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Tag: Riemann

2006 paper nominees

Here are
my nominees for the 2006 paper of the year award in mathematics &
mathematical physics : in math.RA : math.RA/0606241
: Notes on A-infinity
algebras, A-infinity categories and non-commutative geometry. I
by

Maxim Kontsevich
and
Yan Soibelman
. Here is the abstract :

We develop
geometric approach to A-infinity algebras and A-infinity categories
based on the notion of formal scheme in the category of graded vector
spaces. Geometric approach clarifies several questions, e.g. the notion
of homological unit or A-infinity structure on A-infinity functors. We
discuss Hochschild complexes of A-infinity algebras from geometric point
of view. The paper contains homological versions of the notions of
properness and smoothness of projective varieties as well as the
non-commutative version of Hodge-to-de Rham degeneration conjecture. We
also discuss a generalization of Deligne’s conjecture which includes
both Hochschild chains and cochains. We conclude the paper with the
description of an action of the PROP of singular chains of the
topological PROP of 2-dimensional surfaces on the Hochschild chain
complex of an A-infinity algebra with the scalar product (this action is
more or less equivalent to the structure of 2-dimensional Topological
Field Theory associated with an “abstract” Calabi-Yau
manifold).

why ? : Because this paper
probably gives the correct geometric object associated to a
non-commutative algebra (a huge coalgebra) and consequently the right
definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve
tried to explain how this links up with my own interpretation and since
then I’ve thought more about this, but that will have to wait for
another time. in hep-th : hep-th/0611082 : Children’s Drawings From
Seiberg-Witten Curves
by Sujay K. Ashok, Freddy Cachazo, Eleonora
Dell’Aquila. Here is the abstract :

We consider N=2
supersymmetric gauge theories perturbed by tree level superpotential
terms near isolated singular points in the Coulomb moduli space. We
identify the Seiberg-Witten curve at these points with polynomial
equations used to construct what Grothendieck called “dessins
d’enfants” or “children’s drawings” on the Riemann
sphere. From a mathematical point of view, the dessins are important
because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on
them. We argue that the relation between the dessins and Seiberg-Witten
theory is useful because gauge theory criteria used to distinguish
branches of N=1 vacua can lead to mathematical invariants that help to
distinguish dessins belonging to different Galois orbits. For instance,
we show that the confinement index defined in hep-th/0301006 is a Galois
invariant. We further make some conjectures on the relation between
Grothendieck’s programme of classifying dessins into Galois orbits and
the physics problem of classifying phases of N=1 gauge theories.

why ? : Because this paper gives the
best introduction I’ve seen to Grothendieck’s dessins d’enfants
(slightly overdoing it by giving a crash course on elementary Galois
theory in appendix A) and kept me thinking about dessins and their
Galois invariants ever since (again, I’ll come back to this later).

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music of the primes

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
and infuriating
, October 5, 2004
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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pdfsync

I
expect to be writing a lot in the coming months. To start, after having
given the course once I noticed that I included a lot of new material
during the talks (mainly concerning the component coalgebra and some
extras on non-commutative differential forms and symplectic forms) so
I\’d better update the Granada notes
soon as they will also be the basis of the master course I\’ll start
next week. Besides, I have to revise the Qurves and
Quivers
-paper and to start drafting the new bachelor courses for
next academic year (a course on representation theory of finite groups,
another on Riemann surfaces and an upgrade of the geometry-101 course).

So, I\’d better try to optimize my LaTeX-workflow and learn
something about the pdfsync package.
Here is what it is supposed to do :

pdfsync is
an acronym for synchronization between a pdf file and the TeX or so
source file used in the production process. As TeX system is not a
WYSIWYG editor, you cannot modify the output directly, instead, you must
edit a source file then run the production process. The pdfsync helps
you finding what part of the output corresponds to what line of the
source file, and conversely what line of the source file corresponds to
a location of a given page in the ouput. This feature is achieved with
the help of an auxiliary file: foo.pdfsync corresponding to a foo.pdf.

All you have to do is to put the pdfsync.sty file
in the directory _~/Library/texmf/tex/latex/pdfsync.sty_ and to
include the pdfsync-package in the preamble of the LaTeX-document. Under
my default iTex-front-end TeXShop it
works well to go from a spot in the PDF-file to the corresponding place
in the source-code, but in the other direction it only shows the
appropriate page rather than indicate the precise place with a red dot
as it does in the alternative front-end iTeXMac.

A major
drawback for me is that pdfsync doesn\’t live in harmony with my
favorite package for drawing commutative diagrams diagrams.sty. For example, the 75 pages of the current
version of the Granada notes become blown-up to 96 pages because each
commutative diagram explodes to nearly page size! So I will also have to
translate everything to xymatrix&#
8230;

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