The n-Category Café

I might have missed the point where this project of public reviews gained its momentum, but wouldn’t it be helpful to actually start each entry with a (ever so brief) review of the articles being linked to?

Unless I am missing somehting, currently these entries just say: “hey, did you know that there exist four articles with the following titles?”

The standard reaction to that is: no, I didn’t, and I am too busy to read them unless you give me some good reason why I should be interested. Such as a brief summary of the coolest parts to whet my appetite.

What do you think?

Right. Given that these papers are tutorials the titles sounded fairly self-explanatory to me, overlooking the fact that not everyone here is familiar with computer science semantics and quantum computational models. I must say so far this idea of public reviews has proven to be very successful: besides the comments posted here at the cafe there are also those directly emailed to us, which sometimes have been really extensive.

Having said that, here are the blurbs:

Domains, an order-theoretic structure [hence deserving its place here at the cafe], were Dana Scott’s brainchild in his search for a denotational semantics for the Lambda calculus, that is, a calculus of functions which also allows for recursion. Key results of Domain theory include fixed point theorems. One can think of domains as a novel way of doing topology. In particular, they provide topology with an operational interpretation, and also with a corresponding logic, as exposed by Samson Abramsky’s generalisation of Stone duality to Domains. Steve Vickers’ book is the place to read about this.

But it was realised that Domains also provide a powerful framework to study analysis, with work by Keye Martin, Abbas Edalat and Martin Escardo as the most notable examples. Martin’s notion of ‘measurement’ is key to these developments. This aspect of Domains is very present in Martin’s tutorial.

More recently, Keye Martin has been exploring other possible applications of Domain theory, in (quantum) information theory, thermodynamics, and most notably, space-time structure. The joint chapter with Prakash Panangaden provides details on this. A punchline of that chapter is that Dana Scott’s notations for Domain theory and Penrose’s axiomatisation of space-time structure perfectly coincide, and structurally they are basically talking about the same thing. At the time their respective theories were developed they were both at the Mathematical Institute here in oxford - no coincidence it seems.

Here in Oxford, Domain theory is a course on the curriculum for computer scientists and mathematicians, although now that Samson Abramsky and I both have special fellowships which prohibit us from teaching –isn’t that sad– it has been harder to maintain the course.

Here’s the wikipedia entry on domain theory .

Anyonic quantum quantum computing, or topological quantum computing as it also known, is a model of quantum computation, in which unitary gates are encoded in terms of low-dimensional topological properties of particles. More than any model of quantum computing it requires a categorical description, and this is not even controversial anymore. If a quantum computer of this kind would ever been build then this would be a ***killer application*** for category theory.

Here’s the wikipedia entry on topological quantum computing .