[Binghamton Geometry Topology Seminar] Geometry and Topology seminar: John Klein

[Cary Malkiewich]

Hi everyone,

This week we are very pleased to have John Klein (Wayne State University)
speaking in the geometry and topology seminar. The title and abstract are
below. As usual, the talk is at 2:50pm in WH-100E.

Please let me know if you would like to come to the dinner after!

Cheers,
Cary

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Title: Hypercurrents
Abstract: This talk poses the question as to what a higher dimensional
analog of a continuous time Markov chain might be, in which the time
parameter is replaced by arbitrary smooth manifold.

As a partial answer, we introduce the notion of a “protocol,” which
consists of a space whose points are labeled by real numbers indexed by the
set of cells of a fixed CW complex in prescribed degrees, where the labels
are required to vary continuously. When the space is a one-dimensional
manifold, then a protocol determines a continuous time Markov process.

In the presence of a homological gap condition, we associate to each
protocol a ‘characteristic’ cohomology class which we call the
hypercurrent. The hypercurrent comes in two flavors: one algebraic
topological and the other analytical. For generic protocols we show that
the analytical hypercurrent tends to the topological hypercurrent in the
'low temperature' limit.

We also exhibit examples of protocols having nontrivial hypercurrent.
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