[FOM] Manjul Bhargava: A nice exception to a rule

[Joe Shipman]

He's not the only one who went back and improved on Gauss.

As FOMers who were here in 2007 know, I noticed that Gauss's original proof of the Fundamental Theorem of Algebra could be generalized. 

In modern language, that proof shows that a field of characteristic 0 where all equations of odd degree and degree 2 have roots is algebraically closed.

I found out what was really going on combinatorially underneath it all and improved the proof so that you only need to assume prime degree and it works in arbitrary characteristic, and I gave an algorithm for determining all implications between sets of "degree axioms" (axioms that all polynomials of a given degree have roots).

As Bhargava demonstrated, there can be buried treasure in well-plowed fields, if you dig a little deeper.

-- JS


Sent from my iPhone

> On Aug 15, 2014, at 8:19 PM, David Roberts <david.roberts at adelaide.edu.au> wrote:
> 
> Regarding Bhargava, Tim Gowers writes [1]
> 
>> But the first of his Fields-medal-earning results was quite extraordinary. As a PhD student, he decided to do what few people do, and actually read the Disquisitiones. He then did what even fewer people do: he decided that he could improve on Gauss. More precisely, he felt that Gauss’s definition of the composition law [DR: taking 20 pages or so] was hard to understand and that it should be possible to replace it by something better and more transparent.
> 
>> I should say that there are more modern ways of understanding the composition law, but they are also more abstract. Bhargava was interested in a definition that would be computational but better than Gauss’s. I suppose it isn’t completely surprising that Gauss might have produced something suboptimal, but what is surprising is that it was suboptimal and nobody had improved it in 200 years.
> 
> <snip>
> 
>> In this way, Bhargava found a symmetric reformulation of Gauss composition. And having found the right way of thinking about it, he was able to do what Gauss couldn’t, namely generalize it. He found 14 more...
> 
> 
> [1] http://gowers.wordpress.com/2014/08/15/icm2014-bhargava-laudatio/
> 
> Regards,
> 
> David
> 
>> On 14 August 2014 22:04, Colin McLarty <colin.mclarty at case.edu> wrote:
>> 
>> On Wed, Aug 13, 2014 at 3:13 AM, Harvey Friedman <hmflogic at gmail.com> wrote,
>> among much else, that among mathematicians in general when a
>> 
>>> solution uses considerable machinery (t)his is considered an extreme plus
>>> over it being solved by extremely
>>> clever special methods.  There is rationale for this, mainly that if big
>>> machines are used, then that promises
>>> further solutions to further problems more than an extremely clever
>>> special method.
>> 
>> Yes that is common.  But the Fields Medal to Manjul Bhargava shows clever
>> methods without heavy machinery can sometimes also promise further solutions
>> to further problems.
>> 
>> Bhargava cites some mathematicians who use heavy machinery (Langlands, de
>> Jong) but I don't know if he cites the heavier parts of their work.  And
>> what people like about his work is how light weight the machinery is, yet
>> vastly productive, and suggestive of much more.  People compare his work to
>> Gauss's.
>> 
>> Colin
>> 
>> 
>> 
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> 
> 
> -- 
> Dr David Roberts
> Research Associate
> School of Mathematical Sciences
> University of Adelaide
> SA 5005
> AUSTRALIA
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