers
Succinctly naming big numbers: ZFC versus Busy-Beaver
Years ago, I wrote an essay called Who Can Name the Bigger Number?, which posed the following challenge:
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number---not an infinity---on a blank index…
Scott Aaronson
- 9,743
74
votes
4 answers
Fake integers for which the Riemann hypothesis fails?
This question is partly inspired by David Stork's recent question about the enigmatic complexity of number theory. Are there algebraic systems which are similar enough to the integers that one can formulate a "Riemann hypothesis" but for which the…
Timothy Chow
- 78,129
74
votes
7 answers
A learning roadmap for Representation Theory
As Akhil had great success with his question, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enough to care about things like…
Charles Siegel
- 15,805
74
votes
1 answer
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ask if $A \cong A + Z + Z \Longrightarrow A \cong…
Martin Brandenburg
- 61,443
74
votes
22 answers
Essays and thoughts on mathematics
Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written form.
For instance:
W. Thurston wrote the lovely…
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