What I’ve been doing

After a hiatus of 6 months or so, I’m hoping to get back to posting again.

During the past half year I have been living in Berlin as a fellow of the Wissenschaftskolleg zu Berlin. During this time I’ve been involved in plenty of activities both research and non-research. Here is a brief description.

At the WiKo I’ve had the sincere pleasure to meet a great number of extremely talented people from all walks of academic life, from musicians and humanists to social scientists and natural scientists. The most striking thing in all these interactions has been the communication problem that we face in talking to our fellow academics. In listening to the colloquia of the fellows it is remarkable how broadly even basic methodologies differ from discipline to discipline. For example, in the humanist tradition, in contrast to the natural sciences, it is standard practice to sit and read the entire presentation with no visual cues. Also, it is often that case that such things as stating the hypothesis of an investigation is not done in certain disciplines (which causes no small amount of frustration for the natural scientists!). Conversely, many of the humanist and social scientists find the natural scientists “practical approach” somewhat off-putting and over-simplifying 🙂

Things I’ve learnt: (i) the value of a narrative in a presentation; (ii) everything traces back to Aristotle and Plato, and what doesn’t most certainly can be found in the works of Darwin; (iii) a sense of humility.

I also gave a colloquium on my on work, which can be found here: Dienstagskolloquium 2010. In designing the presentation I hoped to made it possible for all of the fellows to understand. I’m not sure if I completely achieved that goal; please judge for yourself.

On the research front it was my objective to immerse myself in a problem I’ve been idly contemplating for many years; my thinking being that this year will be the last chance I’ll get for a while to work on highly speculative research. I’m not sure that it has been entirely successful, but I certainly did learn a lot of interesting stuff.

I’ll briefly sketch the idea here. Basically I was trying to develop a “canonical” procedure to quantise algebraic geometric spaces in positive characteristic. What does this mean? To explain this it is helpful to recall what geometric quantisation means. Here the input is a symplectic manifold (or a Poisson manifold), with a closed two-form which can be thought of as generating hamiltonian flow. Roughly speaking: I really just mean some classical dynamical system. The output of this procedure is a hilbert space {\mathcal{H}}, and a hamiltonian operator {H} on that space which is meant to be associated in some canonical way with the input classical system. The best and most well-known example of this procedure in action is known as Schrödinger quantisation which takes the symplectic manifold {\mathbb{R}^n\times \mathbb{R}^n} with linear dynamics as input and outputs the quantisation {\mathcal{H} \equiv L^2(\mathbb{R}^n)}.

What I initially wanted to do was to replace {X} by an algebraic variety over a field of positive characteristic (actually, a general symplectic scheme) and find out what {\mathcal{H}} ought to be. I believe I pretty much succeeded in this initial goal: the result is what it ought to be in the case of the symplectic scheme {X = k^n\times k^n}, namely, {\mathcal{H} = L^2(k^n)}. Linear dynamics quantise to something familiar in the case of {k = \mathbb{Z}/2\mathbb{Z}}, namely (tensor products of) the pauli matrices. Rather interesting stuff happens for even the simplest nontrivial varieties, but I won’t elaborate here as it is lengthy and assumes rather a lot of background.

Why would anyone want attempt such a baroque thing? Well I’ve had hopes for many years that a natural quantisation program would be a first step in a program to provide another proof of the Weil conjectures in algebraic geometry from the perspective Polya-Hilbert approach. Needless to say, I have not been successful in this objective.

Lately I’ve been thinking about much more physical problems. After a break of several years I’ve been working again on matrix product states and all that stuff. And also thinking a lot about disorder and topologically ordered systems. I hope to report more on all of this in future posts.

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