Title of the course: Riemann Surfaces and Algebraic Curves: Concepts, Classifications, and Elliptic Curves
Instructor: Dr. Özgur İnce
Institution: Sivas Cumhuriyet University
Dates: 16-22 September 2024
Prerequisites: Complex Analysis and Set Point Topology, basic notion about algebraic curves
Level: Graduate, advanced undergraduate
Abstract: Riemann surfaces are obtained by gluing together patches of the complex plane by holomorphic maps, whereas algebraic curves are one-dimensional shapes defined by polynomial equations. (Compact) Riemann surfaces and (complex smooth projective) algebraic curves are equivalent in a precise sense. This means we can study the same objects using both complex analysis and algebra. The course starts with an introduction to the concept of Riemann surfaces, followed by illustrative examples. It then covers the classification of Riemann surfaces using covering space theory and uniformization, based on the fundamental group. Finally, we will explore compact Riemann surfaces of genus 1, specifically focusing on elliptic curves and their details.
Language: TR, EN
Textbook:
– E. Girondo and G. Gonzalez-Diez, Introduction to compact Riemann surfaces and dessin d’enfants, LMS Students texts 79, Cambridge University Press, Cambridge, 2012.
– R. Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics
– H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics
– S. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Mathematics