Friday, December 28, 2018

Topology for artists - workshop sketch 7

Topology for artists, walkers and other non-mathematicians
One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space, Workshop contents sketch. I try to post small updates often, to keep myself working on this.
I browsed XKCD for "topology" I found a few discussions about organizing a workshop for beginners. See the list of sources at the end.

Encouragement for making a workshop
I feel encouraged by this statement. And I still don't know if this has anything to do with the author of the XKCD comic (I guess not, but I'm not sure):

Topology is fun to talk about, since people who don't actually study it have never heard any of this.

I'd like to take this opportunity to encourage any and all grad students to organize their own student seminars. The best way to really learn a topic is to give a talk about it, to force yourself to be able to explain it in understandable terms. Also, the best way to become comfortable with speaking is to give lots of talks. There are many things I'd like to improve about my speaking style, but I think it's safe to say that anything good about it is a result of the many talks that I gave in this seminar over the course of my graduate career.

Some items will be difficult to translate into "normal" (non-mathematical) concepts. I indicate this difficulty with 1 to 3 stars.

Start from our normal space *
Skimp the technical stuff for conceptual understanding. You can't teach topology in an hour, but you can get the students interested in it in that time frame.

Start with a concrete example. We don't need to talk about the general case of a topological space, and it would be pointless to try to do so in an hour-long presentation. One- and two-dimensional space is something everyone is familiar with, so take advantage of that. The entire concept of open and closed sets comes directly from observations in the metrics on R and R^2.

Experiment with spaces *
The intro I like to talk about is gluing together edges of a square in order to make a sphere (all glued into a single point), a torus (opposite sides glued), a Klein bottle (one pair of opposite sides glued with a flip), and real projective space (both opposite sides glued with flips).
What do a sphere, a torus, and a Klein bottle all have in common (but a plane doesn't)?

Open and closed sets *
I think it might be good to show using simple, concrete examples. First, explain what open and closed sets are, in nontechnical terms, and give examples:
  • An open set is a set where every point can be wiggled a little bit and still be inside the set
  • A closed set is a set whose complement is open
  • Examples of open sets in R include (0, 1), (-pi, pi), and (0, infinity)
  • Examples of closed sets in R include [0, 1], [0, 42], [-10, 10]
  • Open and closed are NOT opposites. The interval (0, 1] is an example of a set which is NEITHER open NOR closed. The empty set and the set R itself are BOTH open AND closed.
Structure of space - open sets as building blocks **
A topology is a set along with a collection of open subsets of that set. A topology completely specifies the topological properties of its underlying space. The underlying space is just a convenient metaphor; it is not actually necessary.

The topology doesn't just 'determine' the open sets; it 'is' the open sets. A topology on a set is just a collection of subsets that is closed under arbitrary unions and finite intersections, and which contains the empty set and the whole set. It has no more structure than this. Every topological concept must be definable *entirely* in terms of basic set/logical operations and whether or not certain sets are members of the topology, or it can't be purely topological.

A topology on a set is just a sort of rule telling us which points are 'close together'. It does this by specifying something called a closure operation: the closure of a set A is the set of all the points 'close' to A. Since the idea of topology is to study this idea in general, though, we'll let any operation count as a closure operation, and just redefine 'close' to match, as long as it satisfies a few basic rules: (list the Kuratowski closure axioms).

The hard thing to get used to about topological spaces is just how little structure is imposed by the axioms. For instance, the topology arising from an arbitrary metric space is already much more special and well-behaved than an arbitrary topology.

Neighborhoods *
Go on to explain these concepts generalize into R^2 naturally.
A circle including all points inside and including the boundary is an example of a closed set. A circle including all its interior points, but without the boundary is an example of an open set.
An open set containing a point p is called an (open) neighborhood of p.

Metric spaces and distances ***
If you're talking about distances, you're in a metric space. In what's called a metric space, for example, an open set is one where every point inside has a small region around it that's still inside the set.

