isomorphismes

shapes, figures, and forms.

[isomorphismes]

“Ramified” means that around the point in question the projection map has the same behaviour as does the projection from the parabola to its abscissa.

C. Herbert Clemens, A Scrapbook of ℂ Curves

[isomorphismes]

Test functions and [tempered] distributions require the notion of topological vector space … distributions can be traced back to Green's functions in the 1830’s to solve ordinary differential equations … the 1936 work of Sergei Sobolev on hyperbolic PDE’s.

Laurent Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on.

tempered distribution

topological vector space

(^ if there is a dipole, there must be a notion of subtraction, hence the need for a vector, and to speak of this very conceptually, use a TVS)

[isomorphismes]

“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form solution.” Such a statement seems plausible. Linear differential equations with constant coefficients are the only differential equations for which a general solution is given in closed form. As often occurs in life, however, the exceptions to the rule are sometimes more interesting than the rules themselves. Let us digress from quantum physics to the motion of water, where British shipbuilder John Scott Russell noticed a solitary wave in a canal in August 1834. Neither Airy nor Stokes accepted this observation, yet in 1895 Korteweg and de Vries found an equation for a wave travelling in shallow water in one direction: u̇ + 6•u•uₓ + uₓₓₓ = 0. The KdV equation is easily solved by restricting from two independent space-time dimensions (x,t) to a single dimension x−λt — a frame matching the speed λ of a travelling wave.

Mikhail Ilʹich Monastyrskiĭ, Riemann, Topology, and Physics

[isomorphismes]

Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in 3-space force uniqueness of geodesics joining two points—implying that any segment of geodesic is also a shortest path.

But there is a long way toward the modern statement: “on any complete abstract Riemannian manifold of ≥0 curvature of any dimension, curvature is the quotient of its universal covering by a discrete group of isometries.”

Marcel Berger, Riemannian Geometry during the Second Half of the Twentieth Century

-------

Hadamard, 1898 being Les surfaces à courbure opposées et leurs lignes géodésiques

[isomorphismes]

Hamiltonian mechanics is the feminine side of classical physics. Its masculine side is Lagrangian mechanics, formulated in terms of velocities (tangent vectors) rather than momenta (cotangent vectors). Lagrangian mechanics focusses on the difference of kinetic – potential energies; Hamiltonian mechanics focusses on their sum.

Richard Montgomery, reviewing a book by Stephanie Frank Singer and recalling lectures by Shing-Shen Chern

[isomorphismes]

the cotangent bundle (differential forms) is the feminine side of calculus-on-manifolds; the tangent bundle (vector-fields) is the masculine side.

Shing-Shen Chern, via Richard Montgomery

[isomorphismes]

“Michael Jordan has always got to be beating someone at something.”

— Kareem Abdul-Jabar, as interviewed by Claudia Dreifus

[isomorphismes]

What is “a” homology “theory”?

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.

(If the space is X you will see people write BG(X) – the letter B here means Eilenberg-MacLane topological type.

[isomorphismes]

“the most important part of a principal components analysis is naming the axes”

—

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.

— isomorphismes (@isomorphisms) May 3, 2017

[isomorphismes]

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 

2009 partial update for the 2010 edition  Cut-off date for this update was January 15, 2009.

[isomorphismes]

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.

This image—of the city of Shanghai commissioning a rich ex-pat—a guy who literally wrote I Speak For The Chinese and The Chinese Are Like That—to define, for their foreign targets, the meaning of Shanghai.

[isomorphismes]

Multimaps

Cartesian functions send {A}→{B} with exactly one tail a↦ per a∈{A} connecting to each head ↦b∈{B}.

In other words B has to be equal size or smaller than A.

This is true mapping rings to rings, groups to groups, sets to sets, vector spaces to vector spaces, ... it's just a property of arrows really.

When mathematicians want to talk about "one-to-many" (using the database lingo) or "multimaps" (some stupid word I heard on Wikipedia which absolutely nobody anywhere ever thought was a good term), though, they're not left outside.

If you've got a bundle of arrows ⇶ with tails from {a₀, a₁, a₂, a₃} ⇶ {b₁₄}, then that's a bundle of tails all heading to the same place. If you "grab them all by the head"

So when mathematicians want to talk about a multimap, they use a preimage ƒ⁻¹. Let's say the kernel for example--it's "everything that gets thrown in the trash"---so if multiple things get thrownin the trash, 

(linear subspace / quotient / ring morphism kernel)

So this is how they can associate a bunch of stuff, to one point. For example every point on a manifold gets a tangent space. Maybe this is a vector space for example--which is a lot bigger than just one point.

That would be a problem for 1-to-≥1 functions, so the mathematicians need to turn the arrows around. That's why they define the projection map π:E→B to send a ton of things e∈E onto that one point b∈B i.e. p∈M.

[isomorphismes]

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)

(por mimrir)

[isomorphismes]

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

invention

[isomorphismes]

“The … conception of the person as a bounded, unique, more or less integrated motivational and cognitive universe, a dynamic center of awareness, emotion, judgment, and action organized into a distinctive whole and set contrastively both against other such wholes and against its social and natural background, is, however incorrigible it may seem to us, a rather peculiar idea within the context of the world’s cultures.”

—

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

self-motivated – then where do tropes come from? why are so many comic book authors putting women in refrigerators? Is it because they really individually wanted to do that? Or is there some kind of interpersonal causality at work.

[elanormcinerney]

Star Trek Voyager

[isomorphismes]

“At the turn of the century, the Swiss historian Jakob Burckhardt, who, unlike most historians, was fond of guessing the future, once confided to his friend Friedrich Nietzsche the prediction that the twentieth century would be “the age of oversimplification”. Burckhardt’s prediction has proved frighteningly accurate. Promising a life of bread and bliss, just after the war to end ask wars. Philosophers have progressed daring reductions of the complexity of existence to the mechanics of elastic billiard balls. Others, more sophisticated, have held that life is language, and that language is in turn nothing but strings of marble -like units held together by the catchy connective of Fregean logic. Artists well dished out in all seriousness checkerboard patterns in red wine and blue fetch the highest bus at Sotheby’s. Biologists are mesmerized by the prospect that life may be reduced to a double helix.”

— Gian-Carlo Rota, Husserl and the reform of logic