This is a page to keep track of our working groups and readings related to the project ''Infinity-categories, symbolic dynamics, and mathematical physics''.

== Seminars ==

'''Oct 6, 2011''' Thomas Seiller. Introduction to C*-algebras.

'''Sept 29, 2011''' Guillaume Theyssier talked about some aspects of symbolic dynamics that might benefit from a categorical approach.

== Readings ==

=== C*-algebras and dynamical systems ===

'''[http://arxiv.org/abs/1105.3250 C*-algebras associated with lambda-synchronizing subshifts and flow equivalence]''', by Kengo Matsumoto.
This is the paper we started with. It was probably a bit early to attack, and Tim quickly switched the focus to earlier, more accessible papers.

'''[http://www.uow.edu.au/~dpask/index_files/papers/ckaodg.pdf Cuntz-Krieger algebras of directed graphs]''', by Kumjian, Pask, and Raeburn. Pacific J. Math. 184 (1998), 161–174.

The authors specify, via a universal property, a C*-algebra associated to any given directed graph satisfying a condition called ''row finiteness'' (all vertices have finite input degree).
The idea is that the C*-algebra is a subalgebra of the algebra of bounded operators on an infinite-dimensional Hilbert space. A natural construction, starting from any infinite-dimensional Hilbert space, starts by choosing a specific isomorphism <math>H \cong \bigoplus_{v \in V} H</math>. That is, we specify a way of seeing <math>H</math> as a direct sum of itself over all vertices of the given graph. Then, for each <math>v</math>, choose now a specific isomorphism <math>H \cong \bigoplus_{t(e) = v} H</math>. That is, specify a way of seeing <math>H</math> as a direct sum of itself over all input edges of <math>v</math>.

=== C*-algebras and categories ===

'''[http://www.springerlink.com/content/n23286006t0x2685/ Topos theory and complex analysis]''', by Christiane Rousseau.

In any sheaf topos, there is a natural numbers object, given by the constant sheaf mapping any object of the underlying site to <math>\mathbb{N}</math>.
By adequately quotienting <math>\mathbb{N} \times \mathbb{N}</math>, we get an object <math>\mathbb{Q}</math> of rationals, which similarly maps any object to (the usual) <math>\mathbb{Q}</math>.
There are then at least two natural ways of defining an object of real numbers: as limits of Cauchy sequences, or as Dedekind cuts.
In case the site is spatial, i.e., is the poset of open sets of a space <math>X</math>, Dedekind cuts nicely yield the sheaf of continuous maps to <math>\mathbb{R}</math>.
Anyway, from such an object of real numbers, one may construct an object of complex numbers. Rousseau actually axiomatises such an object, and proceeds to perform a few constructions of analysis.

What might be of interest to us is that the object of complex numbers bears a striking resemblance to C*-algebras.
To start with, the object of complex numbers is a complete metric space object, as well as a ring object.
In particular, if the underlying space is compact, then its global sections are precisely the elements of the classical C*-algebra of continuous maps <math>X \to \mathbb{C}</math>. Furthermore, the internal ring operations agree with the C*-algebra's. And, although Thomas would certainly recall better than me, a Cauchy sequence converges for the usual norm iff it converges for the internal norm (hence in the internal language).

In the general case, continuous maps to <math>\mathbb{C}</math> do not form a C*-algebra, and one usually resorts to some compactification procedure. This is not necessary in the sheaf-theoretic approach: the norm of an internal complex number is a continuous map to <math>\mathbb{C}</math>, i.e., pointwise norm.

Infinity-categories, symbolic dynamical systems, and mathematical physics

2011-10-19T13:12:07Z

Thirs :

Infinity-categories, symbolic dynamical systems, and mathematical physics

2011-10-19T13:03:11Z

Thirs :

CatDyn

2011-10-19T13:01:05Z

Thirs : a déplacé CatDyn vers Infinity-categories, symbolic dynamical systems, and mathematical physics: Je savais pas que c'était forcément le titre affiché en haut de la page.

#REDIRECT [[Infinity-categories, symbolic dynamical systems, and mathematical physics]]

Infinity-categories, symbolic dynamical systems, and mathematical physics

2011-10-19T13:01:05Z

Thirs : a déplacé CatDyn vers Infinity-categories, symbolic dynamical systems, and mathematical physics: Je savais pas que c'était forcément le titre affiché en haut de la page.

== Infinity-categories, symbolic dynamical systems, and mathematical physics ==

This is a page to keep track of our working groups and readings related to the project ''Infinity-categories, symbolic dynamics, and mathematical physics''.

== Seminars ==

'''Oct 6, 2011''' Thomas Seiller. Introduction to C*-algebras.

'''Sept 29, 2011''' Guillaume Theyssier talked about some aspects of symbolic dynamics that might benefit from a categorical approach.

== Readings ==

=== C*-algebras and dynamical systems ===

=== C*-algebras and categories ===

'''[http://www.springerlink.com/content/n23286006t0x2685/ Link to the paper at Springer's]Topos theory and complex analysis''', by Christiane Rousseau.

2009-03-17T10:14:31Z