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Results tagged with big-list
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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.
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 $k((x,y)) = k((x))((y))$ …
85
votes
What are the most misleading alternate definitions in taught mathematics?
I increasingly abhor the introduction of the finite ring $Z_n$ not as $\mathbb{Z}/n\mathbb{Z}$ but as the set $\{0,\ldots,n-1\}$ with "clock arithmetic". (I understand that if you want to introduce m …
55
votes
Fundamental Examples
The Fermat Equation xn + yn - zn = 0.
This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorizati …
52
votes
Most memorable titles
Finding composite order ordinary elliptic curves using the Cocks-Pinch method, by D. Boneh, K. Rubin and A. Silverberg. (To appear in the Journal of Number Theory.)
51
votes
Famous mathematical quotes
Do not ask whether a statement is true until you know what it means. -- Errett Bishop
48
votes
Top specialized journals
The following is my personal (i.e., includes all of my mathematical prejudices) ranked list of subject area journals in number theory.
From best to worst:
1) Algebra and Number Theory
2) Internatio …
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.
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 …
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 …
41
votes
How to present mathematics to non-mathematicians?
For some reason, many mathematicians have trouble with the idea that when some layman asks them about their work, the appropriate response is not to try to figure out how to describe the latest theore …
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 …
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
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 …
27
votes
Can a mathematical definition be wrong?
I think there are many examples, spread out across a continuum of how "wrong" the definition really was. Of course, strictly speaking a definition cannot be "wrong", or can only be wrong in the logic …
24
votes
Accepted
Cryptomorphisms
The phenomenon that I think you have in mind has a name: cryptomorphism. I learned the name from the writings of Gian-Carlo Rota; Rota's favorite example was indeed matroids. Gerald Edgar informs me …