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Mathematics professor at Cambridge

Dec
17
awarded  Nice Answer
Dec
15
awarded  Enthusiast
Dec
8
answered Cardinality of Equivalence Classes of Cauchy Sequences
Dec
6
comment Why is it useful to study vector bundles?
OK, in that case I think one has to turn to more sophisticated answers such as that you can use them to form K groups. If you'll excuse the indirect self-promotion, I'd recommend Burt Totaro's article on algebraic topology in the Princeton Companion to Mathematics, where he has quite a lot to say about bundles and why they are important.
Dec
5
comment What are the most misleading alternate definitions in taught mathematics?
I would almost prefer not even to say what a function is at all. I'd just say that if f is a function from A to B and x is an element of A then f(x) is an element of B. And that's all you need to know. Of course, I'm exaggerating a bit, and this point of view is not sufficient after a while (e.g. how would you decide whether the set of functions from A to B is countable, how would you define function spaces, etc.?) but in some situations this is the most important fact that you need from the basic definition of functions. Of course, one would also give examples, including artificial ones.
Dec
5
answered What are the most misleading alternate definitions in taught mathematics?
Dec
5
comment What are the most misleading alternate definitions in taught mathematics?
I totally agree with this and always tell students to think of "kernel of some homomorphism" as the definition and "closed under conjugation by any element of G" as a fact that can be shown to be equivalent to it.
Dec
5
answered Why is it useful to study vector bundles?
Dec
1
comment k-pseudorandom measures
The problem there is that the sets we are looking at are so sparse that the U^k norm of any function is dominated by the degenerate cubes and so tells you nothing. Also, the control of one norm by another is often OK in the dense case but breaks down if you are sitting inside a sparse random set (because the constant depends on the sparsity of the random set).
Nov
30
awarded  Nice Answer
Nov
29
comment k-pseudorandom measures
It's not true that we removed the correlation condition -- that question is still open. What we did was look at functions bounded by random as opposed to pseudorandom measures, and we obtained best possible results by considering a specially constructued norm rather than the $U^k$ norm. The paper will be posted on the arXiv soon.
Nov
29
awarded  Nice Answer
Nov
29
revised Typical value of totient function
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Nov
28
awarded  Popular Question
Nov
28
awarded  Self-Learner
Nov
28
answered Typical value of totient function
Nov
28
awarded  Editor
Nov
28
answered What is the first interesting theorem in (insert subject here)?
Nov
28
asked Typical value of totient function
Nov
28
revised Intuitive explanation to Probability question
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