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Mathematics professor at Cambridge

May
9
comment Are there any very hard unknots?
Thank you for this example. It's quite interesting as it is in some sense a "product" of smaller knots. I tried replacing the bundles of strands (most of the time four strands) by a single strand and obtained a picture of a knot that I can't instantly see to be the unknot, though I did find a local way of reducing the number of crossings. If this "quotient" knot is not the unknot, then it's a very interesting example.
Mar
30
awarded  Good Answer
Mar
29
comment What can be proved about the Ramanujan conjecture using elementary means?
I don't mind complex analysis, but I'm wondering whether a "non-structural" proof is possible. Without saying precisely what I mean by that, I would say that modular forms are on the wrong side of the boundary.
Mar
29
revised What can be proved about the Ramanujan conjecture using elementary means?
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Mar
29
revised What can be proved about the Ramanujan conjecture using elementary means?
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Mar
29
comment What can be proved about the Ramanujan conjecture using elementary means?
Ah, I see the point now. OK, I'll go back and add a condition.
Mar
29
comment What can be proved about the Ramanujan conjecture using elementary means?
I'm taking $1-q^{a_r}$, and not $1+(-q)^{a_r}$.
Mar
29
awarded  Nice Question
Mar
29
asked What can be proved about the Ramanujan conjecture using elementary means?
Mar
2
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Mar
2
awarded  Good Question
Feb
7
awarded  Good Answer
Nov
28
awarded  Nice Answer
Nov
22
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Oct
26
comment Believing the Conjectures
Going back to your hypothetical Tsirelson story, I still don't see what Maximize adds to it. Why not just say that if you've tried hard to prove a conjecture, it becomes more reasonable to doubt it, regardless of what that conjecture looks like? (This doesn't apply to all conjectures: for example, our failure to prove the twin prime conjecture doesn't suggest that it might be false, since there are good heuristic reasons to expect both that it is true and that it is hard to prove.)
Oct
26
comment Believing the Conjectures
This discussion, which I find very interesting by the way, leaves me with the feeling that I don't understand very well what the rules of thumb are really saying and what their purpose is. I agree about Banach space theory: in some ways it is a very structureless subject, because any old bunch of functionals can be used to define a norm (as long as you've got enough of them that you don't have just a seminorm), but from time to time it springs surprises -- the almost negligible constraints nevertheless interestingly restrict what you can do.
Oct
26
comment Believing the Conjectures
The general point I'm making here is that in this context it's the mathematics that tells us to what extent Maximize is an appropriate principle, rather than the principle that is guiding our mathematical expectations.
Oct
26
comment Believing the Conjectures
Is Tsirelson's space an example of the success of Maximize? Again, I have my doubts. Before his example, it was reasonable to try to prove that every space did contain $c_0$ or $\ell_p$ -- all known spaces did, sometimes for quite non-trivial reasons. I would contend that it was only after (i) a failure, despite considerable efforts, to prove positive theorems and (ii) Tsirelson's example that it became reasonable to believe quite strongly that if you can't easily prove something about a general Banach space, then it is probably false. And there have been counterexamples to that principle ...
Oct
26
comment Believing the Conjectures
As for Maximize, I find it unhelpful in this context, since it is not clear what is "likely to occur". For example, it is still open whether there exists an infinite-dimensional Banach space such that every operator defined on it is a multiple of the identity plus a nuclear operator. I feel as though the answer could go either way, and no principle like Maximize is going to alter that perception. On the other hand, the existence of examples with comparable properties, such as Argyros and Haydon's construction where "nuclear" is replaced by "compact", does have an impact.
Oct
26
comment Believing the Conjectures
The fact that there ought to be a separable example is too trivial to count as a success of Reflection, since if there is any example at all, you can take a separable subspace of it and then you've got a separable example.