Posts tagged with topology

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces.

Jets are a coordinate-free version of … Taylor series.


Pictorial glossary

Bundle:

Sections:

Mapping between spaces:

• Mapping a circle onto a line
• Wrapping a line around a circle
• Associating a number to a Euclidean=flat subregion

• Thinking of knots as mappings of a loop 𝕊¹ into ambient space ℝ³.

• Thinking of homotopies as mappings of cubes into spaces.

• Associating photos to geolocations (Flickr, stochasticplanet)
• Mappings from manifold to manifold—for example the derivative (pushforward) at each point of the surface.
• Associating vectors (based force-arrows) to spots on the planet (2-sphere):

## Extrinsic vs Intrinsic

In The Geometry and Topology of 3-ManifoldsThurston speaks of the “extrinsic” vs “intrinsic” view of {the 3-sphere − a knot}.

I like learning words for something that’s been kicking the walls of my head trying to get out. Sometimes I can look at the world in an objective way, and sometimes everything centres on me.

• Extrinsic I and you and they are cerebral plains-apes, doing all of the things one would expect such to do. Chasing after money, status, sex, it’s all pretty simple when you see things this way.
• Extrinsic your exotic mind-expanding holiday is just a status signal.
• Extrinsic the hobby or job you hang your identity on tells me that you’re bourgeois and therefore define yourself through your activities or achievements.
• Intrinsic reinterprets monkeys’ “social grooming” as an activity that actually feels like something.
• Intrinsic is being in love.
• Extrinsic is love as a neurochemical sequence—dopamine, adrenaline, vasopressin, norepinephrine, oxytocin—generated by chance mutations and selective reproduction in response to an evolutionary problem.
• Extrinsic is 5,000 applicants for 5 jobs = 1‰ chance of getting it.
• Intrinsic my application was rejected so I’m a failure.
• Extrinsic is the pale blue dot, spaceship Earth, a wet rock bearing eukaryotes.
• Extrinsic I’m one of 10000 options s/he has on this dating site.
• Intrinsic is me; my likes, my interests; the homunculus behind the camera, 我, my life. Extrinsic this display is again an outgrowth of my bourgeois background.
• Intrinsic is exotic Peshawar, a world away and uncomfortably hot.

Extrinsic is more primates with a different culture-function applied to them. Going through the same life-stages, wearing different customs. Instead of going to a pub they do some sober, Pakistani alternative—but it’s all the same, all human stuff just expressed differently.
• Extrinsic when Parisians kiss each other on the cheek it’s not romantic, it’s just like a handshake.
• Intrinsic it’s gross that my parents have sex.
• Intrinsic I hate this town I hate the football games I hate the fakeness I hate the parochial small-mindedness there’s something better in store for me I need to get out of here I will get out of here
• Extrinsic is a visitor look how excited these people are about their quaint sports and amusements! That’s so nice.
• Intrinsic is being engrossed in reading.
• Extrinsic is seeing someone sitting still looking at paper with glyphs on it.
• Intrinsic is noticing how Thomas Piketty’s lower teeth are not-at-all straight like George Clooney’s, and being taken back a bit. Extrinsic is knowing that Piketty earns his stature from books and papers (arrayed behind him) whereas Clooney earns based on his looks.
• Intrinsic is waking up in the morning driven by the will to understand pseudo-Riemannian metrics and thereby, the Universe.

Chanting “sheaf cohomology" like a shamanic totem, carving it into my forehead my topknot my yarmulke my niqab, showing the world what’s inside my head and wishing they love me for it.
• Extrinsic I’m yet another plains-ape distracted by ego, status-signalling, and a scientific religion, cultured by stimuli that practically guaranteed I would behave this way.
• Intrinsic my new friend is so cool! She is friends with someone famous!
• Extrinsic of course she leads with her connection to the desirable; it’s all part of her personal branding scheme.
• Intrinsic is the many self-serious songs about my chain, my ding-a-ling, whatever.
• extrinsic is a boy in love with his homeostatic skin flap.
• Intrinsic is watching an ad targeted at you, and just reacting.
• Extrinsic is watching an ad that makes no sense to you, and thinking about the ad on a higher level.
• Intrinsic is appetite; craving; addiction.
• Extrinsic is flavourful sustenance.
• Extrinsic is asthexia; cachenia; syncope; renal failure.
• Intrinsic is the only death that matters (mine).
• Intrinsic is write drunk.
• Extrinsic is revise sober.

Beware of the pursuit of the Superhuman: it leads to an indiscriminate contempt for the Human. —George Bernard Shaw, Man and Superman

## Dummyisation

Statisticians are crystal clear on human variation. They know that not everyone is the same. When they speak about groups in general terms, they know that they are reducing N-dimensional reality to a 1-dimensional single parameter.

Nevertheless, statisticians permit, in their regression models, variables that only take on one value, such as {0,1} for male/female or {a,b,c,d} for married/never-married/divorced/widowed.

