bio | website | jmilne.org/math |
---|---|---|
location | Ann Arbor, MI, USA, and New Zealand. | |
age | ||
visits | member for | 4 years, 6 months |
seen | Oct 18 '10 at 13:35 | |
stats | profile views | 6,151 |
Arithmetic geometry (especially Shimura varieties and abelian varieties).
Aug 16 |
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Inaccessible cardinals and Andrew Wiles's proof
The Stack Project develops a huge amount of Grothendieck style mathematics, including a lot of etale cohomology, using only ZFC (specifically, NOT using universes). If anyone has any doubt that this can be done, I suggest that they look at it. |
Aug 16 |
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Inaccessible cardinals and Andrew Wiles's proof
Actually, Cosmonut misquoted the article by leaving out the rest of the statement: "But there is a general consensus among mathematicians that this was just a convenient short cut rather than a logical necessity. With a little work, Wiles's proof should be translatable into Peano arithmetic or some slight extension of it." |
Aug 6 |
awarded | Enlightened |
Aug 6 |
awarded | Nice Answer |
Aug 4 |
awarded | Good Answer |
Jul 29 |
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How connected are you?
MR collaboration numbers aren't accurate. For example, MR claims my collaboration number with Gauss is an improbable 6. They get that number by claiming Landau and Riemann are coauthors and Riemann and Gauss are coauthors because works by them were included together in reprint collections. Similarly, they incorrectly give my Erdos number as 3 because they count me as a co-author of Gerardin when we only published articles in the same collection. |
Jul 9 |
awarded | Enlightened |
Jul 7 |
answered | Jordan decomposition in a classical group |
Jul 7 |
awarded | Nice Answer |
Jul 7 |
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Does the Tannaka-Krein theorem come from an equivalence of 2-categories?
@Theo et al. --- well, you could try looking up the definition in Catégories Tannakiennes (Saavedra 1972, Deligne 1990) or Tannakian Categories (Deligne and ... 1982, Breen 1994) or .. A Tannakian category over a field $k$ is neutral if it admits a fibre functor over $k$. In general, it only admits a fibre functor over an extension of $k$. There are various expressions of Tannaka duality in 2-category terms in Saavedra, e.g., III 2.3.2, p180. |
Jun 25 |
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Weight filtration over the integers
algori. Thanks! you have helped clarify things for me. |
Jun 25 |
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Weight filtration over the integers
Actually, I'm still confused by Voisin, since in that section she seems to be talking about complex manifolds, not algebraic varieties. Perhaps the assumption is hidden somewhere. |
Jun 24 |
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Weight filtration over the integers
In 8.36 (p.214 of the English version) of her book, Voisin defines a mixed Hodge structure to have a weight filtration on the integral cohomology. She then says that Deligne's theorem shows the existence a mixed Hodge structure on the integral cohomology of the complement $U$ of a normal divisor crossing, and adds that "One can show that this mixed Hodge structure depends only on $U$ and not on its compactification." This seems to contradict what you are saying. Did Voisin mean to define mixed Hodge structure as Deligne does, with a weight filtration defined on the rational cohomology? |
Jun 21 |
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Least collaborative mathematician
Greg, MathSciNet only lists citations after about 1997. For example, they list only 16 citations from references for Weil's Foundations of Algebraic Geometry, the earliest of which is 1997. |
Jun 21 |
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Least collaborative mathematician
There's also Hooley, Crelle 328 (1981), 161--207, which depends crucially on Milne, Crelle 328 (1981), 208--220. Maybe I should have insisted on a joint paper.... |
Jun 15 |
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Dimension of central simple algebra over a global field “built using class field theory”.
See Theorem 2.6, Chapter VIII, of my class field theory notes, but the proof uses the Grunwald-Wang theorem which is not (yet) proved in the notes. |
Jun 14 |
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Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?
This looks wrong to me. Compute the Mumford-Tate group G of the relevant cohomology group (as a rational Hodge structure); then conjecturally, the transcendence degree of the field generated by the periods is the dimension of G (Grothendieck). The point is that algebraic classes force algebraic relations between the periods; hence conjecturally also Hodge classes do (this is known for abelian varieties by Deligne); and Grothendieck conjectured that these are all the relations. So if you want lots of relations, you need to find lots of Hodge classes. |
Jun 12 |
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Hodge Standard Conjecture in Positive Characteristic
Actually the Hodge standard conjecture is not even known in positive characteristic for abelian varieties --- it is only known that it is implied by the Hodge conjecture for complex CM abelian varieties (see my 2002 Annals paper). Of course, the conjecture is trivial for projective n-space. A good reference for these things is Kleiman's article in the proceedings of the Motives conference, Seattle 1991, published by AMS 1994. |
May 31 |
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What was Galois theory like before Emil Artin?
Charles, the independence of multiplicative characters is usually credited to Dedekind. Galois theory is about separable extensions. |
May 31 |
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What was Galois theory like before Emil Artin?
Actually, it wasn't all that different, except that you first proved the primitive element theorem, and then proved things by choosing a primitive element. Artin disliked having to make a choice, and his main contribution was show that you can do Galois theory without choosing a primitive element. It's not obvious to me that this makes things easier or better. You can find the old approach in A.A. Albert's book on algebra. |