The study of mapping class groups began in the 1920s with Max Dehn and Jakob Nielsen. It was about this time that topology was just being developed, so mapping class groups were of immediate interest, being invariants of topological spaces. The works of Dehn and Nielsen were continued by William Harvey in the 1960s and 70s leading to the development of the curve complex, an important construction still very relevant to mathematics today. William Thurston is another important name in this area since he was able to completely classify homeomorphisms of surfaces in 1976, leading to the famous "Nielsen-Thurston Classification Theorem".
Representations were first studied by Carl Gauss in the early 1800s and then explored more thoroughly by Ferdinand Frobenius and Richard Dedekind, among others, at the end of that century. Representation theory has since grown into an extremely important and active area of mathematics today because of its widespread applications to other areas of mathematics and even to other subject areas like physics.
Quantum group theory is the youngest area in which this thesis has its roots. This area was formalized and studied extensively for the first time in the 1980s by such mathematicians as Vladimir Drinfeld, Michio Jimbo, and Nicolai Reshetikhin, and immediately found applications in mathematics and theoretical physics. Like representation theory, the study of quantum groups is currently a highly active area of mathematics due to its widespread applications across the mathematical spectrum.
In this paper I will present two different methods of constructing projective representations of mapping class groups of surfaces. I will then prove some interesting results concerning each of these methods.