Category Archives: MA 59800
In the late 1700’s and early 1800’s, the French mathematician Augustin-Louis Cauchy (1789 – 1857) and German mathematician Bernhard Riemann (1826 – 1866) worked independently to create a theory of differentiating complex-valued functions . In order to define such functions, … Continue reading
In the previous lecture, we discussed how to draw triangles in the plane and on the unit sphere such that the triangles tile these surfaces. In this lecture, we discuss triangle groups in more detail by focusing on discrete symmetries … Continue reading
Over the next couple of lectures, we will generalize cyclic groups and dihedral groups . Given positive integers , , and , formally define the abstract group Such a group is called a Triangle Group, although it is also known … Continue reading
Let denote either , , or . The collection of rational functions which have are called Möbius Transformations. That is, they are in the form where . We denote this collection by . We found in the previous lecture that … Continue reading
Groups were first studied as objects acting on sets. For example, we can consider the group of rotations of a regular polygon. Eventually, we wish to consider a specific type of group acting on the collection of rational functions over … Continue reading
Let be a field contained in , and fix a number in . Recall that is the projective plane, while is the sphere of radius : This assignment is meant to show is not the same as . This assignment … Continue reading
In the previous lecture, we introduced the notion of a non-singular projective curve. Today we discuss examples in detail by focusing on Elliptic Curves, Cubic Curves, and Quartic Curves.