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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 |
awarded | Nice Answer |
Jun
6 |
awarded | Favorite Question |
May
24 |
awarded | Popular Question |
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 |
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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 |
awarded | Popular Question |
Mar
2 |
awarded | Good Question |
Feb
7 |
awarded | Good Answer |
Nov
28 |
awarded | Nice Answer |
Nov
22 |
awarded | Nice Answer |
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. |