Let me

admit it : i was probably wrong in this post to

advise against downloading A walk in the noncommutative

garden by Alain Connes and Matilde Marcolli. After all, it seems

that Alain&Matilde are on the verge of proving the biggest open

problem in mathematics, the Riemann

hypothesis using noncommutative geometry. At least, this is the

impression one gets from reading through The music of the

primes, why an unsolved problem in mathematics matters by Oxford

mathematician Prof.

Marcus du Sautoy… At the moment I’ve only read the first

chapter (_Who wants to be a millionaire?_) and the final two

chapters (_From orderly zeros to quantum chaos_ and _The

missing piece of the jigsaw_) as I assume I’ll be familiar with most

of the material in between (and also, I’m saving these chapters for some

vacation reading). From what I’ve read, I agree most with the final

review at amazon.co.uk

Fascinating, October 5, 2004

and infuriating

Reviewer: pja_jennings

from Southampton, Hants. United Kingdom

This is a book I found

fascinating and infuriating in turns. It is an excellent layman’s

history of number theory with particular reference to prime numbers and

the Riemann zeta function. As such it is well worth the reading.

However I found that there are certain elements, more of style than

anything else, that annoyed me. Most of the results are handed to us

without any proof whatsoever. All right, some of these proofs would be

obviously well beyond the layman, but one is described as being

understandable by the ancient Greeks (who started the whole thing) so

why not include it as a footnote or appendix?

Having established

fairly early on that the points where a mathematical function

“reaches sea level” are known as zeros, why keep reverting

to the sea level analogy? And although the underlying theme throughout

the book is the apparent inextricable link between the zeta function’s

zeros and counting primes, the Riemann hypothesis, I could find no

clear, concise statement of exactly what Riemann said.

Spanning

over 2000 years, from the ancient Greeks to the 21st century, this is a

book I would thoroughly recommend.

Books on Fermat’s last

theorem (and there are some nice ones, such as Alf Van der Poorten’s

Notes on

Fermat’s last theorem) can take Wiles’ solution as their focal

point. Failing a solution, du Sautoy constructs his book around an

April’s Fool email-message by Bombieri in which he claimed that a young

physicist did prove the Riemann hypothesis after hearing a talk by Alain

Connes. Here’s du Sautoy’s account (on page 3)

According

to his email, Bombieri has been beaten to his prize. ‘There are

fantastic developments to Alain Connes’s lecture at IAS last wednesday.’

Bombieri began. Several years previously, the mathematical world had

been set alight by the news that Alain Connes had turned his attention

to trying to crack the Riemann Hypothesis. Connes is one of the

revolutionaries of the subject, a benign Robespierre of mathematics to

Bombieri’s Louis XVI. He is an extraordinary charismatic figure whose

fiery style is far from the image of the staid, awkward mathematician.

He has the drive of a fanatic convinced of his world-view, and his

lectures are mesmerising. Amongst his followers he has almost cult

status. They will happily join him on the mathematical barricades to

defend their hero against any counter-offensive mounted from the ancien

regime’s entrenched positions.

Contrary to physics,

mathematics doesn’t produce many books aimed at a larger public. To a

large extend this is caused by most mathematicians’ unwillingness to

sacrifice precision and technical detail. Hence, most of us would never

be able to come up with something like du Sautoy’s description of Weil’s

work on the zeta function of curves over finite fields (page 295)

It was while exploring some of these related landscapes that

Weil discovered a method that would explain why points at sea level in

them like to be in a straight line. The landscapes where Weil was

successful did not have to do with prime numbers, but held the key to

counting how many solutions an equation such as $y^2=x^3-x$ will have if

you are working on one of Gauss’s clock calculators.

But,

it is far too easy to criticize people who do want to make the effort.

Books such as this one will bring more young people to mathematics than

any high-publicity-technical-paper. To me, the chapter on quantum chaos

was an eye-opener as I hadn’t heard too much about all of this before.

Besides, du Sautoy accompanies this book with an interesting website musicofprimes and several of

his articles for newspapers available from his homepage are

a good read (in case you wonder why the book-cover is full of joggers

with a prime number on their T-shirt, you might have a look at Beckham in his

prime number). The music of the

primes will definitely bring many students to noncommutative

geometry and its possible use to proving the Riemann Hypothesis.

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