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1500 questions
99
votes
9 answers
Theoretical physics: Why not just R4?
You and I are having a conversation:
"Okay," I say, "I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"Something like that."
"...And then we can add these as local symmetries to space…
Tom Boardman
- 3,190
98
votes
6 answers
Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets and not just plain sets; to make…
Akhil Mathew
- 25,291
98
votes
16 answers
What if Current Foundations of Mathematics are Inconsistent?
The title of the question is also the title of a talk by Vladimir Voevodsky, available here.
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, and viewers! --- Here are three more links:
Question…
Pierre-Yves Gaillard
- 4,336
98
votes
10 answers
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like Waldhausen's) and a lot of work devoted to…
S. Carnahan
- 45,116
98
votes
33 answers
Theorems with unexpected conclusions
I am interested in theorems with unexpected conclusions. I don't mean
an unintuitive result (like the existence of a space-filling curve), but
rather a result whose conclusion seems disconnected from the
hypotheses. My favorite is the following. Let…
Richard Stanley
- 49,238
98
votes
17 answers
Google question: In a country in which people only want boys
Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
In a country in which people only want
boys…
nkrkv
- 1,107
97
votes
46 answers
Examples of theorems with proofs that have dramatically improved over time
I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm…
Manya
- 339
97
votes
19 answers
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic a pedagogical speedbump. For example, Serre's…
David Feldman
- 17,466
97
votes
50 answers
Theorems that are 'obvious' but hard to prove
There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would naturally expect a statement in the latter category…
Alec Edgington
- 265
97
votes
10 answers
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space X, we can define the sheaf cohomology of X in
I. the Grothendieck style (as the right derived functor of the global sections functor Γ(X,−))
or
II. the Čech style (first by defining the Čech cohomology groups…
Victoria Flat
- 1,841
97
votes
11 answers
How has "what every mathematician should know" changed?
So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the…
Vipul Naik
- 7,240
96
votes
28 answers
Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should be motivated by the sort of intuition one gains…
Erik Davis
- 1,645
96
votes
17 answers
What's a nice argument that shows the volume of the unit ball in Rn approaches 0?
Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the n-volume of the n-ball. They had finished up talking about finding volume by integrating the area of the…
B. Bischof
- 4,782
96
votes
79 answers
Elementary+Short+Useful
Imagine your-self in front of a class with very good undergraduates
who plan to do mathematics (professionally) in the future.
You have 30 minutes after that you do not see these students again.
You need to present a theorem which will be 100%…
Anton Petrunin
- 43,739
96
votes
8 answers
Which are the best mathematics journals, and what are the differences between them?
Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to?
Coming up with a shortlist isn't very hard. If you look for generalist journals,…
Scott Morrison
- 7,540