I dont like to be too negative in my comments. Li's paper is an attempt to prove a variant of the global trace formula of my paper in Selecta. The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).
QUOTE (Terence Tao+)
The old definition of h was in fact problematic for a large number of reasons (the author was routinely integrating h on the idele class group C, which is only well-defined if h was -invariant). Changing the definition does indeed fix the problem I pointed out (and a number of other issues too). But Connes has pointed out a much more serious issue, in the proof of the trace formula in Theorem 7.3 (which is the heart of the matter, and is what should be focused on in any future revision): the author is trying to use adelic integration to control a function (namely, h) supported on the ideles, which cannot work as the ideles have measure zero in the adeles. (The first concrete error here arises in the equation after (7.13): the author has made a change of variables on the idele class group C that only makes sense when u is an idele, but u is being integrated over the adeles instead. All subsequent manipulations involving the adelic Fourier transform Hh of h are also highly suspect, since h is zero almost everywhere on the adeles.)
More generally, there is a philosophical objection as to why a purely multiplicative adelic approach such as this one cannot work. The argument only uses the multiplicative structure of , but not the additive structure of k. (For instance, the fact that k is a cocompact discrete additive subgroup of A is not used.) Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from ). If the arguments worked, this would mean that the Weil-Bombieri positivity criterion (Theorem 3.2 in the paper) would continue to hold even after deleting an arbitrary number of places. But I am pretty sure one can cook up a function g which (assuming RH) fails this massively stronger positivity property (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)
was originally more deeply flawed than the first published Wiles proof of Fermat's Last Theorem. Two Fields medalists seem of the opinion that it may be unrepairable. Give this a couple months. Four versions of the paper in 3 days seems like excessive churn.http://terrytao.wordpress.com/2008/02/07/s...-prime-numbers/http://noncommutativegeometry.blogspot.com...un-day-two.html
5th July 2008 - 02:39 AM
Thanks for the comments!
Terrance Tao a fields medalist?
5th July 2008 - 01:09 PM
5th July 2008 - 05:23 PM
I've been trying to fill in some understanding of the standing of mathematics from the twentieth century on by going to the math geneology project; but, they havn't updated! I also read "The Honors Class" recently(I actually bought it for like five bucks at a used books store up in the Seatle area while getting some navy gech training for my new squadron; enough of that); but, I suppose I'm still in need of filling in the gaps since the work of E.T. Bell and even Morris Kline(although he tends to avoid talking about much past early twenteith century stuff because he thinks; thought that mathematics was going in bad abstracto lands)
Anyways, yea, Tao definitelly sounds like . . . well, shoot, he's already got a Fields medal! Any idea of what age he got it in?
6th July 2008 - 09:55 AM
A related post from the "Not Even Wrong" Blog of Peter Woit:http://www.math.columbia.edu/~woit/wordpress/?p=707
The paper, apparently, has been withdrawn due to "an error on page 29."
6th July 2008 - 10:47 AM
I'm so glad the popular press isn't running with this story. Otherwise, we would be buried with tortured analogies as to what Tao's objection was.
... and then, having accepted that the "hole in the doughnut" is a different object than what are sold as "doughnut holes," the author withdrew his paper from consideration.
Gah. I made myself sick.
6th July 2008 - 11:16 AM
QUOTE (Raphie Frank+Jul 6 2008, 09:55 AM)
"an error on page 29."
The above should have read...
“due to a mistake on pg. 29"http://www.math.columbia.edu/~woit/wordpress/?p=707
And that's just a minor misquote...
The point being that RPenner has a point regarding the popular press. I have personally been a part of events (typically political in nature...) that were reported upon in the mainstream media. The actuality and the telling of that actuality were so grossly disparate that I sometimes wonder why we even bother to make the literary distinction between fiction and non-fiction. :-)
6th July 2008 - 02:48 PM
riemann's hypotheses may be obtained by the quantum spectrum,that determine the fractional part of the riemann's hypothesy,of where would be originated the zeros of the equation.the the trace of the equations occur the mellin's transformation.the origin the riemann's hypotheses is due the asymmetric part,that
is the imaginary part that are subclasses of the complex vector spaces,that contain the nonassociative and noncommutative properties of the algebra of connes;thence appear the breakdown of the invariance by rotations,but the symmetry is conserved,then the spinors are the metrics of the spaces by rotations,through the entity of the motion(that measure the velocity),the TIME,that is a quantum property
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