Course Identification

Riemann surfaces are complex manifolds of dimension one. They appear in a wide variety of mathematical fields: complex analysis, differential equations, algebraic geometry, flat and hyperbolic geometry. The goal of this course is to give an intuitive and geometric introduction to this notion. Using translation surfaces and geometry of polyhedra, we show how to construct examples of Riemann surfaces by scissors-and-glue techniques. 
We will discuss the following topics (nonexhaustive list):
- Topological classification of surfaces;
- Riemann-Hurwitz formula;
- Genus of a complex algebraic curve;
- Elliptic trigonometry, Elliptic functions, Elliptic curves (and their moduli space).

The student will learn about the basic results of Riemann surface theory with an emphasis on connections with neighboring fields. He will be able to make elementary computations involving ramified covers and will get intuition on the geometric interpretation of complex analytic objects.