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81
votes
4 answers
Can a group be a universal Turing machine?
This question was inspired by this blog post of Jordan Ellenberg.
Define a "computable group" to be an at most countable group $G$ whose elements can be represented by finite binary strings, with the properties that
There is an algorithm (by which…
Terry Tao
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81
votes
19 answers
Reading list for basic differential geometry?
I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is insufficient.
GMRA
- 2,050
81
votes
13 answers
Nontrivially fillable gaps in published proofs of major theorems
Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding Problem for Riemannian Manifolds" (Annals of…
Ali Enayat
- 17,058
81
votes
6 answers
Is data science mathematically interesting?
I have seen a plethora of job advertisements in the last few years on mathjobs.org for academic positions in data science. Now I understand why economic pressures would cause this to happen, but from a traditional view of university organization,…
Monroe Eskew
- 18,133
81
votes
7 answers
Computational complexity of computing homotopy groups of spheres
At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this.
Is there a precise algorithm written down anywhere in the literature? Is there one…
Ryan Budney
- 43,013
81
votes
18 answers
What programming language should a professional mathematician know?
More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable?
If I am right, then please what languages should someone…
Allawonder
- 335
81
votes
3 answers
Intuitive explanation for the Atiyah-Singer index theorem
My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.
I'm trying to learn the Atiyah-Singer index theorem from standard and…
Daniel Moskovich
- 21,887
81
votes
9 answers
What are some examples of interesting uses of the theory of combinatorial species?
This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer.
I've often heard it claimed that combinatorial species are wonderful and prove…
Pietro
- 1,570
81
votes
3 answers
How do I verify the Coq proof of Feit-Thompson?
I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is more "tech support" than math.
But: after…
Nate Eldredge
- 29,204
81
votes
4 answers
Wanted: a "Coq for the working mathematician"
Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar with.
I'm trying to obtain a working knowledge of the…
darij grinberg
- 33,095
81
votes
18 answers
Depressed graduate student.
How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand?
Edit: I am not that grad student; it is a very talented friend of mine.
Moderator's update:
The discussion about this question should…
Anweshi
- 7,272
80
votes
8 answers
Teaching statements for math jobs?
What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these shortly.
Let us assume for the sake of argument that…
Michael Lugo
- 13,858
80
votes
25 answers
More open problems
Open Problem Garden and Wikipedia are good resources for more or less famous open problems. But many mathematicians will be happy with more specialized problems. They may want to find a research theme, e.g. for their PhD thesis, or they may have…
Cristi Stoica
- 4,304
80
votes
22 answers
How would you have answered Richard Feynman's challenge?
Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students of the math department.
I challenged them: "I…
rtsss
- 477
80
votes
12 answers
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his book Topology and Groupoids. The idea of the…
Daniel Moskovich
- 21,887