Abstract: The lecture will be devoted to the geometry of the 3-sphere S3. This is the one-point compactification of the real 3-dimensional space R3, thus obtained from R3 by adding one point ∞ called infinity. By adding to the real line R1 one point ∞, one obtains a circle, this is the 1-sphere; adding to the real plane R2 one point ∞, one obtains the 2-sphere: the Riemann sphere. It is easy to see that the 3-sphere S3 is the disjoint union of circles. It is more challenging (but possible) to write it as the disjoint union of circles which are pairwise linked - this is the result which is the focus of our considerations.
The starting point will be to look for circles on a torus (a torus has the shape of a donut or a lifesaver): the interesting ones are the so-called Villarceau circles: they are named after Villarceau who lived 1813 - 1883, but they have been known already in the 16th century, and are also constitutive for some African wood carvings! Here, we deal with surprising mathematics which can be discovered at the breakfast table (are the circles which we see really circles?)
To use a more fancy language:
it is the Hopf fibration of S3 which we want to discuss.
Let us stress that the Hopf fibration is a very important mathematical object,
its discovery by Hopf in 1931
is the starting point of the stable homotopy theory,
one of the relevant topics of current mathematical research.
Here we are in the realm of topology, but there is also an
algebraic interpretation of the Hopf fibration:
it describes the division of complex numbers.