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Results tagged with big-list
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user 1149
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
6
votes
Favourite scholarly books?
I am very fond of Narkiewicz's Elementary and Analytic Theory of Algebraic Numbers. The bibliography is more than 170 pages long. The end of the chapter notes are wonderful. I find Chapter 1 to be …
11
votes
Favourite scholarly books?
Perhaps the archetypical example of a scholarly work in mathematics is L.E. Dickson's three volume History of the Theory of Numbers. The aforelinked wikipedia page puts it rather well:
"The 3-volume …
30
votes
Consolidation: Aftermathematics of fads
Since Quinn's article is a long opinion piece which he says is 90% complete and welcomes comments, it seems entirely appropriate to contact him for clarification on this point. He would probably be h …
23
votes
Elementary / Interesting proofs of the Nullstellensatz
I have been thinking about the question "What is the best -- i.e., some combination of shortest, most natural, easiest -- proof of the Nullstellensatz?" recently on the eve of a commutative algebra co …
47
votes
Most harmful heuristic?
That there is something weird and unsavory about field extensions that are not separable and that serious contemplation of such things should be put off to the indefinite future.
(In fact, much of th …
185
votes
Examples of common false beliefs in mathematics
Here are two things that I have mistakenly believed at various points in my "adult mathematical life":
For a field $k$, we have an equality of formal Laurent series fields $k((x,y)) = k((x))((y))$ …
46
votes
Examples of common false beliefs in mathematics
"A continuous image of a locally compact space is locally compact."
This is tempting because it is true without the "locally"s and it is often the case that topological properties and statements can …
19
votes
Different ways of proving that two sets are equal
If $A$ and $B$ are both finite, it suffices to show:
(i) $A \subseteq B$ and
(ii) $\# A \geq \#B$.
There are variations on this when the sets have more structure e.g.:
If $A$ and $B$ are finite-dim …
5
votes
What out-of-print books would you like to see re-printed?
Associative Algebras, by Richard S. Pierce. Check out the ridiculous Amazon page for this:
http://www.amazon.com/gp/offer-listing/0387906932/ref=sr_1_1_olp?ie=UTF8&qid=1273632391&sr=8-1&condition=ne …
11
votes
Algebraic geometry examples
One example which is (at least over an algebraically closed field) very classical, but contemporary geometers and number theorists do not seem to be as intimately familiar with is the geometry of curv …
20
votes
How should one present curl and divergence in an undergraduate multivariable calculus class?
I have taught multivariable calculus exactly once, to engineering students at Concordia University in Montreal. I found the course to be replete with expository challenges like the one you mention: n …
- 65.4k
32
votes
Examples of theorems with proofs that have dramatically improved over time
I think that Ax's proof of the Chevalley-Warning Theorem qualifies.
The Chevalley-Warning Theorem is an affirmative solution of a conjecture made by L.E. Dickson in 1909 and taken up more seriously b …
47
votes
Which math paper maximizes the ratio (importance)/(length)?
Noam Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561-568.
23
votes
Approaches to Riemann hypothesis using methods outside number theory
So far as I know, there is no approach to the Riemann Hypothesis which has been fleshed out far enough to get an even moderately skeptical expert to back it, with any odds whatsoever. I think this si …
- 65.4k
32
votes
Accepted
Proofs of the Chevalley-Warning Theorem
I am working on a book-length manusript, Around the Chevalley–Warning Theorem. A complete answer to your question is estimated at about 150 pages!
In terms of what exists at the moment, here are two …
- 65.4k