No one doing this believes that all such people are the same. And anyone who’s done the least bit of data cleaning knows that there will be NA's, wrongly coded cases, mistaken observations, ill-defined measures, and aberrances of other kinds. It can still be convenient to use binary or n-ary dummies to speak simply. Maybe the marriages of some people coded as currently married are on the rocks, and therefore they are more like divorced—or like a new category of people in the midst of watching their lives fall apart. Yes, we know. But what are you going to do—ask respondents to rate their marriage on a scale of one to ten? That would introduce false precision and model error, and might put respondents in such a strange mood that they answer other questions strangely. Better to just live with being wrong. Any statistician who uses the cut function in R knows that the variable didn’t become basketed←continuous in reality. But a facet_wrap plot is easier to interpret than a 3D wireframe or cloud-points plot.

To the precise mind, there’s a world of difference between saying

• "the mean height of men > the mean height of women", and saying
• "men are taller than women".

Of course one can interpret the second statement to be just a vaguer, simpler inflection of the first. But some people understand  statements like the second to mean “each man is taller than each woman”. Or, perniciously, they take “Blacks have lower IQ than Whites” to mean “every Black is mentally inferior to every White.”

I want to live somewhere between pedantry and ignorance. We can give each other a break on the precision as long as the precise idea behind the words is mutually understood.



Dummyisation is different to stereotyping because:

• stereotypes deny variability in the group being discussed
• dummyisation acknowledges that it’s incorrect, before even starting
• stereotyping relies on familiar categories or groupings like skin colour
• dummyisation can be applied to any partitioning of a set, like based on height or even grouped at random

It’s the world of difference between taking on a hypotheticals for the purpose of reaching a valid conclusion, and bludgeoning someone who doesn’t accept your version of the facts.

So this is a word I want to coin (unless a better one already exists—does it?):

• dummyisation is assigning one value to a group or region
• for convenience of the present discussion,
• recognising fully that other groupings are possible
• and that, in reality, not everyone from the group is alike.
• Instead, we apply some ∞→1 function or operator on the truly variable, unknown, and variform distribution or manifold of reality, and talk about the results of that function.
• We do this knowing it’s technically wrong, as a (hopefully productive) way of mulling over the facts from different viewpoints.

In other words, dummyisation is purposely doing something wrong for the sake of discussion.

1. The monad, of which we will speak here, is nothing else than a simple substance, which goes to make up compounds; by simple, we mean without parts.

2. There must be simple substances because there are compound substances; for the compound is nothing else than a collection or aggregatum of simple substances.

Gottfried W. Leibniz, Monadology



Crazy how a “father” of calculus was so illogical in his seminal work of 1714.

• The existence of compound things does not imply the existence of partless atoms.
• He asserts, doesn’t prove, that a compound is “nothing more than" a collection of simple substances. (atoms)

I’ve collected a few tidbits about non-wellfoundedness on isomorphismes:

• the opposite of the idea of “indivisible atoms" at the "bottom" of everything
• turtles all the way down
• (infinite regress is OK)
• a > b > c > a
• (so the two options I can think of for non-wellfounded sets are either an infinite straight line or a circle—which biject by stereographic projection)

as well as examples of irreducible things:

• if you take away one Borromean ring

then the whole is no longer interlinked
• Twisted products in K-theory are different to straight products.

A Möbius band is different to a wedding band.

Yet 100% of the difference is in how two 1-D lines are put together. The parts in the recipe are the same, it’s the way they’re combined (twisted or straight product) that makes the difference.

So Leibnitz’s assertions are not only unsupported, but wrong. (Markov, causality, St Anselm’s argument, conservation of mass, etc. in Monadology 4, 5, 22, 44, 45.)

tl,dr: Leibniz, like Spinoza, uses the word “therefore” to mean “and here’s another thing I’m assuming”.

[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …

A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …

We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory

(I rearranged their words liberally but the substance is theirs.)

in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.

the sine of the reciprocal of [some angle between −1/π and 1/π]

at increasing resolution

s <- function(x) sin( 1/x )
plot( s, xlim=c(-1/pi, 1/pi), col=rgb(0,0,0,.7), type = "l", ylab="output", xlab="input", main="compose [multiplicative inverse] with [vertical rect of a circle]" )



(Source: amzn.to)

Why is the boundary of a boundary always null? What is a coboundary?

Beware, some of the + signs in Ghrist’s Elementary Applied Topology are formal sums, which is pasting things together

rather than umming in the usual dimension-reducing sense.

We often speak of an object being composed of various other objects. We say that the deck is composed of the cards, that a road is [composed of asphalt or concrete], that a house is composed of its walls, ceilings, floors, doors, etc.

Suppose we have some material objects. Here is a philosophical question: what conditions must obtain for those objects to compose something?

If something is made of atomless gunk then it divides forever into smaller and smaller parts—it is infinitely divisible. However, a line segment is infinitely divisible, and yet has atomic parts: the points. A hunk of gunk does not even have atomic parts ‘at infinity’; all parts of such an object have proper parts.