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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.
8
votes
Math for a cake
At Michael Atiyah's 80th birthday conference, the cake had the Atiyah-Singer index formula:
$$\text{Ind}(D) = \int_{T^*M} \text{ch}(\sigma_D) \text{Todd}(TM \otimes \mathbb{C})$$
I can verify that i …
2
votes
Interesting examples of generic behavior of mathematical objects being either unreasonably s...
Atiyah and Hirzebruch proved that if a compact connected Lie group acts smoothly and nontrivially on a compact spin manifold of dimension $4k$ then the $\hat{A}$-genus vanishes. By calculations with …
6
votes
Interesting examples of generic behavior of mathematical objects being either unreasonably s...
The phenomenon of concentration of measure produces many examples wherein a randomly chosen object has an unexpected property with high probability. A classical example is the Johnson-Lindenstrauss l …
7
votes
Blackbox Theorems
C*-algebra theory has a number of good examples of this.
Voiculescu's theorem: an ample representation of a C*-algebra essentially absorbs any nondegenerate representation
Kasparov's technical theor …
16
votes
How do you present a non-existence theorem?
A recurring theme in the answers and comments that you have received so far is that the question is ill-posed since many (or perhaps all) non-existence claims can be recast as universal claims. While …
11
votes
Blackbox Theorems
I think differential topology has dozens of these results. Here are some examples that immediately come to mind:
The tubular neighborhood theorem: every submanifold $N$ of a manifold $M$ has an ope …
8
votes
Examples of famous 'workhorse' theorems
I think it is reasonable to view the h-cobordism theorem (and its relative, the s-cobordism theorem) as a workhorse of differential topology. The statement isn't actually that hard - it simply gives …
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 …
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 …
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 …
14
votes
What areas of pure mathematics research are best for a post-PhD transition to industry?
I have thought a bit about this question because I have been contemplating this transition for myself. Here are a few ideas:
-Stochastic analysis: this is a good area of expertise if she hopes to go …
13
votes
What are some slogans that express mathematical tricks?
There are two interesting tricks in K-theory / operator algebras / homotopy theory - one attached to an amusing slogan and the other with an amusing name - that I think foot the bill.
The first is "u …
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 …
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 …
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 …
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