Low Dimensional Topology

April 9, 2016

Arithmetic Chern-Simons

Filed under: Knot theory,Number theory,Quantum topology — dmoskovich @ 1:48 pm

The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- e.g. this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists. (more…)

September 10, 2009

Pseudo-Anosov automorphisms and curves over finite fields

Filed under: Mapping class groups,Number theory — Nathan Dunfield @ 10:15 pm

During a recent visit, number theorist Jordan Ellenberg told me about a “time-worn analogy” between

(a) A pseudo-Anosov homeomorphism acting on a surface.

(b) The Frobenius automorphism of a smooth algebraic curve X/\mathbb{F}_q.

Jordan has two very interesting posts on this subject, one on what the dilatation should be in case (b) and a recent one where he discusses the finite field analogue of the following question related to the Virtual Haken Conjecture:

Conjecture: A hyperbolic 3-manifold which fibers over the circle has a finite cover with b_1 > 1.

As I noted earlier, this is known when the fiber has genus two, or more broadly if the monodromy is hyperelliptic. Intriguingly, Jordan explains the analogous conjecture in the context of (b) is also known in exactly this case…

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