Predominantly Erroneous (Exohedron nonsense blog)

I know that in theory, here in the US people in grade X are usually about X + 6 years old, but somehow I never manage to grok that. I keep thinking that 4th graders are relatively mature compared to 10-year-olds, possibly because I've only encountered 4th graders in a school setting but I encounter 10-year-olds in other contexts.

 

More paper. I wish I could say that I've gotten to the good stuff, but instead, here we are at

Section 2: Terminology and Notation
Occasionally there might be something interesting in this kind of section, but probably not. It's a problem that is neither really hard nor insightful, it's just a lot. Also I'm not going to use a bunch of it for this lieblog because it's too much effort to find all the special characters.
I guess there's the note that all the vector spaces will be finite-dimensional over C.
The use of lower versus upper-case to indicate chain versus cochain is very strange, but I understand that they want to have more leeway with the index placements. Ehh, I'll get used to it. The homology/cohomology distinction is still sub-versus-superscript.
Distinguishing between the direct sum and the Cartesian product of vector spaces. I like this, because in general products and coproducts are not the same thing, and while they are for vector spaces, it might be useful to pay a bit of attention to the distinction here, because categories.

Okay, done with the substrate, let's do some math.

Section 3: Algebras and States
A bunch of operator algebra stuff, moving from things that look like states to the more general trace-class operators, and things that look like observables to bounded operators. This is good because the real numbers are bad for algebra.
Interesting that the space of bounded operators is isomorphic to the dual of the trace-class operators, but the trace-class operators aren't isomorphic to the dual of the bounded operators when the Hilbert space is infinite dimensional. Duality in the infinite-dimensional case is weird.

Section 4: Building Blocks
Okay, finally a definition of s_p. Which, I mean, I already guessed, I think, but it's good to have confirmation.
I'm so used to projections in representation theory that I was surprised by the difference between GNS(p) = A(H)s_p and Com(p) = s_pA(H)s_p. But of course, we're not just composing commuting projections here, so it does matter.
Wow, the sections are so short now.

Section 5: Building Blocks and Right Essential Equivalence Classes
Okay so here we get a thing that says when two operators are equivalent in a noncommutative generalization of being almost everywhere equal measurable functions. N_p being the stuff that square-integrates to 0 relative to p. N_p = A(H)(1- s_p).
Oh, so we can view the GNS(p) as A(H)/N_p, at least in terms of being a left A(H) module. Also, sections back to A(H), with x + N_p -> xs_p. I'm a little surprised that there is a well-defined section, but I think that if I bothered to unwind things a little bit it will make sense, since s_p is a projection and not just a quotienting map.
We get a similar, if somewhat more complicated, version of Com(p), as the algebra of endomorphisms of GNS(p).

These sections are super short now that we're out of the introduction!

#MathPaperLiveBlog

 

At some point maybe I should actually try putting into words my thoughts about morality and truth and all that stuff, if only so that I can pin it down to look at it from the outside. In particular my beliefs about local consistency versus global incoherence, and the value of principles and theory versus observation and modeling. I get the feeling that most of it is more a reaction to the philosophy that I encountered in college and grad school and to the Rationalist communities to which I am two steps adjacent to.

 

Man, this track is a mess. I have no idea what I'm doing and it just keeps getting more complicated. "Oh, let's add some strings. Oh, let's add some horns. Oh, let's add a new melody here. More snares? More snares." All my stuff is overproduced, sure, but this one feels different.

 

Happy new year!

 

Yup, my metabolism is definitely slowing down.

 

Those jigsaw puzzles where like all the pieces are the same color but they're all different shapes.

 

The sudden realization that you don't need to put croutons in your sandwich if you just toast the bread instead. On the other hand, suppose that when you want to each the sandwich you don't have access to a toaster, and consider that toasted slices of bread are more prone to breakage during transport than untoasted slices; in that case it may be wiser to carry croutons as well, since, being small and roughly cubical, croutons are more transport-ready than slices of toast.

 

More math paper!

Section 6: Bipartite Complexes
So according to the introduction, this is the case where the mutual information actually tells us enough to detect factorizability, without the need for the full cohomological machinery. Let's see how that goes.
Okay, finally some definitions that I was missing.
Reduce a density state by tracing. So this is the "projection" map I was sort of assuming before. I guess this isn't surprising; I mean, what else could it be?
"Pure" means that the state is a product of a tensor p in H_A otimes H_B with p^v. So the thing to keep in mind is that a bipartite state is really a tensor product of four terms: H_A otimes H_B otimes H_A^v otimes H_B^v. Pureness is factorization into H_A otimes H_B and H_A^v otimes H_B^v, while factorizability is factorization into H_A otimes H_A^v and H_B otimes H_B^v. Okay, got it finally.

