MATH 7510

Description in the Notices AMS

A short film depicting the beauty of Moebius Transformations in mathematics. The movie shows how moving to a higher dimension can make the transformations easier to understand. The full version is available at IMA Moebius The background music (from Schumann's Kinderszenen, Op. 15, I) is performed by Donald Betts and available at http://www.musopen.com.

The projective plane is the space of lines through the origin in 3-space. In the projective plane, we have the remarkable fact that any two distinct lines meet in a unique point. Moreover, we may not distinquish between different kinds of smooth conic sections, e.g. between an ellipse and a hyperbola, as illustrated by the animation. Notice that to each line through the origin correspond two antipodal points of a sphere which is centered at the origin. Instead of looking at the lines in total, we may thus restrict ourselves to the points on the sphere. Here, a line becomes a great circle and it is clear that any two such circles meet in a unique pair of antipodal points. Similarly, we may see that a hyperbola is projectively the same thing as an ellipse. Going back to the definition: They are both just a cone in 3-space; each line on the cone is one projective point of the projective conic section. This animation was made by Oliver Labs using surfex.

Pivoted lines and the Moebius Band: This video demonstrates the transformation of the projective plane RP^2 = {lines R^3 which pass thru the origin} into the union of a moebius band and a disc joined along their circle boundaries. The idea of the visualization is to represent the projective as a moving bar fixed at the origin.

The movie shows a generic deformation of the A4-Singularity. One can clearly see the chain consisting of four cycles that is contracted in the singularity: the vanishing cycles. Note that these four cycles are the real part of a chain of four two-spheres in the complex surface. This film was made by Duco van Straten and Oliver Labs using surfex.

In this film we show how discriminants can be used to construct surfaces with many singularities. Our example will be 31-nodal quintics which is the maximum possible number (shown in 1979 by A. Beauville) as already remarked in number 06 of our calendar. The first construction of 31-nodal quintic surfaces was given by E.G. Togliatti in 1940. For the film we follow Barth's construction which is similar to his construction of the 65-nodal sextic.