Slashdot Mirror
Going Beyond Fermat's Last Theorem
amjith writes "An Indian mathematician, Chandrashekhar Khare, is poised to make a significant breakthrough in the field of number theory with his solution of part of a major outstanding problem in algebraic number theory. He is currently an associate professor in Mathematics Department of University of Utah. "
www.math.utah.edu/~shekhar/papers.html
He has proved what is known to specialists in the field as the `level-1 case of the Serre conjecture.' In earlier work done with the French mathematician, J.P. Wintenberger, in December 2004, Dr. Khare outlined a two-part general strategy to prove the Serre conjecture fully. The present result is a first key step.
Wikipedia page for Serre conjecture
the underline appears all the way through " to make a significant breakthrough in the field of number theory with his solution "
even though the word "solution" leads to a different link than all of the preceding words.
It was proved in 1995 by English mathematician Andrew Wiles.
Wikipedia page of the theorem
I don't follow the field close enough to know its relation to Serre's multiplicity conjectures.
From the wikipedia article: "The word algebra itself comes from the name of the treatise first written by a Persian mathematician Al-Khwarizmi 700 AD, who wrote a treatise titled: Kitab al-mukhtasar fi Hisab Al-Jabr wa-al-Moghabalah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr (from which algebra is derived) means "reunion", "connection" or "completion"."
Computers are useless. They can only give you answers.
-- Pablo Picasso
...hundreds of new mathemtical theorems are discovered by people around the world. Many of these become peer reviewed and published. So why is this particular one on the front page? It's basically unknown outside of mathematical circles and is posted on a web site where any crackpot can post. Shall we start having stories about JSH on sci.math?
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Um ... the only one related to crypto would be the theorem that a^p mod p == a if p is prime and a is co-prime to p.
That's not only not the famous Fermat Last Theorem but it's also trivially provable with basic number theory.
Tom
Someday, I'll have a real sig.
Just to speculate on a possible "what use" question that might arise, I can't help but notice the line This is one of the central themes of modern research in number theory and is devoted to the study of the relation between the symmetries of number theory and geometry. . If I may be so bold, anything that ties the study of pure math to geometry probably has implications for quantum mechanics. These objects may lie embedded in higher dimensions, and probably settle into stable configurations from near infinite possibilities. But they still have to satisfy some allowable mathematical model. This is just the type of thing that may allow us to better predict what those allowable states could be.
Letter To Iran
Muslims borrowed heavily from India when they invaded India. The Islamic role in the sciences tended to be more about preserving the best of what they had conquered. As traders, they acted as a point where that knowledge could be disseminated to Europe.
This is my sig.
http://www.fiestasiesta.co.uk/history/jews.html
Great ideas often receive violent opposition from mediocre minds. - Albert Einstein
you got the wrong indians there buddy.
Number theorists are well known to pride themselves on working on things that have no practical application. They actually got mad when cryptography started borrowing some ideas from number theory, and directed their research elsewhere. I'm not kidding.
University of St. Andrews has an excellent math history site full of biographies of many mathematicians:/ history/
~ history/HistTopi cs/Arabic_mathematics.html
http://turnbull.mcs.st-and.ac.uk
They have an excellent section on Muslim contributions:
http://turnbull.mcs.st-and.ac.uk/
They argue that Muslims did far more than preservation.
umm.. no.. 3 of the 4 conjectures have been proven.. positivity of R/p and R/Q is still in question.. and no.. showing that it is non negative is not a proof of positivity.. 0 is not positive.
I am the Alpha and the Omega-3
This site does not have a very good record with mathematical breakthroughs that it runs on the front page. Just to give some examples:
1) A year and a half ago Slashdot ran a story (along with most of the MSM) about a Swedish girl having solved the 16th Hilbert problem. That turned out to be a completely bogus claim - she had, in fact, proved nothing.
2) Slashdot ran with there being infinitely many twin primes. The proof was flawed.
3) No, the Riemann hypothesis (the most coveted result in all of Mathematics) has not been proved.
Those are just the examples I can remember off hand. There have been several more, and I cannot think of a single one that has turned out to actually be true. So please take vague stories about being "poised to make a great story" from local press with a pretty hefty grain of salt...
There's a reason it's called the Hindu-Arabic number system.....m 2217/han.htm
http://scitsc.wlv.ac.uk/university/scit/modules/m
Glancing over the responses so far, I've come across several links to "the" Serre conjecture. Of course, since this is Slashdot (Land of the Karma Whore) it also looks like not a one of those referred to the conjecture relevant to this discussion.
The particular conjecture of Serre that matters here focuses on the two-dimensional representations over a finite field of the Galois group Gal(Qbar/Q). Now since that's not particularly illuminating, let me say a bit more...
First, Qbar denotes the algebraic completion of the rational numbers -- that is, all the stuff you need to add to the rationals so that you can do stuff like factor polynomials with rational coefficients. So things like sqrt(2) are in Qbar, but transcental numbers like pi aren't.
Gal(Qbar/Q) is the group of symmetries of Qbar over Q -- the ways you can map it to itself while still preserving multiplication and addition, and leaving the rational numbers inside Qbar alone. For instance, complex conjugation gives an element of the Galois group.
Now one way to understand any group of symmetries is by looking at its "linear representations" -- basically, ways of assigning matrices to each of the symmetries so that matrix multiplication matches up with the composition of symmetries.
The conjecture talked about here claims to describe (in some sense) all such (irreducible) representations of Gal(Qbar/Q), at least if you limit yourself to 2x2 matrices and coefficients in a finite field.
This is similar to the Langlands Correspondence, which (among other things) deals with representations of Gal(Qbar/Q) by complex matrices (though not just 2x2).
"Of the six (trigonometry functions), five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India."
/ Islamic%20astronomy.htm
http://faculty.kfupm.edu.sa/phys/alshukri/PHYS215
I have to add, while the role of Islamic science in preserving and transmitting knowledge from Greek, Egyptian, Chinese and Indian
societies was important, it wasn't the only one. Muslim scientists did a lot of ground breaking work in astronomy, medicine, geometry and algebra.
Abu Bakr al-Razi (844-926) made the first connection between bacteria and infection.
Omar Khayyam better known as a poet, made significant contributions to the solution of cubic equations
by geometric methods involving the intersection of conics.
Nassereddine al-Tusi is credited as the founder of modern trigonometry, separating it from astronomy.
Google for more if you are interested.
As a side note, any time you hear a scientific type term starting with 'al' it came from Arabic. That
includes algebra, algorithim, alcohol, alkali
They surely are Americans as well... "technically". For some odd reason the world has equated American as people who are from the United States of America only but America is actually North, Central and South America. Chile is in South America which is part of America thus they are Americans, although we really don't refer to them as Americans :P