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Mathematics professor at Cambridge

Jun
13
comment Are there any very hard unknots?
I drew the "quotient" knot and the picture has been sitting on my desk for about a month. At first it looked hard to simplify, but then I saw that one could make a "hole" in the middle and take a chunk of knot and pass it up through the hole and back down again. This kind of global untwisting would, I think, have to be part of any unknotting procedure of the kind I fantasize about. At some point I might make the knot out of string and see whether I can indeed untie it fairly straightforwardly starting with that move.
Jun
12
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Jun
6
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May
24
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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
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Mar
29
asked What can be proved about the Ramanujan conjecture using elementary means?
Mar
2
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Mar
2
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Feb
7
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Nov
28
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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.