Section 6.1: Cochain complexes
It continues to bother me that the chain complexes start at degree -1. I mean, it makes sense if you view these things as Cech complexes, but still. I guess it's like how you can view a category as containing (-1)-morphisms if you really, really want to, they're just trivial to the point where you usually don't.
Section 6.2: Chain complexes
The mix of oplus and - is confusing.

Short one today, since it looks like 6.3 will be kind of beefy.
#MathPaperLiveBlog

 

I got all fired up to to be an adult today and file my taxes, but it turns out that the firm I use to handle investments hasn't posted the tax documents for this year yet. Argh. All that work of convincing myself to get this out of the way and I can't even do it.

 

Today's entertainment was listening to my coworkers react to the dry erase crayons with varying combinations of shock, incredulity, and disgust.

 

Gated community college

 

Section 6.3: Basic Properties of the Bipartite Complex
Oh boy (not sure if sarcastic)

6.3.1: Descent to Support Equivalence Classes
I used to wonder about descent in the topological/algebro-geometric context. Like Grothendieck is pretty famous for his stuff on descent, and indeed a lot of his work is motivated by the idea of descent. But I could never get myself to really care enough to read about it. Then I read a wonderful paper by Schreiber and Waldorf, "Parallel Transport and Functors", that talked about translating between local trivializations and descent data. Suddenly it's like "oh, so that's what they mean by descent."
I hope that vague intuition about descent holds here.
I guess it's good that "support equivalence" is the only thing it can be given that name. I'm not sure how happy I am with the fact that the cochain complexes only depend on the support equivalence classes, though. Does the distribution information really not matter?
Hmm. The idea that the cochain complex only depends on subspaces of H_AB still feels a little suspect, but perhaps less so. Not sure yet.

Section 6.3.2: Trace duality for Chains and Cochains
The duality is an isomorphism (because we're working over C, so it kind of has to be an isomorphism) so we only look at cohomology. Everyone likes cohomology better anyway, right?

Section 6.3.3: Equivariance under Permutation of Tensor Factors
No braiding.

Section 6.3.4: Equivariance under Local Unitary/Invertible Transformations
Okay, so local does mean on a single tensor factor.
On no, a bunch of typos in the definition of a local invertible transformation. I mean, they're obvious copy-paste errors, but still.

Section 6.3.5: Poincare Polynomials as SLOCC Invariants
We have two equivalences: support equivalence, and local invertible transformation equivalence.
LOCC = local operations and classical communication.
For bipartite pure case, Schmidt rank determines SLOCC class. Not true for multipartite.

Section 6.3.6: Comparison of the Bipartite GNS and Commutat (Co)homologies
Natural inclusion of com into gns, and natural surjection of GNS into Com. Blah blah blah long exact sequence.

Interestingly, the maps from gns onto com and Com into GNS don't produce suitable maps. Multiplication by s_p is not functorial?

That's it for section 6.3, so I'm stopping here. Next section, Factorizability and Cohomology, sounds like main-result-ish, so that will come later.

#MathPaperLiveBlog

 

Probably overthinking Alita: Battle Angel, but I'm trying to figure out why it worked for me. Western, live action movie adaptations of anime or manga rarely work for a number of reasons that don't always involve racism or lack of cultural context, but often do.

Spoilers for good chunks of the movie, I guess.

Starting with the elephant, generally I would rather actors of an underrepresented race or ethnicity get the chance to play characters of that race or ethnicity over white actors when there aren't extenuating circumstances, if only because it might help mitigate the underrepresentation. "The best actor for the job" is usually not a good excuse for whitewashing.
On the other hand, this is not exactly that. The original manga character of Alita wasn't really Japanese, at the very least not fully; her progenitor had a German last name and a Japanese first name. The actor voicing and doing mocap for her, Rosa Salazar, is of Peruvian descent. I don't have much to say here, because it's good that nonwhite actors are getting titular roles, but I'm not sure how I feel about nonwhite actors playing nonwhite actors of different forms of nonwhiteness.
The character who really got whitewashed was Ido, who in the manga was Daisuke Ido, not Dyson Ido.
This sort of ties in to the other thing I mentioned, cultural context. In this setting, Japan doesn't exist as a cultural or ethnic or national entity, and the story, setting and writing could be anything written by a Western author. Sure, in the later parts there are certain examinations of selfhood and conflict that invoke ideas from Zen Buddhism and philosophies associated with Karate and similar martial art forms, but in terms of the material that the movie covered, no, it's mostly just "girl with amnesia hurts a lot of people".

