Entropy and Algebra and Topology = ?
math3ma.com
math3ma.com
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 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.
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.
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.)
Yes, that is a good point.
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.
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.
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 answer at the end is that we get a derivation (Shannon Entropy) on the operad of simplices. What is the connection to knots and chaos?
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?
In fact I can tell you a lot about its ultimate fate if you just tell me its geometry.
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