Context: The Riemann-Hurwitz formula is a fundamental result in algebraic geometry and the theory of Riemann surfaces. It relates the genera of two Riemann surfaces connected by a holomorphic map, taking into account the ramification of the map. Problem Statement: Let f CD be a non-constant holomorphic (branched covering) map between compact, connected Riemann surfaces C and D. Suppose f has degree d, and let go and gD denote the genera of C and D, respectively. Tasks: 1. Definitions and Preliminary Concepts: a. Genus of a Riemann Surface: Define the genus g of a compact Riemann surface. Explain its topological significance in terms of the surface's connectivity and "holes." b. Holomorphic Maps and Degree: Define what it means for a map f : C→ D between Riemann surfaces to be holomorphic. Explain the concept of the degree d of the map f. c. Ramification Points and Indices: Define ramification points of f and ramification indices. Explain the ramification divisor associated with f.

Complex Sets 1. Sketch the following sets and determine which are domains: (a) |z – 2 + i| 1; (e) 0 4; (d) Im z = 1; (f) \z – 4| > |z]. 2. Which sets in Exercise 1 are neither open nor closed? 3. Which sets in Exercise 1 are bounded? 4. In each case, sketch the closure of the set: -T 0. 5. Let S be the open set consisting of all points z such that |z| < 1 or |z – 2| < 1. State why S is not connected. 6. Show that a set S is open if and only if each point in S is an interior point.

1. Describe geometrically the sets of points z in the complex plane defined by the following relations: (a) 221 222] where 21, 22 € C. (b) 1/2 = 7. (c) Re(z) = 3. (d) Re(z) > c, (resp., 2 c) where e ER. (e) Re(az + b) > 0 where a, b € C. (f) |z| = Re(z) + 1. (g) Im(z)e with CER. 2. Let (,) denote the usual inner product in R². In other words, if Z= (x₁, y₁) and W = (2, 92), then (Z, W) = x₁x2 +Y₁Y2. Similarly, we may define a Hermitian inner product (,) in C by (z, w) = zw.

3. A quadric surface is a surface in R³ defined by an equation of the form ax² + by² + cz²+dxy + exz + fyz + gx +hy+iz + j = 0. After a suitable rotation and a translation, it is possible to convert the equation into one of the standard forms listed in the table here: https://en.wikipedia.org/wiki/Quadric#Euclidean_space The general method for converting the equation is as follows. First, rewrite the equation in matrix notation as where (x y z) A Then a d/2 A = d/2 b X 2 e/2 f/2 C + B X e/2 ƒ/2 B=(g_hi). 2 Z Since A is real and symmetric, by the spectral theorem there exist an orthogonal matrix Q and a diagonal matrix D such that QtAQ = D. One may choose Q with a positive determinant, so that it represents a rotation by Euler's rotation theorem. Let + j = 0, 0--0 = Qt 1 X (x y z) A · · * (?) + ² (1) − × ( ) +- B +j= (x' y' z') Dy' + j اج where B' BQ. Finally, find a suitable translation by completing the square to put the equation into the standard form. Convert the following two…