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.

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…

J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f…

(2000) J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atta.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Rea! and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function…