Wednesday, January 12, 2011

If you ask me where I’m going, I’ll tell you where I’ve been.

This post is a brief recap of my work on my MSci project at UCL since July 2010. This post was hastily written. Please excuse any grammatical errors or typos.

Since I've now stated the significance of the Bost-Connes system, its time to consider the next step. But it's now mid-January, and due-dates are fast approaching for my MSci project. Thus, I thought I would take some time to write a short summary of everything I've covered since I started in July. I've actually not done much, and I'm ashamed of how little work I managed to do in such a long time. In any case, lets go over where I've been:

• Quantum Statistical Mechanics
Basic definitions (no examples), KMS states, symmetry groups. There's a lot of details I missed, particularly about the relationship between GNS representations and extremal KMS states. I have books by Bratteli and Robinson on Operator Algebras and Quantum Statistical Mechanics and Haag's Local Quantum Physics to help me out there.

• $\mathbb{Q}$-lattices and Commensurability
Basic definitions, and conversations with Professor Kim to help build some intuition, inversion and division.

• $C^\ast$-algebras
Basic definitions and few examples, though I'm missing quite a few details that I should like, particularly GNS representations. (Gelfand-Naimark Theorem would be nice, but not really needed here.)

• Profinite Completion of $\mathbb{Z}$
I spent quite a few hours trying to figure out Pontryagin duality and how $\text{Hom}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$. I would like to go over this again, using more sophisticated methods (category theory) and better understand how $C(\hat{\mathbb{Z}}) = C^\ast(\mathbb{Q}/\mathbb{Z})$

• Category Theory, Groupoids, and \'{E}tale Groupids
Categories, Functors, Groupoids, a few examples, \'{E}tale Groupids (a topology imposed on a groupoid), which I don't think I fully understand and would like to review.

• Convolution Algebras and $C^\ast$ groupoid algebras for an \'{E}tale Groupid
Basic definition on convolution, general discussion of the $C^\ast$ completion of a group/oid algebra via a Haar measure (without rigorously defining the Haar measure), discussion of why an Etale Groupid does not need this, and the completion of a $C^\ast$ groupoid algebra this way via a maximal norm of the image of a representation onto a Hilbert space.

• Generators and Relations for Bost Connes System via semigroup crossed products
Proof that the $C^\ast$ groupoid algebra for the Bost Connes System is a semigroup crossed product (I don't yet understand the proof fully, despite conversations with Dr Javier Lopez), which is vitally important, because later research shows that this is the right'' way to describe Bost-Connes type systems. Proof of the generators and relations via the semigroup crossed product.

• (Briefly) Hecke Algebras and the Bost Connes System
Brief definition and description of the Bost Connes system via Hecke Algebras, as they were used in the original paper.

• (Briefly) The Arithmetic Subalgebra and its analysis inspired from Eisenstein Series
I didn't spend too much time on this, because I \underline{think} it's more important for generalizations of the Bost-Connes system then for the 1-dim case, and I don't plan to discuss the generalizations in detail. In any case, I went over the results of trigonometric analogue of the Eisenstein series to study the arithmetic subalgebra of the Bost-Connes system. Its easiest to see this algebra exist from the Hecke algebra perspective (I think). Weil's boook on Elliptic Functions contains a lot of the ideas needed to do this in detail.

• Class Field Theory and the action of the KMS states of the Bost Connes System
Very brief discussion of adeles, ideles, and the global Artin map, and statement about how the Bost Connes System can recover a presentation of the maximal abelian extension of $\mathbb{Q}$ without reference to field extensions. Plan to generalize this result. See my last post, as well my notes from my October talk at the Undergrad Maths Colloquium

So what's next? Besides filling in the details for the above (I'm embarrassed that despite taking so long to cover that, I glossed over a lot of details), I essentially have two items left I want to cover: 1) the proof of the theorem in my last post and 2) a summary of the on-going work on generalizations of the BC system. The latter may require learning about Shimura varieties. Both will certainly require a better understanding of class field theory: the adeles, ideles and Artin maps. So that is where I'm going!