I think I can answer what "?" means in simple terms, since I've been conjecturing very this for some time. Homology (abstract simplex algebra) sits in between representation theory (abstract linear algebra) and knot theory (fundamental topology).
Entropy enters the equation when you frame all three in terms of information theory, dynamical systems, and chaos theory. I can only put this into terms on my own conjectures, but the question for me was how can you represent, encode, and operate algebraically on information and state of a system? Or in other words, can you represent dynamical information (work) over time. More specifically, you want to know when the homological representational model becomes chaotic and to do that you need a way to measure its entropy.
My conjecture was along the lines, that higher ranking primality of the knot simplex of a dynamical model of some datum of the system implies less entropy, but only if state dictates the representation, then the operations that take the ambient isotopy (let's call it a "polymer") from state k to k' that doesn't result in any new topology (but doesn't violate it either) is the unknot and represents no new or "surprise" entropy, and there no new information. Or at least that's how far I was at as of the other day, but the question was always how to model this entropy in terms of homology to start.
The practical implications of this paper are profound, but probably won't see application in software and networking until a number of conjectures in prime number theory are answered. But still, this was long over due. HN is certainly more interesting at night, ha.
To me, representation theory means a lot more than abstract linear algebra. Of course, one can define terms however one wishes, including in such a way that these two become synonymous—and, also of course, maybe you were intentionally using the terms informally to give an idea of your meaning without going into too much technicality—but, to me, abstract linear algebra is the study (for each field $k$) of the category whose objects are $k$-vector spaces and whose morphisms are $k$-linear maps, whereas representation theory is the study of the category whose objects are triples $(G, X, \alpha)$ of a group $G$, a space $X$, and an action $\alpha$ of $G$ on $X$ (let's stipulate—though not everyone restricts representation theory to this setting!—that $X$ is a $k$-vector space, and $\alpha$ is a linear action), with morphisms being intertwiners in an obvious sense.
That is, in some sense, I view representation theory as somehow 'reifying' the morphisms of abstract linear algebra to become objects (a perspective which could, and sometimes does, lead one down the path of $n$-categories).
I would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc.
I think going down the route of trying to identify fields of maths with specific “things that can be stated in category theoretical language” is a bit wrong and bordering on hubris - sure, all group representations can be stated as some kind of functor between categories, but this is just convenient compact language for describing the definition. Any deep theorems that follow can almost certainly not be deduced from the categorical description, and are many times inhibited or obscured by it.
> I think going down the route of trying to identify fields of maths with specific “things that can be stated in category theoretical language” is a bit wrong and bordering on hubris - sure, all group representations can be stated as some kind of functor between categories ….
I missed this point in my earlier reply, and it's too late to edit. I certainly agree that it's easy to be misled about the utility or appropriateness of a category-theoretic taxonomy, but I wasn't attempting that here. Rather, I was just looking for a way to capture the intuition I have about the different way in which representation theory looks at what are, in some sense, the same kinds of objects as those considered in abstract linear algebra.
(Come to that, I was specifically avoiding looking at group representations as functors; not that it much matters, since, again, the category theory was just a way to try to formalise intuition rather than an attempt to prove anything, but I was actually going the other way by regarding the representations as objects.)
> I would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc.
That's a very good point about other kinds of 'representers' —though I think it doesn't change the central point that, in some sense, linear algebra seems to focus on "spaces as objects", whereas representation theory, loosely, focuses on "morphisms as objects".
I agree that most people will mean when they refer to representations to refer to linear actions, but every so often one does encounter references to, say, actions on sets as 'permutation representations' (although that also has a, closely related, linear-action meaning). I just meant to acknowledge that occasional extra inclusiveness.
Representation theory by virtue I feel has more angles than most branches of mathematics, but your definition isn't off at all. I tend to focus in particular in terms of information theory which is where the combinatorial and symmetric groups, to name a few sides of representation theory, are of particular interest, but isn't the entire picture by a long shot.
> probably won't see application in software and networking until a number of conjectures in prime number theory are answered
The thing about conjectures like those and their relevance to practical applications is you can just assume whichever ones you like.
That's because either your choice is correct (or, for the pedants, at least 'not inconsistent') in which case great you didn't need to wait for a proof, or your choice is incorrect. In that case, then if that wrecks your application you've just found some excellent observational evidence towards resolving it. If it doesn't wreck your application, then that means you didn't need to have considered in in the first place.
Topology is often used to prove existence or non-existence. If you base that on a shaky conjectures you’re not any wiser.
If you apply it to algorithms that means the difference may be that the algorithm encrypts your data safely every time or just 99.99% of the time. Or you can crack it by precomputing something for 100k CPU years.
Please name some actual examples in which the truth or falsity of any particular "conjecture[] in prime number theory" has a observable bearing on the performance of any algorithm.
I felt it was clear from context that the OP meant one of the serious conjectures like GRH, which is an awful long way away from being 'shaky'.
The paper didn't mention knot theory or dynamical topology because, like I mentioned, those are areas that my research bring into the picture, but was the result of (what I suspect) is the same end goal. I read through the author's blog and my jaw kind of dropped, it was uncanny how similar paths we went down. It's clear he's interested in a general theory of linguistics that can be described in the language of algebraic topology among other mathematical branches and theories, so information theory was obviously involved. However, he's way ahead of me in respect to the subject of the paper; most of my spare time has been dedicated to the greater evil, which is prime number theory, but that paper (although has proof of concept) is still far from polished but I know who's perfect now to review it when it's ready.
I really liked this article. It's a nice concept to write an informal introduction for a paper and go over the backgrpund a bit. It makes it much more accessible to non phd's such as my self!
>>> entropy behaves a little like "d of something" for some suitable (co)boundary-like operator d
Even if a coincidence, this is an engaging interactive presentation of "functional" Shannon Entropy. Reminds me of Loop Quantum Cosmology. What happens when we quantize spacetime? Does entropy emerge from quantum probability operads? And is it predictive? Can we determine the ultimate fate of the universe from the current state?
Many websites have that problem nowadays. This is why I prefer RSS anyway. But RSS feeds have been degraded to the first paragraph on many sites, so you have to use a full-text RSS service or host your own.
If you want a clutter free version of this post, simply add https://morss.it/https://www.math3ma.com/blog/rss.xml to your RSS reader of choice or click on the link and read it in the browser. However, in this particular case there's a problem with the inline math expression, because they are rendered with js, but I find it still readable.
Entropy enters the equation when you frame all three in terms of information theory, dynamical systems, and chaos theory. I can only put this into terms on my own conjectures, but the question for me was how can you represent, encode, and operate algebraically on information and state of a system? Or in other words, can you represent dynamical information (work) over time. More specifically, you want to know when the homological representational model becomes chaotic and to do that you need a way to measure its entropy.
My conjecture was along the lines, that higher ranking primality of the knot simplex of a dynamical model of some datum of the system implies less entropy, but only if state dictates the representation, then the operations that take the ambient isotopy (let's call it a "polymer") from state k to k' that doesn't result in any new topology (but doesn't violate it either) is the unknot and represents no new or "surprise" entropy, and there no new information. Or at least that's how far I was at as of the other day, but the question was always how to model this entropy in terms of homology to start.
The practical implications of this paper are profound, but probably won't see application in software and networking until a number of conjectures in prime number theory are answered. But still, this was long over due. HN is certainly more interesting at night, ha.