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1500 questions
74
votes
21 answers
How should one present curl and divergence in an undergraduate multivariable calculus class?
I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and divergence. This course follows Stewart's book and…
Kevin H. Lin
- 20,738
74
votes
4 answers
What is the amplituhedron?
The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces a remarkable new way for computations in quantum…
Gil Kalai
- 24,218
74
votes
14 answers
Why is Set, and not Rel, so ubiquitous in mathematics?
The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling out a particular kind of relations, namely the…
Qfwfq
- 22,715
74
votes
4 answers
Groups that do not exist
In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and other such precise information which usually can…
Mariano Suárez-Álvarez
- 46,795
74
votes
7 answers
How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".
I'm a graduate student with interest in number theory. I feel like I'm in danger of losing the big picture…
Lepidopterist
- 937
74
votes
11 answers
Does War have infinite expected length?
My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers.
The question is: Is the expected length of the game…
Joel David Hamkins
- 224,022
74
votes
16 answers
Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 on a Tamagawa number, GH…
Tom Copeland
- 9,937
74
votes
5 answers
Is there a "geometric" intuition underlying the notion of normal varieties?
I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me is that the definition of normality is so…
aglearner
- 13,995
73
votes
1 answer
Derived Functors Versus Spectral Sequences
Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.
Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious natural transformation $\zeta_{G,F}:R(GF)\Rightarrow…
Steven Landsburg
- 22,477
73
votes
30 answers
What are some examples of ingenious, unexpected constructions?
Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by taking a new, "higher" perspective. The most…
Robert Kucharczyk
- 1,288
73
votes
10 answers
Why does the Gamma-function complete the Riemann Zeta function?
Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or intuition, even if it cannot be made into a proof…
Peter Arndt
- 12,033
73
votes
6 answers
whence commutative diagrams?
It seems that commutative diagrams appeared sometime in the late 1940s -- for example, Eilenberg-McLane (1943) group cohomology paper does not have any, while the 1953 Hochschild-Serre paper does. Does anyone know who started using them (and how…
Igor Rivin
- 95,560
73
votes
3 answers
Is there a "purely algebraic" proof of the finiteness of the class number?
The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching soon.
I just inserted the statement of the…
Pete L. Clark
- 64,763
73
votes
6 answers
Still Difficult After All These Years
I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were cryogenically frozen today, and thawed out in one…
arsmath
- 6,720
73
votes
7 answers
Roots of truncations of $ e^x - 1$
During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has pure imaginary roots.
If you plot the roots of…
Vectornaut
- 2,224