Harmonic analysis and the topology of spheres (2019)
The goal of this undergraduate thesis is to study the harmonic analysis in homogeneous spaces, based on the theory of representations of compact Lie groups, and how we can use this theory to find topological invariants of those spaces. Specifically, it deals with the case of spheres S^n, which can be seen as homogeneous spaces by means of the quotient of special orthogonal groups SO(n + 1) / SO(n), and explicit calculations of its Euler character are presented. The latter with the help of the Hodge decomposition theorem and the index theorem for the Dirac-de Rham operator on the sphere