In this mini-course, we explain how to do symplectic and Poisson geometry in the non-commutative "Kontsevich world". The plan is as follows:
Lecture 1: We'll develop differential calculus on free associative algebras, define vector fields and differential forms in this framework and introduce a notion of Poisson brackets (Van den Bergh double brackets).
Lecture 2: We'll revisit some classical examples of symplectic and Poisson manifolds including coadjoint orbits and moduli spaces of flat connections and show that they can be treated using the non-commutative calculus of the previous lecture.
Lecture 3: We'll introduce the notion of a non-commutative divergence and show how it can be applied to the Kashiwara-Vergne problem in Lie theory.
Lecture 4: Our final example is the Goldman-Turaev Lie bialgebra defined in terms of 2d topology. We'll explain that the Kashiwawra-Vergne theory and moduli spaces are naturally unified in this construction.
The mini-course is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.
It may be viewed as one of the intellectual scandals
of present-day theoretical physics that, more than
ninety years after the discovery of matrix- and wave
mechanics, there is still an enormous amount of
of confusion surrounding the deeper meaning of quantum
mechanics. This mini-course is inspired by the desire
(or illusion?) that we are actually able to do something
about this problem, and that it is possible to come up
with a clear framework enabling us to interpret this
wonderful theory.
I start my course with a definition of isolated (but open)
systems in quantum mechanics. I then discuss a fundamental
mechanism of "Loss of Access to Information" exhibited by a
large class of isolated systems, and I outline some of its
implications. This will enable me to introduce a suitable
notion of "events" into the theory and to show that only
isolated systems exhibiting "Loss of Access to Information"
can feature events.
I then develop the quantum theory of direct (von Neumann)
measurements and attempt to clarify the "ontology" under-
lying quantum mechanics. (This part is the one that usually
triggers controversies and confusion. I hope my discussion
will alleviate some of this confusion.)
In a last part, I develop the theory of indirect (Kraus)
measurements and discuss various concrete examples. The
results discussed in this part may be of interest also
for theories of perception.
Various very concrete open mathematical problems will be
described.
Notes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Classical field theory is studied using the language of connections on principal bundles. We will begin this minicourse by reviewing this formalism, especially focusing on eletromagnetism as a U(1) gauge theory. We will see how the example of the dirac monopole forces us to "twist" our bundles by a gerbe. After a quick introduction to the differential geometry of gerbes, I will explain recent work on how to interpret abelian monopoles as connections on meromorphic sections of gerbes.
TBA