Why Mathematics?
[1997 -- under construction]
Why would one be interested in mathematics? A random image tour through some of my own interests
Although the problem of classifying all knots is easy to state, it has been of the hardest mathematical problems around. First of all, incontrast with the knot that you tie in your shoe laces, a mathematical knot cannot be untied. The proper defintion of a mathematical knot is a closed curve embedded in space without intersections, such as the following pictures (all taken from the beautiful Knotplot site)
One can also consider links that consist of several components
Links can have wonderful properties. For example the Borromean rings
cannot be pulled apart, although the individuals rings are not pairwise linked. A nice example of a higher-order relation in mathematics and often used as a symbol of cooperation (Olympic flag).
Mathematicians wonder: Is there such a thing as a `telephone book' that lists all possible knots in some systematic way? It would look like this
This issue has only been (partly) settled very recently. Knot theory has obtained a great impetus from developments in particle and statistical physics. Particles that move in space and time knit large knots
One can assign a number to a knot (the knot invariant) by computing the quantum mechanical amplitude of an electron that follows the path of the knot.
This approach is at present the best way to tackle the problem of classification.
Knot theory has application is such unexpected areas as molecular biology where it is used to study mRNA strands. Here is a picture of such a strand. Knot theory helps to explain what kind of enzymatic reactions take place
Of course they are also simply beautiful objects
Let's consider the equation
x3+y3=1.
It describes a so-called elliptic curve in the xy-plane.
We can now go to complex coordinates. There it will describe a complex curve or real surface. It looks like a doughnut
We can also go to projective coordinates, use ratios x/z and y/z. Then we have the equation
x3+y3=z3
in projective space. Now consider the same equation over the integers: we are in the world of number theory. Fermat's Last Theorem dates from the 17th century. It states that the equation
xn+yn=zn
has no integer solutions if n is greater than 2. This theorem was only recently proven by Andrew Wiles in one of the greatest (and dramatic) struggles in mathematics. Surprisingly, his method uses again elliptic curves.
String theory is based on the (deceptively simple) premise that at Planckian scales, where the quantum effects of gravity are strong, particles are actually one-dimensional extended objects. Just as a particle that moves through spacetime sweeps out a curve (the worldline)
string will sweep out a surface (the world-sheet)
In contrast with particle theories, string theory is highly constrained in the choice of interactions, supersymmetries and gauge groups. In fact, all the usual particles emerge as excitations of the string and the interactions are simply given by the geometric splitting and joining of these strings:
In this way the usual Feynman diagrams of quantum field theory are generalized by arbitrary Riemann surfaces
Physical processes are obtained by averaging over all Riemann surfaces. This link with string theory has made some new break-throughs possible in the mathematics of Riemann surfaces.
The world has always been fascinated with polyedra, in particular the Platonic solids
 |
 |
 |
 |
 |
| tetrehedron | octahedron | cube | dedecahedron | icosahedron |
This fascination was not always scientifically sound, it was linked with the four elements of the ancient world (plus one for the Universe)
Mathematician always want to generalize. You can think of polyedra in higher-dimensions, such as the hypercube in four dimensions. Here are some visualisations
By the way, only in dimension three and four do the exotic polyhedra such as the icosahedron and its four-dimensional cousing the 600-cell exist. This is a reflection of the fact that four-dimensional geometry is very special.
Polyedra are succesfully applied in chemistry and biology. The buckyball or Buckminsterfullerene C60 molecule (Nobelprize 1996), the truncated icosahedron, is one of the `hottest' molecule around
A picture of the common cold virus (Rhinovirus 14). It is patterned after the icosahedron
You find the Platonic solids everywhere in mathematics, usually under the technical name ADE classification.