Speaking of, I can't help but compare the bar scene with the bar scene from Serenity. There are a number of dramatic differences, once you get beyond the fact that it's a small, waifish girl beating up a lot of hardened mercenaries in a bar. In particular, motivation, context, and outcome. River Tam beating up mercenaries in a bar against her will because she was remotely triggered is very different from Alita is disappointed and also likes getting into fights.

Indeed, I think the movie got Alita's character pretty faithfully: impulsive, mouthy, violent, wants to believe the best of people but isn't averse to attacking those who go against her notion of "best". She's an unrepentant weapon of war and also a young girl with all that entails.

Anyway, so now we come to two of the main issues that have little to do with Japan and mostly to do with movie adaptation of things that are longer than two hours of material. Namely pacing and writing.
One thing I think the movie did well: not having Alita take down the main bad guy. Like, she takes down the guy who's directly trying to kill her, and the guy who is directing the guy who's directly trying to kill her, but not the guy who is actually in charge of both of them. This both leaves room open for a sequel, and also deals with the fact that trying to reach him would be impossible on a narrative level within a roughly-two-hour timeframe.
She takes down a lot of small fry and one medium fry, and we end with her vowing to reach the big fry eventually. That's a good set of accomplishments for a two hour movie that needs to introduce a world and a bunch of characters basically from scratch. Often, movie adaptations try too hard to cram a lot of plot into two hours merely because the original work had that much plot. Better to realize that the original material had narrative arcs and that you can in fact end on the end of one of those arcs instead of trying to encompass too much.
Also, another thing the movie did well: the writing was better than the original manga, because the original manga had bad writing. It had a lot of cool ideas, but the writing wasn't great. Which is not to say that the movie had great writing; it had decent-to-good writing. But in terms of plotting and scene-choice and character development, I think it was definitely a step up from the original manga. Especially Hugo.

Okay, so. Overall I enjoyed the movie. I don't think it was game-changing in any real way, and I would be surprised if it got a sequel, but I enjoyed it. It was fun, it was kind of stupid but didn't introduce any stupidity that wasn't in the source material, and indeed fixed some of the issues with the source material.

 

Apparently my stomach does make noise when it wants to be filled, I just don't get any internal signals.

 

I think my tendency to attach "up to isomorphism" to everything makes it very difficult for me to take analytical ontology seriously as a discipline. Why should I care about what "really exists" or what something "actually is" when they give equivalent models? Along with my tendency to think of things as models rather than as actual things; there are no actual things, merely models of models.
My issues with analytical epistemology stem from a combination of Kantian "pure reason can't tell us anything about the real world" and Godelian "pure reason can't tell us enough about synthetically-defined worlds" along with not caring about the parts of analytical philosophy that aren't pure reason.
Ultimately it's probably all just contrarianism, but at least if someone asks I can point to the fact that in the framework that I tend to think in, the only correct way to do philosophy is an infinite tower of (meta)ⁿphilosophy with each level aimed at dismantling the assumptions baked into the language and formulation of the level below.

This reminds me of the list of translations I've seen for the opening of the Tao Te Ching, all trying to elucidate the main idea with faltering, insufficient words. Except instead of words it's philosophical frameworks.

 

I wish there were a way to transliterate noises better, because a lot of my brain noise is just that: sounds that my brain thinks I should hear. Sometimes these are sounds that my brain thinks I should incorporate into whatever music project I'm working on, and sometimes it's just "here's a weird sound effect I came up with."
And I wish to share these experiences with the world without having to figure out how to produce them and then record them and then upload them somewhere. If it were like a melody or something then I could just type that out as text, but if it requires like a filter-sweep and modulated distortion and very careful gating then that's a bit harder to transcribe.

 

I wish salsa weren't so salty in general, because otherwise I would just drink it.

 

The GPT-2 stuff makes me somewhat confident that we're approaching the point where our machine-learning capabilities are able to produce strings of text that are, to a human, indistinguishable from actual research papers on machine-learning.
This should be distinguished from the machines actually being able to do research on machine-learning, or even having any concept of papers, research or otherwise.

 

So Captain Marvel was pretty good. I'm not sure I have all that much to say about it.