Charles Matthews
  • Member for 11 years, 6 months
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10 answers
61 votes
9k views
Teaching proofs in the era of Google
65 votes

How would you teach anything in an age when the "arcana" or guild secrets had been made public? Well, you would teach. And you would not ask questions that had answers that could be called "answers" ...

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8 answers
35 votes
9k views
What is the proper initiation to the theory of motives for a new student of algebraic geometry?
29 votes

Asking for the moon, in my view. Here are 10 "heuristics" that try to place the theory. NB that many people stop at #1, as if this were enough. None of these points is particularly easy to track in ...

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6 answers
13 votes
4k views
Gauss's views on pure mathematics
26 votes

Quotation from Gauss: "...the greatest thing is purely mathematical thinking: this is worth much more than the application of mathematics." In conversation in 1854, a few months before his death, ...

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2 answers
111 votes
26k views
Why is the Hodge Conjecture so important?
26 votes

Here are three points, and you'd have to care about at least one of them, I think. (1) A (co)homology class is better understood if it is represented geometrically in some way. This point really ...

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6 answers
61 votes
7k views
Origin of terms "flag", "flag manifold", "flag variety"?
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25 votes

Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good ...

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9 answers
23 votes
8k views
How to motivate and present epsilon-delta proofs to undergraduates?
22 votes

Epsilon-delta represents a pair of quantifiers (for all ... there exists ...). Challenge-response. The discrete mathematics take ought to be "hey, this is like a game", because if you iterate the ...

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16 answers
219 votes
52k views
What elementary problems can you solve with schemes?
22 votes

The "classical" example is surely duality of abelian varieties. If you want this duality to work over finite fields (or in characteristic p generally), it becomes apparent that you can't work with ...

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10 answers
67 votes
11k views
When have we lost a body of mathematics because errors were found?
18 votes

I feel the answer is obviously "yes", and indeed that much of 19th century mathematics was lost, in a serious sense, for much of the 20th century. I was struck recently by discovering that Henry Fox ...

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6 answers
15 votes
22k views
How to understand the concept of compact space
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18 votes

Some heuristic remarks are helpful only to a subset of readers. (Maybe that's true of all heuristics, as a meta-heuristic - if everyone accepts a rough explanation, it's something rather more than ...

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3 answers
3 votes
1k views
prime powers between n and 2n
17 votes

Actually there is a power of 2. It goes to show the power of binary arithmetic ... : write 2n in binary and write zeroes after the initial one.

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3 answers
17 votes
5k views
The multiplicative order of 2 modulo primes
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17 votes

The answer is "yes" - the order mod p of 2 is almost always as large as the square root of p (actually you get epsilon less than this in the exponent). If you take r multiplicatively independent ...

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5 answers
85 votes
16k views
What is sheaf cohomology intuitively?
17 votes

I'm going to stick my neck out, rather than recycle some well-known phrases in sheaf theory. I suggest trying to answer this as two possibly simpler questions: 1) What is the intuitive meaning of a ...

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2 answers
17 votes
3k views
Why were Abelian functions so important in the 19th century?
16 votes

Recall C. L. Siegel's rant, about the modern theory of abelian functions not having any functions in it. From a point of view that would have made sense to Weierstrass, mathematics has "addition ...

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8 answers
19 votes
5k views
To what extent is it true that "number theory = mathematics"?
16 votes

I just don't think it's true, despite my own tastes in topics. Such formulations are substantially a matter of fashion. There is one basic axis, running from very detailed information at one end (...

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13 answers
20 votes
6k views
Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)?
15 votes

Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables by C. L. Siegel is a standard reference using complex function theory. There are older works (e.g. H. F. ...

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4 answers
-3 votes
4k views
What is the situation with Hilbert's Fifth Problem?
14 votes

http://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem is a decent survey. In general in the discussion of "status" of the Hilbert problems, there are at least two recognisable routes. Route A is the ...

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4 answers
28 votes
4k views
Gossip about Grothendieck and distributive lattices
14 votes

It's a tendentious question, certainly. It might mean, if Bourbaki, let us say, had had more of an interest in lattice theory, that the French word for "lattice" of this kind would be more familiar at ...

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17 answers
19 votes
6k views
What is your favorite isomorphism?
14 votes

I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with ...

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2 answers
7 votes
3k views
4900, a particularly square number
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14 votes

This is a classical Diophantine equation (Mordell, Diophantine Equations, p. 258). Apart from n = 0, 1, -1, there is only the solution n = 24. Proofs by G. N. Watson (1919), W. Ljunggren (1952). ...

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4 answers
13 votes
5k views
Au revoir, law of excluded middle?
14 votes

I don't know whether this will be helpful, but here goes. There used to be things called the "Laws of Thought", and they used to be equated (tendentiously) with sort-of axioms for rationality, when "...

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6 answers
37 votes
4k views
Elementary examples of the Weil conjectures
13 votes

The grandfather of all examples is by Gauss: http://en.wikipedia.org/wiki/Weil_conjectures#Background_and_history Of course Gauss didn't mention finite fields other than the prime field. I think it ...

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1 answers
1 votes
1k views
Ado's Theorem Proof
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13 votes

Your argument fails because the bracket can (sometimes) take pairs of elements into the centre. Therefore the direct sum as vector spaces isn't necessarily a direct sum of Lie algebras. For nilpotent ...

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1 answers
6 votes
760 views
When is a ring the ring of adeles of some global field
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13 votes

Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ...

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6 answers
19 votes
2k views
Theorems which say "such and such method cannot possibly prove FLT"
13 votes

The motivation "because I'd like to know [...] if it is impossible to prove FLT using elementary methods" seems to require comment. It is much more likely (in my view) that it is true that FLT can be ...

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2 answers
6 votes
637 views
To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?
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12 votes

There are perhaps three or four themes lurking in here. NB that "undergraduate linear algebra" is perhaps an artificial construct, and examples set to test whether students understand basic concepts ...

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8 answers
42 votes
19k views
Approaches to Riemann hypothesis using methods outside number theory
12 votes

I think, tautologously, any method proving the Riemann Hypothesis (or even seriously improving our knowledge on the zeroes) becomes "number theory" immediately. That said, I know what the question ...

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7 answers
17 votes
6k views
Grothendieck on Topological Vector Spaces
11 votes

It seems clear enough to me that Grothendieck was (perhaps is) sui generis as a mathematician, something that can be said of a few other mathematicians in each of the 19th and 20th centuries (e.g. ...

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2 answers
1 votes
1k views
Is Algebraic Geometry really natural?
11 votes

Algebraic geometry has been an important part of mathematics since Descartes, who pretty much invented it. In other words it is part of 17th century mathematics, like calculus. It happens that there ...

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5 answers
34 votes
4k views
What is the general geometric interpretation of modules in algebraic geometry?
11 votes

As you say, projective modules correspond in a highly moral way to vector bundles. Bundles pull back but do not push forward, in topologists' terms. This might be a good point at which to start. ...

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2 answers
2 votes
1k views
Solve in positive integers $n!=m^2$
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11 votes

Bertrand's postulate (http://en.wikipedia.org/wiki/Bertrand%27s_postulate).

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