Introduction to Riemann Surfaces and Elliptic Curves

The study of Riemann surfaces lies at the beautiful intersection of complex analysis, topology, and algebra. For students of mathematics and physics, understanding how these geometric objects behave is essential for advanced work, yet standard textbooks often skip the vital intuitive steps. This text-based course bridges that gap, guiding you through the foundational theory of complex 1-tori and elliptic curves with clarity and precision. You will transition from basic complex analysis to a structured understanding of algebraic curves, learning how geometric properties translate into algebraic equations. The course starts with essential terminology, basic topological concepts, and foundational definitions of complex manifolds before moving into practical mathematical derivations. What you'll learn: - Understand the foundational definitions of complex manifolds and Riemann surfaces - Construct and analyze complex 1-tori as quotient spaces of the complex plane - Apply the Weierstrass elliptic functions to parameterize algebraic curves - Map the deep connection between complex tori and cubic curves in projective space - Explore modern applications of elliptic curves in cryptography and number theory Starting with fundamental definitions, this written course systematically builds your mathematical maturity, moving from the topology of tori to the algebraic equations that define elliptic curves. It is designed for beginners to the subject of algebraic geometry, requiring only a basic background in undergraduate complex analysis. Start reading today to unlock the elegant connections of modern geometry.