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Stream: theory: applied category theory

Topic: Invitation: CT approach to discrete differential geometry

Eric Forgy (Dec 13 2020 at 19:42):

Hello :wave:

I introduced myself here so won't repeat myself, but in a nutshell, I am working on an updated version of an old preprint I wrote with Urs Schrieber:

• Discrete Differential Geometry on Causal Graphs

The goal is to:

• Be more expository.
• Include concrete applications / simulations with numerical results.
• Demonstrate the naturalness / functorality of the approach.

It is the last part, I could use help with :pray:

In our paper, we start with sets, build up vector spaces, build up algebras, introduce tensor / direct products, adjoint / dagger objects and inner products that, in the end, results in a form of discrete differential geometry that, under certain conditions converges to usual continuum differential geometry. From a numerical analysis perspective, this is kind of a holy grail because it means any discrete solution is guaranteed to converge to the continuum solution because the mathematical framework itself converges.

Each step of the way "feels right". Although we never used the word "functor", I nevertheless feel each step is functorial and want to make that explicit. If you've read any of the topics here with me asking questions (thanks John!), it is painfully obvious my CT chops are pretty weak, but I do have a knack for solving real-world problems and building numerical models.

In October, I got the crazy idea to attempt a return to academics and apply for a faculty position in engineering at San Jose State University. While working on my research statement (I haven't published any serious research in nearly 20 years!), I decided to try to whip this paper out first to demonstrate "I still got it". It is taking me longer than I expected (maybe I don't "have it" :sweat_smile: ) and the window for me to apply is closing soon.

I'll continue to ask questions here, but I am also open (nay happy) to collaborate if anyone is interested in working on this together.

In a nutshell, I hope to say something like

Discrete differential geometry is simply a functor from $\mathsf{DiGraph}$ to $\mathsf{DGHilbMod}$.

Applications include (but not limited to):

• Computational finance (derivative pricing, security modeling, market projections, etc)
• Computational physics (electromagnetic theory, fluid dynamics, heat / thermodynamics, etc)
• Computer graphics (surface modeling, video games, etc)
• Computational linguistics (natural language processing, knowledge graphs, etc)
• Open dynamical systems (I haven't looked into this much yet, but I expect DDG should be natural in network theory, Petri nets, etc.)

So far, I have functors for the first couple of steps:

• Directed graphs: $X\overset{\mathcal{G}}{\to}\mathsf{Set}$
• Free vector spaces on graphs: $X\overset{\mathcal{G}}{\to}\mathsf{Set}\overset{\mathcal{F}_V}{\to}\mathsf{Vect}$

I'm not sure where to go from here, but I'm thinking

• Free monoids on graphs: $X\overset{\mathcal{G}}{\to}\mathsf{Set}\overset{\mathcal{F}_V}{\to}\mathsf{Vect}\overset{\mathcal{F}_M}{\to}\mathsf{Mon(Vect)}$

I "think"

$\mathcal{F}_M\circ\mathcal{F}_V\circ\mathcal{G}: X\to\mathsf{Mon(Vect)}$

gives me a tensor algrebra on graphs, but I still need to understand this better.

I feel like this is all "natural" and should have a nice CT presentation in terms of free functors, but I'm struggling to make it work.

If this sounds interesting to anyone, please feel free to reach out. I could use any help I can get.

Thanks and happy holidays! :santa: :holiday_tree:

Eric Forgy (Dec 13 2020 at 20:49):

Btw, the paper I started to work on was less ambitious, but I think could still be interesting. While writing my research statement, I did a quick catch up on some of the cool things that happened while I was away and I am still blown away by:

• Quantum Reality: Space, Time, and Entanglement (link points to the part where the mind blowing point starts )
  

I don't know if there is any meaningful relation to my work, but something there really resonated. Entanglement on a boundary QFT gives rise to geometry in the bulk space. Removing entanglement leads to splitting up of the bulk geometry until, in the extreme, geometry disapears when entanglement is removed completely.

Something maybe related happens in discrete differential geometry. A fundamental "discrete 1-form" we denote $G$ and call the "graph operator" gives both the differential structure as well as the geometry (at least on "diamond graphs", i.e. cubic graphs with time flowing along the major diagonal). Discrete 0-forms $f\in\Omega^0$ are basically elements of the free vector space on the set of nodes of a directed graph written as

$f = \sum_{i\in\mathcal{G}^0} f(i) e^i$

with unit element $1 = \sum_{i\in\mathcal{G}^0} e^i$ and discrete 1-forms $\alpha\in\Omega^1\subset\Omega^0\otimes\Omega^0$ are basically elements of the free vector space on the set of directed edges written as

$\alpha = \sum_{(i,j)\in\mathcal{G}^1} \alpha(i,j) e^i\otimes e^j$

with the graph operator

$G = \sum_{(i,j)\in\mathcal{G}^1} e^i\otimes e^j$

which plays a role similar to an adjacency matrix.

When the directed graph is complete, i.e. there is an edge $i\to j$ for every pair of nodes $i$ and $j$, the graph operator can be written as

$G = 1\otimes 1.$

The discrete differential geometry resulting from this graph operator is the "universal differential envelope" (see e.g. this). This DDG is not very interesting (similar to how bulk geometry is not very interesting when you remove all entanglement).

Now, if you consider $\Omega^0$ to be a Hilbert space (because it is), then $\Omega^1$ is a subset of the product Hilbert space $\Omega^0\otimes\Omega^0$ (Note: When $G=1\otimes 1$ we have $\Omega^1=\Omega^0\otimes\Omega^0$) and you can interpret $G = 1\otimes 1$ as having no "entanglement" (because it isn't entangled). You introduce entanglement by removing directed edges from $G$. In doing so, you start introducing meaningful geometry to the resulting DDG.

I don't know if this DDG version of $\mathsf{Entanglement}\leftrightarrow\mathsf{Geometry}$ has anything to do with the holographic version of $\mathsf{Entanglement}\leftrightarrow\mathsf{Geometry}$, but I think it would still make a nice short article looking at "graph entanglement".

Eric Forgy (Dec 13 2020 at 21:45):

Btw^2, although I'll probably continue to ask general questions in #learning: questions , I might try to work out some things in a stream I created #working: discrete geometry and entanglement. Feel free to join / subscribe if interested :blush:

Simon Burton (Dec 13 2020 at 22:41):

Interesting.... Have you seen this book: https://www2.math.upenn.edu/~ghrist/notes.html ?

Eric Forgy (Dec 13 2020 at 22:42):

Hi Simon. No I haven't, but thanks. I will check it out :blush:

Eric Forgy (Jan 09 2021 at 09:38):

If anyone is interested, I've written up a summary of my progress over the last month or so here.

Thank you to everyone who has helped me so far :pray: I would have made zero progress without John Baez. Amar and Dan Doel also made helpful comments that influenced my thinking a lot :raised_hands:

Still a lot of work to do though :muscle: :runner: