Linked Questions
16 questions linked to/from Trichotomies in mathematics
333
votes
34
answers
96k
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Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
256
votes
16
answers
71k
views
Why worry about the axiom of choice?
As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...
89
votes
9
answers
13k
views
Why should I believe the Mordell Conjecture?
It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and ...
- 2,827
33
votes
10
answers
6k
views
Is the empty graph a tree?
This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The ...
- 3,463
34
votes
5
answers
4k
views
The origin(s) of the word "elliptic"
The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ...
- 11.9k
44
votes
5
answers
5k
views
Groups, quantum groups and (fill in the blank)
In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
- 85.5k
27
votes
8
answers
2k
views
Examples of sequences whose asymptotics can't be described by elementary functions
It is somewhat miraculous to me that even very complicated sequences $a_n$ which arise in various areas of mathematics have the property that there exists an elementary function $f(n)$ such that $a_n =...
22
votes
4
answers
3k
views
What is the virtue of profinite groups as mathematical objects?
In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite ...
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.3k
33
votes
4
answers
4k
views
Can the difference of two distinct Fibonacci numbers be a square infinitely often?
Can the difference of two distinct Fibonacci numbers be a square infinitely often?
There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and $F_{13}-F_{11}=12^2$...
- 25.4k
44
votes
2
answers
3k
views
Why can't we take three loops?
Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:
No ...
- 118k
8
votes
1
answer
2k
views
How "frequent" are smooth projective varieties with (anti-)ample canonical bundle?
How "frequent" are smooth projective varieties $X$ with (anti-)ample canonical bundle $\omega_X = \bigwedge^d \Omega^1_{X/k}$?
E.g. for curves $C/k$, the canonical bundle is ample iff the genus $g(C) ...
user19475
8
votes
3
answers
2k
views
Rational exponential expressions
Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...
- 2,283
17
votes
2
answers
1k
views
Examples of triality in mathematics
There are tons of interesting examples of duality in mathematics (Poincaré duality, Verdier duality, Stone duality, s-duality, Tannaka duality, Koszul duality, Spanier-Whitehead duality ... ). What ...
7
votes
4
answers
772
views
Trig functions based on convex curves
Pardon my naivety, but I wonder if
much use has been found for
trigonometric functions
defined in terms of a centrally symmetric convex curve $K$ replacing
the circle $C$.
For example, here is the ...
- 151k