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Stream: deprecated: discrete geometry and entanglement

Topic: Invitation

Eric Forgy (Dec 17 2020 at 00:59):

I recently wrote an invitation in #practice: applied ct that explains a bit of what this stream is about, but thought it would be good to have some info here as well (especially since that invitation is likely to scroll into oblivion).

In a nutshell, I am hoping to write a more expository follow up to:

• Discrete Differential Geometry on Causal Graphs

As part of that, I'm hoping to express some of the constructions functorially. The difficult thing is that my CT is pretty weak, so what would probably be trivial for most people here to see is not obvious to me.

Most of what happens here is me rambling as I wander in the dark. Any statement I make should be understood as an implicit question with me expecting to be wrong and hoping to be corrected.

So far, I am happy with my progress in Bourbaki III 10.10: Universal problem for derivations. At least to first order. The punchline is here, but I'll repeat it below:

To summarize, given an $A$-bimodule morphism $g: I\to E$, we get a unique derivation $d: A\to E$ given by

$d := g\circ\tilde d.$

Conversely, given a derivation $d: A\to E$, we get a unique $A$-bimodule morphism $g: I\to E$ given by

$g: \sum_i x_i\otimes y_i \mapsto \sum_i x_i d y_i.$

That can be summarized in the follow commuting diagram:

image.png

First order is already interesting and there are lots of applications, e.g. stochastic processes, finance, heat, fluid dynamics, etc, that only require the first order calculus, but I want to work out higher orders as well.

Some references that discuss the higher order calculus include:

• Remarks on the differential envelopes of associative algebras
• Differential Calculi on Commutative Algebras
• Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics

I am super happy if anyone pops in to ask questions or has suggestions and I am also very happy if anyone is interested in digging deeper and being a coauthor.

Eric Forgy (Jan 22 2021 at 00:21):

Update: 2021-08-01

Eric Forgy (Jan 22 2021 at 00:26):

Update: 2021-21-01