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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.
3
votes
Comments and reference-request on books for KK-theory
Here is a very rough outline of the proof of the index theorem using KK-theory:
Define $KK_G(A, B)$, where $G$ is a Lie group and $A$ and $B$ are
[adjectives] C*-algebras, and the Kasparov product be …
- 29.2k
9
votes
What is an important mathematical question?
Words like "important" and "natural" are obviously subjective, and that's OK! It shouldn't surprise you to learn that other people have opinions about the value of certain mathematical ideas, just as …
- 29.2k
13
votes
Theorems with many distinct proofs
There is an old MO question about Bott periodicity which lists 9 fairly different proofs, and it missed a few (heat equation, extensions of $C^*$-algebras, Toeplitz operators, coarse geometry...)
- 29.2k
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 …
- 75
8
votes
Is $C^{*}$-algebra the most modern way to study QFT?
My PhD work used C*-algebras quite heavily, so I guess I can claim some expertise there, but I'm not an expert in QFT. That will be the main perspective of my answer.
A good starting point for this d …
- 29.2k
7
votes
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed t...
Lots of great answers so far, but I'll add another (hopefully at least good) answer: the notation $v^T u$ makes it somewhat difficult to speak of collections of bilinear pairings depending on a parame …
- 29.2k
2
votes
Compact spaces whose compactness does not come from a product of compact spaces
Elaborating on Michael Greinecker's comment: if $X$ is a compact Hausdorff space then the map $i \colon X \to \Pi_{C(X,I)} I$ given by $i(x)_f = f(x)$, where $I = [0,1]$, is a homeomorphism onto its i …
- 29.2k
7
votes
Unconventional examples of mathematical modelling
There are a lot of applications of ideas in dynamical systems to social media for things like event detection and forecasting. Some of the literature has to be taken with a grain of salt because of t …
- 29.2k
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 …
- 29.2k
8
votes
Accepted
What is the Essential Difference Between Random Matrices and Random Graphs?
I suppose the main point is that the typically studied random graph models are not directed or weighted and they generally don't have self loops. Under your correspondence, this means they are limite …
- 29.2k
12
votes
What programming language should a professional mathematician know?
I'll start with a meta-answer: given the large (and growing) number of programming languages out there, how do you decide where and how to invest your time?
The answer turns out to be quite simple, b …
- 29.2k
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 …
- 29.2k
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 …
- 29.2k
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
8
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
I think Fourier analysis generates a lot of interesting examples of this phenomenon; it is an extremely powerful tool, but it's not always easy to see what the tool is actually doing.
A good case stu …
- 29.2k