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Graph Theory, unlike number theory, is a strangely shallow subject. By this I mean that a lot of the results don’t require a large number of previous results (t

by moomin 9y ago

Graph Theory, unlike number theory, is a strangely shallow subject. By this I mean that a lot of the results don’t require a large number of previous results (there’s definitely some). Compare this to number theory, where every interesting extant problem appears to require ten years study. (I realise people may infer a value judgement from shallow/deep, but none is intended.)

deleted 9y ago
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vinchuco 9y ago
Ultimately, naming is pointing at clouds, but number theory IS older: https://books.google.com/ngrams/graph?content=graph+theory%2Cnumber+theory%2Clinear+algebra&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cgraph%20theory%3B%2Cc0%3B.t1%3B%2Cnumber%20theory%3B%2Cc0%3B.t1%3B%2Clinear%20algebra%3B%2Cc0 https://books.google.com/ngrams/graph?content=graph+theory%2... Afterthought: I wonder what caused the decline in the 80's (also true for other branches of math)
- Cyph0n 9y ago
  That's around the time public-key crypto was invented. Coincidence?
daniel-levin 9y ago
>> Compare this to number theory, where every interesting extant problem appears to require ten years study. I don't often come here to comment but as someone in progress on an original research masters in number theory I can say this is utter bullshit. I assume your 'interesting' qualification (somehow) excludes obvious candidates like Landau's problems [0]. Some examples. I was taught about the ABC conjecture as an undergrad. You can easily teach the Brun sieve [1] method of working out that the sum of the reciprocal of the twin primes converges. Novel solutions to Diophantine problems are sometimes accessible to undergrads. Richard K. Guy wrote a whole book on unsolved problems in NT, some of which have been solved using undergraduate number theory and someone's upper bound you can just use (as easy as apt-get installing this_dope_bound). You can start reading papers without a PhD, never mind ten years of study. I think it's possible to get an utterly unrepresentative sample of either field by only sticking to "elementary" results. There are some extraordinarily subtle results in graph theory! Conversely, you can find NT problems amenable to elementary techniques [2]. [0] https://en.wikipedia.org/wiki/Landau%27s_problems https://en.wikipedia.org/wiki/Landau%27s_problems [1] https://en.wikipedia.org/wiki/Brun%27s_theorem https://en.wikipedia.org/wiki/Brun%27s_theorem [2] https://en.wikipedia.org/wiki/Quadratic_reciprocity https://en.wikipedia.org/wiki/Quadratic_reciprocity
- moomin 9y ago
  I’ll defer to your superior experience here. Only done fairly introductory number theory and graph theory stuff. That was just the impression I got from what I’d studied.
amai 9y ago
Isn't graph theory basically linear algebra, because every graph can be represented as https://en.wikipedia.org/wiki/Adjacency_matrix https://en.wikipedia.org/wiki/Adjacency_matrix ?

by moomin 9y ago

deleted 9y ago

[deleted]

vinchuco 9y ago

Ultimately, naming is pointing at clouds, but number theory IS older: https://books.google.com/ngrams/graph?content=graph+theory%2Cnumber+theory%2Clinear+algebra&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cgraph%20theory%3B%2Cc0%3B.t1%3B%2Cnumber%20theory%3B%2Cc0%3B.t1%3B%2Clinear%20algebra%3B%2Cc0 https://books.google.com/ngrams/graph?content=graph+theory%2... Afterthought: I wonder what caused the decline in the 80's (also true for other branches of math)

Cyph0n 9y ago
That's around the time public-key crypto was invented. Coincidence?

daniel-levin 9y ago

>> Compare this to number theory, where every interesting extant problem appears to require ten years study. I don't often come here to comment but as someone in progress on an original research masters in number theory I can say this is utter bullshit. I assume your 'interesting' qualification (somehow) excludes obvious candidates like Landau's problems [0]. Some examples. I was taught about the ABC conjecture as an undergrad. You can easily teach the Brun sieve [1] method of working out that the sum of the reciprocal of the twin primes converges. Novel solutions to Diophantine problems are sometimes accessible to undergrads. Richard K. Guy wrote a whole book on unsolved problems in NT, some of which have been solved using undergraduate number theory and someone's upper bound you can just use (as easy as apt-get installing this_dope_bound). You can start reading papers without a PhD, never mind ten years of study. I think it's possible to get an utterly unrepresentative sample of either field by only sticking to "elementary" results. There are some extraordinarily subtle results in graph theory! Conversely, you can find NT problems amenable to elementary techniques [2]. [0] https://en.wikipedia.org/wiki/Landau%27s_problems https://en.wikipedia.org/wiki/Landau%27s_problems [1] https://en.wikipedia.org/wiki/Brun%27s_theorem https://en.wikipedia.org/wiki/Brun%27s_theorem [2] https://en.wikipedia.org/wiki/Quadratic_reciprocity https://en.wikipedia.org/wiki/Quadratic_reciprocity

moomin 9y ago
I’ll defer to your superior experience here. Only done fairly introductory number theory and graph theory stuff. That was just the impression I got from what I’d studied.

amai 9y ago

Isn't graph theory basically linear algebra, because every graph can be represented as https://en.wikipedia.org/wiki/Adjacency_matrix https://en.wikipedia.org/wiki/Adjacency_matrix ?