6 ms·

Entropy and Algebra and Topology = ?

104 points by gbrown_ 5y ago

danuker 5y ago
I am appalled by the number of third-party requests this site makes. * ajax.googleapis.com * cdnjs.com * cloudflare.com * cloudflare.net * d231vab146qzfb.cloudfront.net * c.disquscdn.com * d3e54v103j8qbb.cloudfront.net * disqus.com * disqus.map.fastlylb.net * disquscdn.com * doubleclick.net * edgecastcdn.net * facebook.com * facebook.net * fastly.net * fbcdn.net * fonts.googleapis.com * ggpht.com * google-analytics.com * google.com * accounts.google.com * apis.google.com * gstaticadssl.l.google.com * fonts.gstatic.com * static-doubleclick-net.l.google.com * static.doubleclick.net * www-google-analytics.l.google.com * www.google-analytics.com * www-googletagmanager.l.google.com * www.googletagmanager.com * youtube-ui.l.google.com * www.youtube.com * www.google.com * googletagmanager.com * www.googletagmanager.com * googleusercontent.com * gstatic.com * twitter.com * viglink.com * webflow.com * youtube.com * ytimg.com
- hutrdvnj 5y ago
  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 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.
tobbe2064 5y ago
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!
poletopole 5y ago
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.
- JadeNB 5y ago
  > representation theory (abstract linear algebra) 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).
ArtWomb 5y ago
>>> 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?
- Y_Y 5y ago
  > 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.
andrewalan 5y ago
I would suggest the help of a GREAT Hacker to you whose email is geekhackers247@gmail.com… I have utilized him a lot of times and he has never disappointed me.He does a wide range of versatile hacks, fix school score or grades, he additionally have financing programs you could profit with if searching for an advance to begin little or enormous size of business , clear DMV,get unhindered and unnoticeable admittance to your Partner/Spouse(PHONE TAP), Skype, Facebook Account, Email(s), Whatsapp, Instagram, Text messages, In coming and Out going calls, Twitter, Snap Chats, Bank accounts, Viber, Deleted documents, Emails and IPassword and so forth .Taking care of business is pretty much as straightforward as sending a message via Text/Whatsapp 714-576-0359 stating what you need to do.

104 points by gbrown_ 5y ago

I am appalled by the number of third-party requests this site makes. * ajax.googleapis.com * cdnjs.com * cloudflare.com * cloudflare.net * d231vab146qzfb.cloudfront.net * c.disquscdn.com * d3e54v103j8qbb.cloudfront.net * disqus.com * disqus.map.fastlylb.net * disquscdn.com * doubleclick.net * edgecastcdn.net * facebook.com * facebook.net * fastly.net * fbcdn.net * fonts.googleapis.com * ggpht.com * google-analytics.com * google.com * accounts.google.com * apis.google.com * gstaticadssl.l.google.com * fonts.gstatic.com * static-doubleclick-net.l.google.com * static.doubleclick.net * www-google-analytics.l.google.com * www.google-analytics.com * www-googletagmanager.l.google.com * www.googletagmanager.com * youtube-ui.l.google.com * www.youtube.com * www.google.com * googletagmanager.com * www.googletagmanager.com * googleusercontent.com * gstatic.com * twitter.com * viglink.com * webflow.com * youtube.com * ytimg.com

hutrdvnj 5y ago
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 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.

tobbe2064 5y ago

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!

poletopole 5y ago

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.

JadeNB 5y ago
> representation theory (abstract linear algebra) 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).

ArtWomb 5y ago

>>> 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?

Y_Y 5y ago
> 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.

andrewalan 5y ago

I would suggest the help of a GREAT Hacker to you whose email is geekhackers247@gmail.com… I have utilized him a lot of times and he has never disappointed me.He does a wide range of versatile hacks, fix school score or grades, he additionally have financing programs you could profit with if searching for an advance to begin little or enormous size of business , clear DMV,get unhindered and unnoticeable admittance to your Partner/Spouse(PHONE TAP), Skype, Facebook Account, Email(s), Whatsapp, Instagram, Text messages, In coming and Out going calls, Twitter, Snap Chats, Bank accounts, Viber, Deleted documents, Emails and IPassword and so forth .Taking care of business is pretty much as straightforward as sending a message via Text/Whatsapp 714-576-0359 stating what you need to do.