Explore the intricacies of differential Galois theory and its application to algebraic solutions in this comprehensive lecture from the Workshop on "Algebraicity and Transcendence for Singular Differential Equations" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the fundamental concepts and advanced techniques presented by Michael Singer as he examines the relationship between differential equations and their solutions from an algebraic perspective. Over the course of 89 minutes, gain valuable insights into the theoretical foundations and practical applications of this specialized field, enhancing your understanding of both differential Galois theory and the algebraicity of solutions in mathematical analysis.