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Eric Forgy (Oct 30 2023 at 18:16):Hello! :wave:
I am a bit of a scientific mutt. My day job is in the field of decentralized finance building the financial primitives needed to free us from the zombie banks who still exist today only because of government bailouts spread over decades that have eroded the well being of most people while lining the pockets of fat cat banking executives. We don't need them anymore.
My startup (which I promise not to shill here), CavalRe, is based on the Arxiv preprint:
The basic ideas of discrete (noncommutative stochastic) calculus that motivated this come from the beautiful pioneering work of Dimakis and Mueller-Hoissen.
After a PhD in computational physics from UIUC back in 2002, I spent 2 years at MIT Lincoln Laboratory before selling my soul to Wall Street (so I have been in the belly of the beast). The highlight of my academic research career came when I published this article with Urs Schreiber (whom I admire deeply):
Now close to 20 years ago although it feels like yesterday.
Despite having some level of scientific / commercial success outside category theory, when it comes to (higher) category theory, I've always felt a bit like a jester in a court of kings and queens. I can see glimpses of its beauty through a thick veil of fog just out of reach to me (so far).
When John Baez (whom I've admired and followed since the early days of sci.physics.research) switched fields to climate science, I wanted to see if I could help. So far, the most interesting result to come out of that from me is summarized in an old blog post:
It might sound a little crazy, but Navier-Stokes can be thought of as a kind of "stochastic" Maxwell's equation.
When I'm not trying to create a new decentralized financial system, my thoughts inevitably return to the beauty of foundational physics. In that regard, I am in the "finitist" camp and have always enjoyed following progress in causal set theory. If you are not familiar, this is a beautiful introduction to the subject:
I popped back into this Zulip shortly after having my mind blown by the video (link points to where things get interesting):
featuring Brian Greene (moderator) and Mark Van Raamsdonk, Gerard ’t Hooft, David Wallace, Birgitta Whaley.
I would feel like a crackpot trying to explain how deeply profound this is so I won't even try, but have a look for yourself.
In my opinion, I think the intersection of entanglement <-> spacetime, causal set theory and the work that Urs and I started based on pioneering work of Dimakis and Mueller-Hoissen would be a fruitful one. All of this is screaming to have someone wave the wand of arrow theory over it to help make it all come together in one beautiful coherent theory. I am trying, but not making much progress, so if you see me rambling here, this is what I'm trying to do.
Eric Forgy (Nov 10 2023 at 21:33):Little update...
Thanks to some helpful comments from @JR in response to my question, I think I have a concrete way to start connecting causal set theory and what I call "discrete (noncommutative) differential geometry".
In our paper, we show that on special directed graphs, namely those with no intermediate edges and no opposite edges, we can construct a mimetic version of differential geometry that captures most of the salient feature of the continuum version including the formulation of discrete gauge theory among other things.
It turns out that those conditions, i.e. no intermediate or opposite edges, correspond to the Hasse diagram, or transitive reduction, of a poset. Very cool.
We called our directed graphs (and the resulting cell structure): "diamond complexes".
Starting with a poset, you can forget edges to get the transitive reduction. Similarly, given a special kind of directed graph, the kind we care about, you can generate a poset. This sounds like a forgetful functor and its free functor.
Eric Forgy (Nov 10 2023 at 21:42):In our paper, we restricted attention to directed hypercubes, which are a simple class of directed space, but since then I've wanted to extend it to more general directed spaces. I am pretty confident this can be done now (although intuitively I always felt it should be possible).
Eric Forgy (Nov 25 2023 at 18:18):Update:
It has been a blast the last few weeks reconnecting with old friends / acquaintences, making some new ones and thinking about discrete noncommutative differential geometry (DNDG) again. Helpful discussions here have added to my understanding and I think we made some real progress.
Thank you John, Todd and Mike for infinite patience as always. Thank you JR for your interest and pointing me in the right direction. Thank you David and Jason for your interest and for being good sports about finding a 3-diamond smaller than a cube. That was fun :blush: Thank you Jean-Baptist for interest and questions :blush:
To highlight some progress:
We now have a definition of diamond graph. We understand that the transitive reduction of a poset is a diamond graph, but I think more importantly, given a diamond graph, we can construct a poset. This provides a direct link between DNDG and causal set theory (CST).
However, note that the link was already there in our 2004 paper, but only for the special case of hypercubic graphs. At the bottom of page 56:
Causal sets. The above shows that the graph operator singles out a causal structure on our discrrete space. We could more generally consider scalar multiples of the preferred metric operator (4.45):
These describe geometries where each discrete lightcone is identified by and carries a spacetime volume given by . This is the data used in causal set theory [24] to describe spacetime geometry. We here see that the formalism used here, with the preferred role that the graph operator plays in it, naturally makes contact with concepts known from causal set theory.
There are so many remarkable results in that paper (most due to Urs) :blush:
We almost have a good definition for general diamond complexes that is not restricted to hypercubic graphs. Instead of merely enforcing to impose equivalence classes of diamonds, we also need to impose that the boundary of an -diamond consists of -diamonds. An actual definition is within reach (and close enough already by my standards :sweat_smile: )
Since an -diamond is an equivalence class of paths of length , that also means (thanks to Jason!) that we can think of them as multipartite graphs where each "time step" is a "part". This observation should help me communicate more clearly :+1:
Just as I am about to go back into hiding again, I stumbled across some more beautiful research.
First, this old paper that I'm sure I would have seen back in grad school, but failed to connect to my work back then:
John had this to say about it:
Ted Jacobson's paper contains one of the most amazing results I know connecting thermodynamics to general relativity. He's a cool guy, too.
I agree, but I would also rank this one (also by Ted!) as pretty high up there as some of the coolest stuff I've seen:
Here is an awesome related video:
from Lecture at the Quantum Gravity Foundations: UV to IR held at KITP, Mar30-Jun19, 2015.
It is fun to hear comments from Mark Van Raamsdonk in the audience, because Mark's work on entanglement is what inspired my last round of soul searching.
My indulgence for this round of "Math is good for the soul" is coming to a close and I need to get back to focusing on my day job.
I have the Wolfram dream. Hopefully I make enough money from my day job so I can retire and spend the rest of my days thinking about this stuff :blush:
PS: I am similarly open / transparent in my "day job" so if you're interested in following along with that (which I think is pretty cool actually) please feel free to join our Discord: https://discord.gg/VA6MhZ8cVb :cowboy:
I'm more than happy to answer questions there.