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1500 questions
75
votes
7 answers
What items MUST appear on a mathematician's CV?
Writing a CV makes me paranoid that I'm failing to abide by unwritten rules. Of course CVs are flexible to capture the diversity of accomplishments someone might have. But there must be plenty of things a hiring committee absolutely expects. So I'm…
anon
- 51
75
votes
9 answers
Why is an elliptic curve a group?
Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the neutral element being the point at infinity and…
Harald Hanche-Olsen
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75
votes
13 answers
Counterexamples in PDE
Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their detailed derivations.
Please give one example per…
timur
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75
votes
8 answers
What is an intuitive view of adjoints? (version 1: category theory)
In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.
I know the definition (several of them), I've read the nlab page (and any good…
Andrew Stacey
- 26,373
75
votes
10 answers
What is a Lagrangian submanifold intuitively?
What are good ways to think about Lagrangian submanifolds?
Why should one care about them?
More generally: same questions about (co)isotropic ones.
Answers from a classical mechanics point of view would be especially welcome.
Jan Weidner
- 12,846
75
votes
13 answers
What precisely Is "Categorification"?
(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
Gil Kalai
- 24,218
75
votes
1 answer
A number theory problem where pi appears surprisingly
For a given positive integer $M$, the sequence $\{a_n\}$ starts from $a_{2M+1}=M(2M+1)$ and $a_k$ is the largest multiple of $k$ no more than $a_{k+1}+M$, i.e.
$$a_k=k\left\lfloor\frac{a_{k+1}+M}{k}\right\rfloor,\quad k=1,2,\cdots,2M.$$
The original…
Andrew Sakura
- 701
75
votes
3 answers
Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs.
I encountered other analogies as well.
$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\ln xy}}\,dx\,dy\\&=\sum…
Anixx
- 9,302
75
votes
5 answers
When the automorphism group of an object determines the object
Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M, N\in \mathbf{Man}$ there is the following…
Muhammed Ali
- 771
75
votes
1 answer
What is an étale theta function?
Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, they should be regarded as 'introductions to…
Minhyong Kim
- 13,471
75
votes
4 answers
What are reasons to believe that e is not a period?
In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are. The former claim is repeated at many other…
Vincent
- 2,437
75
votes
53 answers
Free, high quality mathematical writing online?
I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or…
Kim Greene
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75
votes
19 answers
What are some deep theorems, and why are they considered deep?
All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number theorem, and the Poincaré conjecture. I am…
John Stillwell
- 12,258
75
votes
6 answers
Errata to "Principles of Algebraic Geometry" by Griffiths and Harris
Griffiths' and Harris' book Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My suggestions:
Page 10 at the top, the definition…
SpecR
- 1
75
votes
3 answers
Does a power series converging everywhere on its circle of convergence define a continuous function?
Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges .
We thus obtain a complex-valued function $f$…
Georges Elencwajg
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