The Laplacian: An Exploration and Historical Survey Tailored for Translation Surfaces

Abstract

This thesis is a historical survey of the Laplacian as an operator on $L^2$-functions specifically geared towards building the understanding necessary to define a Laplacian on a translation surface. The author explores the role the Laplacian has played historically in analysis and geometry, with a particular interest in the connections between the Laplacian and the geodesics. The primary thread the author follows develops a representation-theoretic perspective of the Laplacian, which proves advantageous when working on symmetric spaces. The other appeals to a functional-analytic perspective in more abstract settings. In the final section, the author proposes a starting point for defining a Laplacian on a translation surface.