The June 2010 edition of Discover Magazinehas an article on Harvard mathematician, Shing-Tung Yau. The article describes Dr. Yau as the man who has devised the math to describe string theory. String theory postulates that at the deepest reality the universe is composed of 10 dimensional vibrating strings. His particular branch of mathematics is topology.

Topology is the mathematical study of the way in which the properties of objects are preserved through the deformation, twisting, and stretching of objects. Tearing, however, is not allowed. In topology a circle is equivalent to an ellipse since you can turn the circle into an ellipse by stretching the circle. The set of all possible positions of the hour hands on a clock is equivalent to a circle, a one dimensional closed curve with no intersections that can be embedded into a two dimensional space.

A central idea in topology is that spatial objects like circles and spheres can be treated as objects in their own right and that the knowledge of these objects is independent of how they are represented or embedded in space.

The article in Wolfram Mathematica gives the following examples of the following studies in topology: the space-time of general relativity including fractals, knots, and manifold which have some of the same basic properties as the universe, phase spaces in physics such as the positions of the hands of a clock, and symmetry groups such as the collection of ways to rotate a top.

In the interview in the article Dr. Yau explains that geometry is specific whereas topology is general and that topologists study large patterns and categories of shapes. The example given in the article is that in geometry a cube and a sphere are distinct but not in topology. In topology the two objects can be deformed into each other without cutting through the surface. A torus, a sphere with a hole in the middle is different. A torus is distinct from a sphere because one cannot deform a torus into a sphere no matter how much one twists it. The interesting graphic above depicts the twisting of a torus into the shape of a coffee cup.

In mathematics topology highlights the importance of non linear equations in nature. He notes that even though the stock market is traditionally described in straight lines and linear equations that is not a correct depiction of reality. The stock market fluctuates up and down in a nonlinear way. Topology constrains geometry in the physical world. He notes that if water flows around a sphere, there must be two points at which the water is perfectly still. If the ocean flows from east to west, it cannot flow in the same direction everywhere without hitting a snag. By way of contrast in another topology, that of the torus, the water can flow around and around. There is no point at which the flow must stop because the hole eliminates the impasse.

These views of topology lie at the heart of string theory which postulates the possibility of a world composed of many more dimensions than three. The article requires some rigorous attention but contains some fascinating ideas and is well worth reading.