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Mathematics professor at Cambridge

Jun
18
comment Are there very strongly pseudorandom permutations?
Good point -- thanks for the tip.
Jun
18
comment Are there very strongly pseudorandom permutations?
I have now found a source that seems to suggest that the Luby-Rackoff construction won't give hardness greater than $2^n$. So it looks as though a different idea would be needed. But maybe there are some different ideas out there.
Jun
18
asked Are there very strongly pseudorandom permutations?
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
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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?
added 164 characters in body
Mar
29
revised What can be proved about the Ramanujan conjecture using elementary means?
added 619 characters in body
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
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Nov
28
awarded  Nice Answer