Being a mathematician requires a careful balance of two important character traits: tenacity and laziness.
A mathematician will not let go of a problem until an answer has been found, but equally a mathematician would rather find an easy answer than a difficult one, even if finding the easy answer takes longer than the more direct one.
Laziness also means that mathematicians are extremely reluctant to give up a good theory.
If a theorem was useful in one area, it is reasoned, it ought to be useful in others as well - even if there is no apparent connection between the two.
One of the most successful, certainly the most practical, areas of mathematics is calculus.
And so mathematicians have spent hundreds of years pushing calculus into areas that it should never have been taken to so that now calculus appears in just about every area of mathematics, from number theory to discrete dynamics.
Differential topology is one of the milder outposts of calculus, yet even there one can encounter spaces that would make Isaac Newton's head spin.
In some spaces, the back of your head is the front, and the middle of the top of the right hand side of the front is the back.
Even Einstein would get lost here (though due to relativity, it would not be him that was lost but the universe around him would be lost).
In these spaces, it's wise to have a look first from a safe distance before venturing further in.
This talk was given as a series of animations.
The original avi
format is rather large for the web so these are in
mpeg
.
Even so, the original size still produces files of nearly a mega-byte
in size so I've made smaller versions as well.
The animations were designed to loop indefinitely until I moved on to the next one.
The running order, together with a brief explanation as to the purpose
of each animation, follows.
- Title sequence title (532K) title (182K)
A 4-link borromean ring borromean
(564K) borromean (294K)
The purpose of this animation and the following three was to illustrate the type of answer that one is likely to get if one asks a differential topologist for examples of applications of differential topology: knots, robot arms, and black holes.
A robot arm robotArm (314K) robotArm (70K)
- A black hole.
This was just a black picture on the screen.
- A robot arm in a knot near a black hole knottedArm
(0) knottedArm
(0).
Technically, this was just a static picture made into a movie.
The background picture is from the Nasa gallery and depicts the centre
of our galaxy.
Mottos for science and mathematics motto
(376K) motto (80K)
The role of science is to observe the world with a view to changing it (having an effect) for the better: thus "To Observe and Effect".
To do this properly, scientists need a language in which to record their observations and build their models of reality.
This is mathematics.
Thus mathematics needs to take the scientific model and solve it, whence it can predict the outcome.
(The original motto is that of the Toronto police department.)
Modelling a process by a function process
(514K) process (114K)
The basic "thing" that we wish to model is a "process", which we model by a "function".
It's not hard to argue, for practical reasons, that this function should be continuous and differentiable.
The Leibniz quotient quotient
(366K) quotient (84K)
As calculus has been such a successful part of mathematics, it's natural to want to push it as far as possible.
Studying the Leibniz quotient is a good place to start.
From looking at the "limit" part, we realise that we only need our spaces to look like Euclidean space locally.
A close-up of a sphere sphereZoom
(774K) sphereZoom (322K)
A sphere looks like Euclidean space when examined sufficiently closely.
Making a sphere from a piece of paper sphere
(236K) sphere (50K)
A sphere can also be made by taking a piece of paper and deforming it slightly.
Making a torus from a piece of paper torus
(458K) torus
(76K)
As can a torus.
Life on a sphere onSphere
(350K) onSphere (66K)
But life on a sphere can look very strange.
As something moves away from you, it starts to get small but then gets very big again when it nears the antipodal point (the sphere is 2D here, the 3rd dimension is Euclidean).
Life on a torus onTorus (706K)
onTorus (180K)
Whereas on a torus, stuff just gets small.
A water-ride at a theme park waterCoaster
(732K) waterCoaster (158K)
This animation, and the following ones, imagines the following problem: you want to build a "water ride" at a theme park based on a manifold.
So the dingy in the ride is pushed around by the flow of water.
This flow is assumed to be static, in that the speed of the water at a particular point doesn't change.
There are two ways of powering this ride.
At every point we can put a fan, and we can put a pressure regulator.
The fan is more general but, for the sake of this illustration, more expensive so ideally one wants to build the ride using pressure alone.
The question is: when can it be done?
The answer is: when there are no "Escher staircases" in the ride, since if the ride is driven by pressue alone then one can never go against a pressure gradient.
Finding potential staircases in a given design can be difficult, but there is an easy class of them which can be found: infinitesimal staircases.
On a sphere, this turns out to be enough since any large-scale staircase can be collapsed to an infinitesimal one.
On a torus, this is not enough since there are flows with large-scale staircases but no infinitesimal ones.
Slightly more mathematically, the question is: when is a vector field X on a Riemannian manifold of the form X = ∇f for some function f?
There is a necessary condition which is easier to express in the language of forms: if α_{X} is the 1-form corresponding to X then we need d α_{X} = 0.
This is also sufficient if and only if the first cohomology group of the manifold vanishes, which is true for the sphere but not for the torus.
However, even when the first cohomology group does not vanish, this significantly simplifies the problem as it tells you where to look for the large-scale staircases.
Fans versus pressure waterFan
(712K) waterFan (158K)
Illustrating the two methods of powering the flow.
Escher's staircase staircase
(116K) staircase (52K)
Illustrating a non-trivial cycle in the flow.
Collapsing the staircase on a sphere staircaseCollapse
(582K) staircaseCollapse (120K)
A "large scale" cycle in the flow on a sphere can be scaled down to an infinitesimal one.
A non-collapsable staircase on a torus staircaseNoCollapse
(116K) staircaseNoCollapse (34K)
A "large scale" cycle in the flow on a torus that cannot be scaled down to an infinitesimal one.
A water-ride on a torus with no "mini staircases" but with a
large-scale staircase torusFlow
(638K) torusFlow (122K)
An example of a flow with a large-scale cycle that cannot be scaled down to an infinitesimal one.
Technical notes: the animations were prepared using blender.
The method for displaying them was perhaps slightly complicated: I wanted each animation to seemlessly loop until I indicated that it was time for the next one.
The simplest way of doing this was to use totem in fullscreen mode with repeat mode selected.
Totem can be controlled by shell scripts, even to the extent of changing the currently playing movie.
Thus with a script, I could replace the current animation by the next one (or previous one) and play/pause.
The only snag was that I had to have both totem and the xterm on display, fortunately my computer can handle split screens and I could get totem to display on the projector screen and the xterm on the computer screen (it took a couple of iterations to get them the right way around!).