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noncommutative Fourier transform

At the
noncommutative algebra program in MSRI 1999/2000, Mikhail Kapranov gave
an intriguing talk Noncommutative neighborhoods and noncommutative Fourier transform
and over the years I’ve watched the video of this talk a number
of times. The first part of the talk is about his work on Noncommutative geometry
based on commutator expansions
and as I’ve once worked through it
this part didn’t present problems. On the other hand, I’ve never
understood much from the second part of the talk which claims to relate
these noncommutative formal neighborhoods to _noncommutative Fourier
transforms_. The string coffee table has a post Kapranov
and Getzler on Higher Stuff
linking to two recent talks by Kapranov
on noncommutative Fourier transforms at the Streetfest. Marni
Sheppeard made handwritten notes available. I definitely should find the time
to get through them and have another go at the Kapranov-video…

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non-(commutative) geometry

Now
that my non-geometry
post
is linked via the comments in this
string-coffee-table post
which in turn is available through a
trackback from the Kontsevich-Soibelman
paper
it is perhaps useful to add a few links.

The little
I’ve learned from reading about Connes-style non-commutative geometry is
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
as topological information about this bad orbit space. An intruiging
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry
from this perspective are the Lecture on
Arithmetic Noncommutative Geometry
by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry
enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities
notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh
. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry
by Paul Smith who
maintains a
noncommutative geometry and algebra resource page
page (which is
also available from the header).

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz
and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces
by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper
. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine
being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.

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something to think about

This is
not going to be the post I should be writing (this morning I found out
that the last post
must have been rather cryptic as I didnt manage to get it explained to
people who should know at least half of the picture, so at the moment
Im writing out a short note giving the dictionary between the Kontsevich-Soibelman
approach
and my
own
. I’m still undecided whether this will make it here, or to
the arXiv or to my dustbin…).

Instead I want to draw your
attention to one of the best posts I’ve read lately. It’s
called A man’s character is his fate and it’s from
Christine C. Dantas’ blog Christine’s Background
Independence
and clearly has a history which you may know if you
somewhat followed (some) physics blogs this week or which you may
reconstruct from this and this from her site and something else.

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