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bio website maths.ox.ac.uk/people/…
location Oxford, United Kingdom
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visits member for 4 years, 6 months
seen Mar 9 at 23:06

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Mar
28
awarded  Nice Answer
Mar
16
awarded  Favorite Question
Mar
10
awarded  Nice Question
Mar
7
asked Classical and Quantum Chern-Simons Theory
Oct
19
awarded  Yearling
Oct
16
awarded  Good Answer
Mar
4
awarded  Pundit
Jan
29
comment What, precisely, does Klein's Erlangen Program state?
@Terry Tao: I don't know if Klein would have agreed, but I'm sympathetic to the view that category theory inherits the Erlangen programme. If one allows the categorical generalization of symmetry, a version of the 'recover space from symmetries' view that is almost tautologically true is the Yoneda Lemma. In a different direction, there are a number of interesting theorems in algebraic and arithmetic geometry that allow the reconstruction of a space from an associated category.
Jan
22
awarded  Great Question
Jan
21
awarded  Stellar Question
Jan
16
comment What, precisely, does Klein's Erlangen Program state?
This answers seems to capture the rough understanding I had of the programme. I recall hearing an even more simplistic formulation to the effect that 'a geometry is determined by its symmetries.' If some idea of this sort really was articulated in the Erlangen programme, then it seems to have rested on a very restricted (one might even say old-fashioned) notion of geometry. From the Riemannian viewpoint, most geometries have no symmetries at all. That is, from a modern perspective, such a statement is clearly wrong.
Dec
30
awarded  Nice Answer
Dec
5
comment Why is algebraic de Rham cohomology via completion independent of embedding?
There are a number of approaches using tubular neighborhoods. But a nice way is to use the *infinitesimal topology,' a characteristic zero version of the crystalline topology. The ideas are explained in Illusie's article in Arcata 1974.
Dec
5
comment What is the geometry of an undecidable diophantine equation?
I never checked this myself, but Mazur told a number of us many years ago that he had looked at Matiyasevich's equations, and found that they all had plenty of rational solutions. Someone should ask him about this. Alternatively, you could look at the equations yourself. This doesn't answer your question, but it's a start.
Nov
26
comment Examples where adding complexity made a problem simpler
My favorite elementary example is the computation of $1+2+\cdots +n$, which is easier to do twice than once.
Nov
13
awarded  Notable Question
Nov
8
comment Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?
(That should have been integral points.) It's easy to forget now that the significance of proving the finite generation of points on something as exotic as a Jacobian, not corresponding in any obvious way to an actual equation, must not have been obvious in the early twentieth century. Siegel's concrete theorem was what made it clear that this was a useful thing. It's also easy to forget that Siegel's theorem was essentially the best thing Diophantine geometry had to offer for many decades preceding Faltings, and probably one of the best theorems in all of number theory.
Nov
8
comment Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?
Dear Paul, This is a very interesting answer, and I think I understand your caution with respect to the larger-than-life people. However, I would have to agree with quid in this case: Weil speaks highly of Siegel in a number of different places. John Coates points out to me that Siegel was the person who really thrust Weil's name into the mathematical landscape by using the Mordell-Weil theorem to prove the finiteness of points on affine hyperbolic curves.
Nov
5
comment Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?
Pete's answer demonstrates very well the correctness of this expectation, but I agreed even before I saw it.
Nov
5
comment Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?
This is a very nice answer and, a bit paradoxically, illustrates how the question is not very strange. On the one hand, I do understand how some people may be wary of questions that somehow feel 'gossipy.' However, I understood Jonah's question to be something like this: 'Of course Siegel is great, but there are many great mathematicians. What special circumstance made Weil name Siegel in particular? I think a good answer to this question by someone knowledgeable in number theory and some history would be quite illuminating from a mathematical point of view.'