Low Dimensional Topology

May 4, 2014

Low dimensional topology of information

Filed under: Logic,Misc. — dmoskovich @ 5:32 am

Is information geometric, or is it fundamentally topological?

Information theory is a big, amorphous, multidisciplinary field which brings together mathematics, engineering, and computer science. It studies information, which typically manifests itself mathematically via various flavours of entropy. Another side of information theory is algorithmic information theory, which centers around notions of complexity. The mathematics of information theory tends to be analytic. Differential geometry plays a major role. Fisher information treats information as a geometric quantity, studying it by studying the curvature of a statistical manifold. The subfield of information theory centred around this worldview is known as information geometry.

But Avishy Carmi and I believe that information geometry is fundamentally topological. Geometrization shows us that the essential geometry of a closed 3-manifold is captured by its topology; analogously we believe that fundamental aspects of information geometry ought to be captured topologically. Not by the topology of the statistical manifold, perhaps, but rather by the topology of tangle machines, which is quite similar to the topology of tangles or of virtual tangles.

We have recently uploaded two preprints to ArXiv in which we define tangle machines and some of their topological invariants:

Tangle machines I: Concept
Tangle machines II: Invariants (more…)

November 26, 2013

What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”.  It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

Conference banner

October 18, 2010

Homogeneous hyperbolic groups

Filed under: Geometric Group Theory,Logic — Henry Wilton @ 11:15 pm

I just got back from an extremely enjoyable meeting in Montreal, where I learned some nice new results about homogeneity, a natural logical property of groups. Now, I realise that logic may seem like a distant and irrelevant subject to many topologists, but I hope you’ll bear with me, as in this case I think there’s a very interesting relationship between logic and geometric group theory. If you need to be further convinced, perhaps it would help if I told you that this circle of ideas is intimately connected to Sela’s solution to the homeomorphism problem for hyperbolic manifolds. (Perhaps I’ll write some more about that on another occasion.) (more…)

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