Lectures on Riemann Surfaces

The book provides an introduction to Riemann surfaces, emphasizing their role as covering spaces and the foundational topology necessary for understanding them. It explores the construction of Riemann surfaces through analytic continuation and multi-valued functions, including algebraic functions. The second chapter focuses on compact Riemann surfaces, presenting classical results like the Riemann-Roch Theorem and employing sheaf cohomology, particularly the first cohomology groups, to derive key theorems. The approach is accessible, blending complex manifold theory with elementary methods.