Closed versus bounded *
Closed and bounded are unrelated, in the sense that we can find examples of subsets of topological spaces for every combination of {closed, not closed} and {bounded, not bounded}. In fact, we can find subsets of all four types just within the real line:
  • [0,1] as a subset of the real line is closed and bounded, and its complement (-infinity,0) U (1,infinity) is neither closed nor bounded.
  • (0,1) as a subset of the real line is bounded but not closed, and its complement (-infinity,0] U [1,infinity) is closed but not bounded.
Mappings and functions ***
Then, it is nice to produce a useful result with the definitions. The topological definition of a function is a really good example.
A function f maps points from R^2 to R^2. We can also think of f mapping subsets of R^2 to subsets of R^2 in a natural way: If X is a set, then f(X) is the set {f(x) for x in X}.
Then, finally, a continuous function from R^2 to R^2 is one where the following is possible. Given any open set in R^2 called Y, we can find an open set in R^2 called X such that f(X) is a subset of Y.
The reason for using R^2 is it makes for a nicer picture. Wikipedia has a nice picture of what I mean (though, the names of the variables are different):

Pathologies ***
Finally, some pathology: you could display the multitude of limits of a sequence in a non-T2 space etc. If you really want to, you can talk intuitively about the long line (although I doubt there's anything intuitive about it ).

If your audience is at a high enough level, it might be fun to show some "strange" examples. Define a homeomorphism and talk about why we require a continuous inverse to a continuous mapping (ask them for an example; if they've only ever seen real functions of a real variable, they'll have to think about it), show them some homeomorphic spaces (specifically do the stereographic projection of a sphere onto a plane), maybe show some manifolds (they look nice; it might be worth it to mention fundamental polygons here, at least for the torus and the Möbius strip, perhaps the projective plane).

Further study ***


Topology for artists - workshop sketch 5

Topology for artists, walkers and other non-mathematicians
One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space. I try to post small updates often, to keep myself working on this.

I read a lot of discussions on topology. Below is a summary of what I learned, structured as a workshop outline. See the list of sources at the end.

The nature of space
Topology studies the nature of the space itself. Does it have holes? Is it close together (compact) or spread out? This allows higher math to define calculus over strange spaces.
For workshop:
  • Think about the difference between the sphere and the torus.
  • Discuss the possibility of living in finite but unbounded space (asteroids game). 
(Point set) Topology as basic theory
It's like calculus, linear algebra, set theory or most other fields of mathematics that are taught in college or the first year of graduate school. You have to know them because you'll be using them in whatever you will be doing as a mathematician. You probably won't be doing research within one of these fields, but they're all used in current research.

I’m not aware of any applications of pure point set topology. Usually, applications involve more complicated topological structures, like manifolds or knots. However, if you want to really understand these more complicated structures, you need to understand the more basic theory, particularly because point set topology shows you just how unintuitive general topology is. In the general setting, there are all sorts of weird counterexamples that don’t feel like they should exist, but you need to be aware of these if you want to have the right intuition of how much information you need to be able to solve various problems. 
For workshop:
  • Discuss the extremely basic concepts introduced in topology, that demonstrate that "things could have been totally different and very weird" (like the multiverse).
  • Try to show some weird spaces, very different from ours (maybe using separation properties).
Structures on a space
Point set topology is the basic study of topological spaces. A topological space is sort of the minimal amount of structure that you need in order to be able to talk about convergence, connectedness, and continuous functions.

