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1500 questions
185
votes
62 answers
Interesting mathematical documentaries
I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I am in charge of nourishing our departmental math…
Ricardo Menares
- 243
183
votes
127 answers
Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view was my motivation for asking this question…
Suvrit
- 28,363
182
votes
19 answers
How do I make the conceptual transition from multivariable calculus to differential forms?
One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on the algebra $C^{\infty}(M)$ of smooth functions $M…
Qiaochu Yuan
- 114,941
181
votes
33 answers
What should be learned in a first serious schemes course?
I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (most of all the students, who
were smart,…
Ravi Vakil
- 3,837
181
votes
60 answers
Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where 'reasonable' is open to interpretation, and similar…
Q.Q.J.
- 2,083
178
votes
11 answers
Knuth's intuition that Goldbach might be unprovable
Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really puzzles me. (See page 321 of…
AgCl
- 2,675
177
votes
80 answers
Best online mathematics videos?
I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.
Randomblue
- 2,937
175
votes
7 answers
Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability space rather than Lebesgue measurable functions.…
Mark
- 4,804
173
votes
8 answers
How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.
As an undergraduate we are trained as mathematicians to be universalists. We are expected to embrace a wide spectrum of mathematics. Both algebra and analysis are presented on equal footing with geometry/topology coming in later, but given its fair…
anon
- 1
173
votes
39 answers
Most harmful heuristic?
What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
Michael Hoffman
- 1,785
173
votes
3 answers
Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an even…
alex alexeq
- 1,871
171
votes
7 answers
Proofs of Bott periodicity
K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological…
Eric Peterson
- 6,218
171
votes
7 answers
Does $\DeclareMathOperator\Aut{Aut}\Aut(\Aut(\dots\Aut(G)\dots))$ stabilize?
Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form
$$ {\Aut}^n(G):= \Aut(\Aut(\dots\Aut(G)\dots)). $$
Some quick results:
For finitely-generated…
Greg Muller
- 12,679
170
votes
36 answers
Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by Tim Gowers and is in line with other forms of…
Gil Kalai
- 24,218
169
votes
8 answers
The "Dzhanibekov effect" - an exercise in mechanics or fiction? Explain mathematically a video from a space station
The question briefly:
Can one explain the "Dzhanibekov effect" (see youtube videos from space station or comments below) on the basis of the standard rigid body dynamics using Euler's equations? (Or explain that this is impossible and that would be…
Alexander Chervov
- 23,944