Riemann Surfaces - Algebraic Equations and Geometric Analysis

Explore the fundamental theory and applications of Riemann surfaces through this comprehensive lecture series that delves into the geometric properties of solution spaces for algebraic equations in two complex variables. Master the basic tools developed by Riemann for studying algebraic equations and understanding the geometry of compact Riemann surfaces, starting from polynomial equations P(x,y) = 0 in C×C. Learn to analyze the topology and geometry of these surfaces while developing skills in integrating differential forms along closed contours. Discover the structure and properties of moduli spaces of Riemann surfaces, including their dimension and topological characteristics. Study compact Riemann surfaces through charts, atlases, and topology, examining meromorphic functions and one-forms alongside theorems on poles and residues using Newton's polygon. Investigate integrals, periods, Abel maps, Jacobians, and divisors while exploring theta functions, prime forms, and fundamental forms within the framework of cycle bases, homology, and cohomology. Examine moduli spaces of Riemann surfaces including Deligne-Mumford compactification, Chern classes, and the tautological ring, with connections to Kontsevich integrals and the KdV hierarchy. Gain exposure to advanced topics including fiber bundles, Hitchin systems, and their relationships to integrable systems, drawing from foundational texts including Mumford Tata lectures, Fay lectures, and the Farkas-Kra book.