Most modern mathematics concerns "sets with structure" of various sorts. The problem is that the structure of "topological space" by itself isn't really strong enough to produce much interesting mathematics--topological spaces by themselves are too wild to make many interesting statements that hold for all topological spaces. Point-set topology studies potentially pathological topological spaces from an essentially set-theoretic point of view.
For workshop:
  • Think about the building blocks of our daily space.
  • Discuss bases.
  • Discuss the standard (open balls) and the discrete (points) topology on a space.
  • Discuss how different topologies lead to different and weird properties of a space (Sorgenfrey line is much too complicated for workshop).
  • Try to present a few "wild" spaces (if possible).
Geometry without a metric
Topology seems to have arisen to answer the question "What is geometry without a metric, without a way of measuring distance?" It's a study of geometry where there isn't a metric -- a way to measure the distance between two points. Several lines of thought from different areas converged into questions about the nature of the space itself.
For workshop:
  • Think about the different practical metrics in our daily (Euclidean) space.
  • Discuss measuring distance in kilometres versus measuring distance in time or in difficulty.
  • Discuss the difference between car, bicycle or walk distance.
  • Discuss manhattan-block distance (pixels).
  • Discus distance through and on a sphere.
  • Discuss the space of music or literary styles.
Closeness, nearness
In the broadest sense, topology formalizes the notions of nearness and continuity. Reality appears to behave in a mostly continuous manner, at least at the scale on which humans and human intuition evolved. We tend to think of things in a continuous manner, we tend to interact with reality in a continuous manner, and now we have topology to formalize all of these continuous intuitions and interactions.

It is like geometry in that studies space, but it does not worry about the nuisances of angles and distances (even if metric spaces are touched on, which do use the latter), instead simply speaking of “closeness” (adjacency).
For workshop:
  • Think about continuity and discontinuity in our daily surroundings.
  • Discuss the absence of infinities, and infinite gradients, in daily experience.
Open and closed
Topology (point-set or general topology, that is) fundamentally captures the idea of continuity in a general setting, via open sets (or closed sets, if you like) which remain open after pullback.
For workshop:
  • Think about open and closed sets.
  • Discuss the openness or closedness of concepts like "marxist" or "city centre".
Topology is also cool in its own right. It is fascinating that we can rigorously consider objects continuously deformable into each other to be equivalent (c.f., homeomorphism & homotopy equivalence) and get something out of that. I am sure you have heard of the donut and the coffee cup, which gets the vague idea across.
For workshop:
  • Think about mappings between different domains.
  • Think about difference between map and reality. 
Countable and uncountable
For workshop:
  • Countable and uncountable sets (maybe mention Cantor and infinities).
  • You can cover an uncountable space with a countable set of open sets (I think).
  • How to make this practical? I find Q versus R fascinating, but is it explainable to outsiders?
(Point set) Topology as a mature subject
Point-set topology is a “dead” field the same reason classical real variables or elliptic integrals are: the subject has been studied extensively, the hidden nuggets have been extracted, and further study of it is largely considered uninteresting. Everything in it has already been studied and proven.

Point set topology is basically a dead field, and is really not a popular research topic. For most interesting problems that use topology, it is necessary to assume spaces are nice enough that stronger techniques--such as those from algebraic topology, differential topology, or functional analysis--become available. Certainly point set topology forms a foundation for a lot of really interesting mathematics, but it isn't really studied much for its own sake anymore.
For workshop:
  • Since 1900 mathematicians have thought about the structures and properties of space. This was done to produce a solid theoretical underpinning of "normal" mathematics.
  • Topology is more philosophical than practical. It presents us with puzzles and makes us think.
For further study
You must be comfortable with concepts like compactness, connectedness and homeomorphisms. You also need partitions of unity, subordinate to an open cover (very important!) and for this you have to be familiar with (something like) the smooth Urysohn lemma. Finally, some understanding of quotient topology will be a plus, since it gives many important examples (Klein bottle, Projective space).
For workshop:
  • Find the connection between point set topology and the weird structures of the klein bottle or the projective plane. I haven't found that yet.
Importance of topology:
Non importance of topology:
For later research:

Wednesday, December 26, 2018

UFO landscapes - 2

The classic UFO landscapes are extremely romantic. They're even better when combined with classic UFO reports. It's a pity that the time of UFO's is past. They were so much fun. Now we have only ISIS, NSA, China and Russia to speculate about. No fun mysteries.
Peter Day ... was between Thame and Aylesbury close to Cuddington on the Oxfordshire border … He had seen the strange thing in the sky about two minutes before he was able to stop the car … Using the cine camera that he carried on a seat of the car he filmed the mystery round orange object. In less than 15 seconds the object had gone and Peter Day was a very puzzled man. …

