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Tag: noncommutative

time for selfcriticism

The
problem with criticizing others is that you have to apply the same
standards to your own work. So, as of this afternoon, I do agree with
all those who said so before : my book is completely unreadable and
should either be dumped or entirely rewritten!

Here’s what happened :
Last week I did receive the contract to publish _noncommutative
geometry@n_ in a reputable series. One tiny point though, the editors
felt that the title was somewhat awkward and would stand out with
respect to the other books in the series, so they proposed as an
alternative title _Noncommutative Geometry_. A tall order, I thought,
but then, if others are publishing books with such a title why
shouldn’t I do the same?

The later chapters are quite general, anyway,
and if I would just spice them up a little adding recent material it
might even improve the book. So, rewriting two chapters and perhaps
adding another “motivational chapter” aimed at physicists… should
be doable in a month, or two at the latest which would fit in nicely
with the date the final manuscript is due.

This week, I got myself once
again in writing mode : painfully drafting new sections at a pace of 5
to 6 pages a day. Everything was going well. Today I wanted to finish
the section on the “one quiver to rule them all”-trick and was
already mentally planning the next section in which I would give details
for groups like $PSL_2(\mathbb{Z}) $ and $GL_2(\mathbb{Z}) $, all I
needed was to type in a version of the proof of the last proposition.

The proof uses a standard argument, which clearly should be in the book
so I had to give the correct reference and started browsing through the
print-out of the latest version (about 600 pages long..) but… _I
could not find it!???_ And, it was not just some minor technical lemma,
but a result which is crucial to the book’s message (for the few who
want to know, the result is the construction and properties of the local
quiver at a semi-simple representation of a Quillen-smooth algebra). Of
course, there is a much more general result contained in the book, but
you have to be me (or have to be drilled by me) to see the connection…
Not good at all! I’d better sleep on this before taking further
steps

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master class 2007

Next
week our master programme on noncommutative geometry
will start. Here is the list of all international mini-courses (8 hours
each) and firm or tentative dates. For the latest update, it is always
best to check with the Arts seminar
website
.

  • Hans-Juergen Schneider (Munich) “Hopf Galois extensions and
    quotient theory of Hopf algebras”. February 20-23 each day from
    10h30-12h30.

  • Markus Reineke
    (Wuppertal) “Representations of quivers”. February 27-28, March 1-2
    each day from 10h30-12h30.


  • Arthur Ruuge
    (Moscow) “Semiclassical approximation of quantum
    mechanics”. March 6-9 each day from 10h30-12h30.
  • Rupert Yu
    (Poitiers) in March or April.
  • Isar Stubbe (Antwerp) in April.
  • Fred Van Oystayen (Antwerp) in April.
  • Raf
    Bocklandt (Antwerp) in April or May.
  • Goro Kato (Los Angeles)
    in May.
  • Florin Panaite (Bucharest) in May.
  • Pjotr
    Hajac (Warsaw) in June.

Apart from these mini-courses
there will be four regular courses (approx. 30hrs each) during the whole
semester.

  • Raf Bocklandt “Knot theory”.
  • Lieven Le Bruyn “Noncommutative geometry”.
  • Geert Van
    de Weyer “Quantum groups”.
  • Fred Van Oysyaeyen
    “Noncommutative algebra”.

<

p>Dates and places of all
lectures will be made available through the Arts seminar
site
.

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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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