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Results tagged with soft-question
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user 4362
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
135
votes
Why should one still teach Riemann integration?
From a conceptual standpoint, I think that there are three things one asks of an approach to integration
1) An easily accessible geometric interpretation
2) A readily available computational toolbox …
115
votes
36
answers
31k
views
Quick proofs of hard theorems
Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later t …
67
votes
Quick proofs of hard theorems
Here is my example. In the 1930's (I think), Wiener gave a proof that if $f$ is a continuous nonvanishing function on the circle with absolutely convergent Fourier series, then so is $1/f$. The proo …
59
votes
What are the most misleading alternate definitions in taught mathematics?
I normally won't bother with a 5 month old community wiki, but someone else bumped it and I couldn't help but notice that the significant majority of the examples are highly algebraic. I wouldn't wan …
51
votes
When and how is it appropriate for an undergraduate to email a professor out of the blue?
I am a graduate student, so I can only provide you the point of view of someone who has (and still does) asked vast amounts of questions to all kinds of people, both in person and by e-mail. I think …
49
votes
Are there examples of non-orientable manifolds in nature?
It seems that nature "is" a Klein bottle in the following sense. There is a growing field in applied topology (yes, I said that) which goes by "topological data analysis" or sometimes "persistent hom …
33
votes
Why have mathematicians used differential equations to model nature instead of difference eq...
First, a historical remark: it was not until relatively recently in the history of science that people were convinced that the atomic theory of matter is correct. I believe the tide was turned by a p …
33
votes
Should computer code be included within publications that present numerical results?
My answer is:
Don't put code in your paper. Do: put pseudocode in your paper, version control your code on Github, and add a link to your Github repository to your paper.
The purpose of a pape …
30
votes
Why higher category theory?
Here is a fairly naive answer from a non-expert.
If you've studied algebraic topology then you can probably appreciate the philosophy that for the purposes of homotopy theory you should not consider …
28
votes
Is there a nice application of category theory to functional/complex/harmonic analysis?
I've never completely understood what counts as "an application of category theory". With other areas of mathematics an "application" of area A to area B is generally a result which translates a prob …
27
votes
What problem in pure mathematics required solution techniques from the widest range of math ...
I'm not qualified to certify optimality, but I've always thought that the Mostow rigidity theorem is a good candidate. The theorem says that every isomorphism between the fundamental groups of two fin …
26
votes
Accepted
'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math
As a (slowly) recovering category-phobe, allow me to suggest that you change the way you think of category theory. Specifically, don't think of category theory as a "theory". A theory in mathematics …
25
votes
How many quit mathematics because they (are afraid that they) can not find a permanent job?
The question asks for data on why people leave research mathematics. I don't have any, so in a sense I can't really answer the question. What I have is personal experience and a reasonable number of …
21
votes
Is there a database for tracking the dependencies of mathematical theorems?
The Stacks Project provides an example of what you're looking for. Every definition, lemma, theorem, etc. is given a tag, and the tags are used as references in proofs. They even provide an API for …
- 29.2k
19
votes
Theorems that are 'obvious' but hard to prove
I think that the ergodic theorem is a good example of this. In down-to-earth terms it says that if you have a box full of gas then the average velocity of all of the gas particles at a given time (th …