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Eric Forgy (Nov 28 2020 at 23:56):Matteo Capucci said:
Eric Forgy said:
(one may want to replace HoTT with Cubical Type Theory though)
Or "diamond" type theory :blush:
Is this a thing or just a meant to be suggestive? :laughing: Googling 'diamond type theory' I found this book and it took me some time to understand is probably crackpot 'mathematics'
The diamonds I had in mind don't seem to have anything to do with that article :blush:
My motivation has always been (maybe semi-secretly) about foundational questions in physics, e.g. my dissertation was trying to build a fundamentally discrete "differential geometry" hiding behind the fact that any such framework could be used for practical purposes in computational electromagnetics. So I felt that since differential geometry was beautiful and differential geometry did such a good job of describing continuum physics, that if physics was actually discrete (and the continuum emerged as some kind of limit), there should be an equally beautiful language to describe it.
In the preprint with Urs, we found there is a special structure we call "diamond" where the algebraic structures on finite / countable sets seem to mimectically line up nicely with continuum spacetime geometry (and there are some eerily cool applications in finance and climate science). A diamond is like a minimal causal set connecting two causally connect points in spacetime.
The results we wrote up were restricted to globally cubic directed lattices, which I'm afraid could make people think the results are limited to cubic lattices, but diamonds can discretize (a conjecture that still needs proving) any smooth manifold of the form . The idea is that since any smooth manifold can be triangulated, then extruding this triangulation and "lifting" edges so that every edge connects a node on one copy of to a subsequent copy, i.e. a directed edge consumes a step in time. I wrote up an old sketch on the nLab here.
Eric Forgy (Nov 28 2020 at 23:59):A (global) 2-diamond lattice is basically a binary tree, which is used in finance to model stochastic processes.
Matteo Capucci (he/him) (Nov 29 2020 at 00:17):This is superinteresting. I've been scavenging this stuff for the last hour and it's very intriguing. The fact that Ito's formula is just non-commutative differential geometry blew my mind.
Eric Forgy (Nov 29 2020 at 00:18):Yeah. Me too. That's why I decided to switch careers and go into finance :sweat_smile:
Matteo Capucci (he/him) (Nov 29 2020 at 00:21):Sounds like a profitable decision, too
Eric Forgy (Nov 29 2020 at 00:21):It also blows my mind that discrete Ito formula also falls out of discrete calculus on a binary tree (2-diamond). There are two limits of discrete calculus on a 2-diamond. One leads to stochastic calculus and one leads to exterior calculus (and it is tantalizing to think of how they can be connected)
Eric Forgy (Nov 29 2020 at 00:21):I call the whole framework a "meta" algorithm because it produces algorithms.
Matteo Capucci (he/him) (Nov 29 2020 at 00:22):What do you mean exactly by 'differential calculus' on a 'binary tree'? I downloaded some stuff but I've yet to delve into it
Eric Forgy (Nov 29 2020 at 00:23):I am working on a new preprint now revisiting the stuff I did with Urs and one point I like to highlight is that we effectively found a map (functor?) from directed graphs to differential graded algebras.
Eric Forgy (Nov 29 2020 at 00:26):"Discrete calculus" describes the result of applying this to a given directed graph, i.e. a discrete calculus is a differential graded algebra arising from a directed graph.
Eric Forgy (Nov 29 2020 at 00:27):There are different kinds of "calculi", e.g. Netwonian and stochastic, so you can start to ask questions like, "Given a special calculus, what is the directed graph that gives rise to this calculus?"
Eric Forgy (Nov 29 2020 at 00:29):I still remember Urs and I were working on our paper and he went offline for a week on a bike ride and when he came back, we had both independently found that the directed graph that gives rise to stochastic calculus is the n-diamond.
Eric Forgy (Nov 29 2020 at 00:29):In hindsight, it is obvious I guess :sweat_smile:
Eric Forgy (Nov 29 2020 at 00:33):There are a series of blog posts that go through a lot of this stuff that I was hoping to make into a book someday:
Eric Forgy (Nov 29 2020 at 00:47):This looks like a good place to start if interested :blush:
Eric Forgy (Nov 30 2020 at 00:55):More diamonds :large_blue_diamond:
This looks cool and relevant:
https://arxiv.org/abs/1904.01034
[Edit: Also https://arxiv.org/abs/1710.07379 ]
Eric Forgy (Nov 30 2020 at 01:21):Yes. So what Urs and I call "diamonds" correspond to elementary discrete "causal diamonds" in the sense of those :point_up: papers, i.e. Alexandrov intervals.
Notification Bot (Nov 30 2020 at 10:46):This topic was moved here from #general > Spans in 2-categories (sidetopic: Diamonds) by Matteo Capucci