Posted 12/01:
This is a famous question which has been around in mathematics for fifty years. What does it mean? To get into the right frame of mind, here is a little exercise.
Step 1. Using the non-writing end of a pen or pencil, tap a few
of the objects around you - tables, books, chairs, chair legs, ... (NOT people).
You will notice that different objects produce different sounds.
Step 2. Choose four or five different objects to tap. Some of
them will probably produce high pitched sounds while some
will produce low pitched sounds. The pitch may depend on where you tap
the object. Choose a single tapping spot on each object where you
will always tap it, and try to list the objects in order of pitch.
Important note: In fact there are many pitches involved in a single tap, not just one.
It may seem that a tapping sound has a definite pitch, but there are almost
always lots of other
quieter pitches in the note as well. It is very
hard to pick these out with your ear,
but these quieter pitches contribute to the overall sound and
they help your ear differentiate between the sounds of different objects.
Step 3. This is the main step and you will need a partner.
One person is the tapper and the other person is the guesser.
The guesser closes their eyes while the tapper chooses an object and taps it.
The guesser listens and trys to figure out what the object is.
Try with a few different objects, then agree on four or five objects for the
tapper to tap, and practise until the guesser always gets them right.
Switch roles so that you both have a chance to do the guessing.
After this game, perhaps you understand the question: `can you here the shape of a drum?'
Suppose that instead of just everyday objects you have a collection of
drums with different shapes. Is it possible to tell which drum someone
is playing just from the sound it makes? Put another way, the question
is: `do drums with different shapes make different sounds?'
Mathematicians had a certain set of drums in mind when they first asked
this question, namely the collection of all planar drums.
A (simply connected) planar drum is
obtained by taking a loop of wire, bending it into any shape that
can be laid flat on the table, and stretching a drum skin across it.
There are infinitely many different shapes for planar drums,
and the question is whether these all sound different.
We should clarify what it means for drums to sound different.
Some people have better hearing than others, and whereas a person with
good hearing might be able to tell two drums appart, they might
sound identical to someone whose hearing is not so good.
It seems that the answer to the question `can you hear the shape of
a drum?' may depend on how good your hearing is.
Mathematicians try to answer the question in an idealized situation, where
an idealized note is determined from the shape of the drum by a mathematical process.
The mathematical process approximates the way the note is actually produced in
real life, so the real life note will be approximately the one given by the mathematical
process.
Whereas in the real world there is a limit to how high notes can be,
in the idealized model, each tap on a drum contains an infinite list of pitches which
get arbitrarily high. The precise mathematical question is: Are there two drums with
different shapes which give rise to the same infinite list of pitches?
Planar drums are two dimensional. In my work I study drums
of higher dimension which cannot be built
in the classroom.
Even the three dimensional drums I study
cannot, because the three
dimensional space we live in does not have room for
the way the drums twist and turn around. Nevertheless these drums
can model phenomena in the real world where more
than just spacial position is involved. A few examples of other
physical variables are velocity, energy, time and magnetic field.
One of the main questions I am studying is:
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