isomorphismes

It’s impossible to get far in reading 20th-century mathematics without encountering the word cohomology. Cohomology & schemes are the subject of Hartshorne’s classic, where you can find out (Appendix C) that the Weil conjectures were resolved by defining a thing called l-adic cohomology.

(Cohomology even showed up in economics, information theory and computer theory — although here it’s clear that the influence of this ide has been less pervasive, and unclear why.)

Schemes are like varieties = cycles. And cohomology is a way (ok, apparently various ways!) of “calculating” shape.

So what does it mean to “define” “a” cohomology “theory”? What does it mean that Dror Bar-Natan is fascinated by Khovanov homology? That Dale Husemöller wants to interpolate beween different cohomology “theories”—crystalline, étale, Hochschild, and so on?

Mathematicians drop the word “theory” like rappers drop the “N” bomb.

One starts with simplicial homology

This “theory” takes triangular decompositions of a space into triangles and returns a chain complex with a (everywhere ∂²=0) boundary map.

Why do chain complexes come up in this topic? To algebraicise the geometric idea here, you set up maps from the higher-dimensional things to lower-dimensional things. (In case of simplicial homology, the “things” are . In keeping with Eilenberg & Mac Lane’s fundamental rule of homology, ∂² needs to always zero out.

(We are doing this in a “formal” sense—meaning that an entry might be like that blue <small>(solid)</small> pyramid over there, for example in the sentence −3 of that blue (solid) pyramid over there (hey, they came from the (3,5) position of the filtered complex − 2 of that blue (solid) pyramid over there + 5 of that blue (solid) pyramid over there.

(Hey — you already knew that mathematical sentences get too long — just like you knew that topological (not algebraic) functions can get so wiggly that it’s not worth trying to explicitly describe them.

The fact that these homology sentences can be so horrid, and yet the cancellation criterion ∂²=0 is so simple, is–I think–why mathematicians regard the Eilenberg-MacLane condition as “deep”.

According to Matt H, the boundary map is what makes a homology-theory be different from other homology-theories.

You can have twists in the homology-theory

At some point people figured out that you can cover the possible topological types …………… and thus be able to say something about X, again without having to go into painfully boring detail. —- For applications people, this may mean that the things we want to talk about fall ………… or it may not.

—

William Kruskal

(source: I heard this second-hand but I don’t know if it’s written down anywhere)

(For those not in-the-know,

Principal components are composite dimensions, like 5×faculty pay + 3×library size + … ÷ 10 or 3×vote on bill 3 + 8×vote on bill 12 + … ÷ 100

The hope is that,

by using linear algebra

, you can present many things as fewer things. [beta vs p examples]

The reasons this hope might have some possibility of working are two: (1) covariation and (2) small contributions. If two of your data fields do similar things (1), you can combine them into one dimension which acts pretty much the same. (in mathematical terms, by rotating the basis). If many of your data columns make a small contribution, why not smush them into one composite dimension (eg irrelevant_1 + irrelevant_2 + … + irrelevant_N)

You want dimension names to look like this

but how do you get them from reality = messy data which wasn’t gathered well, isn’t necessarily defined how you want, isn’t defined the

You lie.

Product idea: instead of giving people what they want, lie—saying you’re giving them what they want—then deliver something else.

Notice how some of the historical error bars do not contain the future “right answer”.

These historical data (thank you to C. Amsler et al. for compiling them from across many articles!) of particle physics measurements show not only

the epistemic nature of probability estimates and confidence intervals,

but also the difference between

probability as computed within one experiment and

overall, actual, total, legitimately objective certainty.

Since this is physics we dont’ have to worry about the usual social-science problems like the property in question not existing, or not having experimental data and thus needing to infer from examples

Picture by C. Amsler et al. (Particle Data Group), Physics Letters B667, 1 (2008) and 

Carl Crow’s map of Shanghai

I came to the story of Crow through Hua Hsu story about expertise about “China”. Hsu was lecturing as well about the origins of pleasure—how marketing shapes desire, and (defensively) how criticism—to praise and to blame—can serve as a counterweight to the mind-share that corporations vie for.

What are sites and sheaves?

Internal Sieve is a downward-closed collection of open subsets of a topological space

“Downward-closed” means all smaller subsets wholly contained in any part of the sieve, also are in the sieve–and smaller subsets wholly contained in those are also in the sieve, and so on.

External Sieve is the following:

Map the category of open sets on 𝒳𝐗𝒪𝓞𝑿  𝓞(𝐗) onto the category of functors from 𝓞(𝐗) to Set. 𝓞(𝐗) → {𝓞(𝐗)^opp, Set} (category of Set-valued presheaves on 𝐗)

an individual open set ↦ Hom_{𝓞(𝐗) } (—, the individual open set)

That is: embedding some stuff 𝐗 in a gigantic category (presheaves)

The embedding relates the stuff we started with 𝐗, their internal organisation 𝓞(𝐗) and their relationships to each other Hom(𝐗)

the inventor of Swype

went to graduate school to try to communicate with dolphins → help alleviate various human conditons that impair communication → more usable phone interfaces for everyone else

lives in Nevada City, Calif. (home of Joanna Newsom and Terry Rile)

https://www.youtube.com/watch?v=2OnHlC7zi_8

—

Clifford Geertz, From the Native’s Point of View

Grateful Geertz has gathered these assumptions together in one places where they can be conveniently interrogated.

bounded – where did my thoughts come from? environmental factors - parents - chauvinism

unique – no, there are others like me

integrated – ignores situational psychology - stimulus/response – so without the chance to be kill I won’t do it (nad won’t have that future life history) – without the chance to work at an investment bank – without the chance to be brave