The basic details of the observation by Peter Day are quite simple, he was driving along a road he knew well when for a couple of brief minutes he saw a totally unfamiliar event. The round orange object he estimates was at least three-quarters of a mile away and travelling in a level plane and horizontal to the tree lined horizon. … He describes the disappearance of the object as "one minute it was there and the next minute it was gone". In fact the film shows the object to be clearly visible in the penultimate frame and not in the final frame.
The second case, the John Flattley film ... is still under investigation. … The events of that memorable evening began when a portable television flickered and the picture died. ... Shortly afterwards the first of many strange lights was observed for a brief moment. It looked like an orange in the sky, it meandered about gently and then disappeared. Soon it was followed by more, and it is these that are recorded for all the world to see.

The low light film, Kodak Ektachrome 160, shows the scene as dusk fell. The foreground clearly visible, and the strange orange light in the sky moving through the field of view, slowing and stopping. … This being a film intended for use in daylight the dusk was insufficiently light to produce a fully illuminated picture, so only the brightest are as are recorded, in this case the lights. This section shows lights appearing 'from nowhere', and melting back again in to nothing.
I have tried to reproduce this atmosphere using a modern digital camera. In the right place, under the right circumstances it is possible. You get nice photographs of mysterious spooklights. Of course they are not situated in the sky. But they still fit the narrative.
On October 6th 1971, by sheer good fortune, an experienced film crew witnessed a UFO event whilst preparing a sequence of film for a television programme. They were able to take a good length of colour film of the phenomenon … Unfortunately … the event filmed was of limited value. What is recorded generally resembles a high f1ying aircraft accompanied by a vapour trail, and looks most unlike a typical UFO event. The film has proved extremely difficult to analyse, although considerable efforts have been made. … Had the film not been taken there is little doubt in our minds that the case would now be forgotten as fairly uninteresting and quite possibly representative of an aircraft seen under unusual conditions.
UFOs: A British Viewpoint, Jenny Randles and Peter Warrington, 1981, Book Club Associates, Robert Hale Ltd. - Chapter: Instrumentally Detected Cases

Cat apparitions

Cats can do the weirdest things. The can assume the shape of inanimate objects.
They can even assume the shape of absences.

(Den Bosch, Rotterdam)

Tuesday, December 25, 2018

Pirated booklist poetry

A very short poem

Anglicanism Hieroglyphs
Weapons Testament
The Spanish Politics
Nuclear Ireland

Fascism Quakers
Russian Relativity
The Dead Sea Scrolls
African History

Continental Autism
Buddhist Ethics
American Vikings
Marxist Geopolitics

Medieval Film
Schopenhauer - Wittgenstein,
The World Trade Tragedy

British Kabbalah
Stuart Britain Socialism
Capitalist Crusades
Literary Journalism

Egyptian Theory
Free Will Nothing
Quantum Philosophy

Revolution Music
Brain International
Physics Logic

Pre-Socratic Feminism
Marquis de Sade
Modern Photography
The Kafka Particle

Roman Dreaming
Civil War Philosophy
Christian Myth
Egyptian Psychiatry

Renaissance Art
Platonic Mormonism
Renaissance Design
The Old Postmodernism

The History of Time
The Philosophy of Law
The First World War
Northern Foucault

A discussion with a friend (E.L.) about the "Very short introductions" series.
My friend then sent me a list of the booklets that were illegally downloadable from the Internet.
I edited the list of available titles. It has that "intellectual yet idiot" feeling.

Fortean Times Forum poetry

Dream of darkness

I once saw a man
walking in the distance in a dream.
He turned slightly and saw me
but carried on walking.

The place was peaceful and familiar to me.
I saw distant blinking lights in darkness.

A cheerful man was singing on the radio.
And I heard someone clearly shout my name.

I heard a woman's voice say:
"pull the sheet over your face and stare death in the face".
Woke up screaming about giant spiders
and floating plasma balls.

Forum comments, edited and rearranged. Source: