MY-ASSIGNMENTEXPERT™可以为您提供sites.google MA307 Riemann surface黎曼曲面课程的代写代考和辅导服务！
这是印度科學理工學院黎曼曲面课程的代写成功案例。
MA307课程简介
Course description: Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps. This course will be an introduction to the the theory of Riemann surfaces, with an emphasis on analytical and topological aspects. After describing examples and constructions of Riemann surfaces, the topics covered would include branched coverings and the Riemann-Hurwitz formula, holomorphic 1-forms and periods, the Weyl’s Lemma and existence theorems, the Hodge decomposition theorem, Riemann’s bilinear relations, Divisors, the Riemann-Roch theorem, theorems of Abel and Jacobi, the Uniformization theorem, Fuchsian groups and hyperbolic surfaces.
Prerequisites
There will be no class on January 5th (Tuesday).
There will be no class on December 17th (Thursday).
There will be no class on November 19th (Thursday). Also, no lectures would be held in the midterm week of Nov 23 -27.
A new Teams code has been emailed to the mailing list.
There will be weekly homework also.
There will be no lecture on Tuesday, October 6th.
Lectures will be held on Tuesdays and Thursdays, 3:30-5pm.
Our first meeting will be held on Thursday October 1st , 3:30-5pm.
Classes will be online, at the beginning over Microsoft Teams. The first meeting link will be put up here, and on the IISc Intranet Bulletin board, before the meeting time.
MA307 Riemann surface HELP（EXAM HELP， ONLINE TUTOR）
Suppose $X$ and $Y$ are compact Riemann surfaces, and suppose $\mathcal{C} \subset X$ and $\mathcal{D} \subset Y$ are finite subsets such that $X \backslash \mathcal{C}$ and $Y \backslash \mathcal{D}$ are conformally equivalent. Then show that $X$ and $Y$ are conformally equivalent.
Suppose $\left[f_0, p_0\right]$ is an algebraic germ that satisfies $P\left(z, f_0(z)\right) \equiv 0$ in a neighborhood of $p_0$, where $P(z, w)$ is an irreducible polynomial. Show that $\left[f_0, p_0\right]$ admits analytic continuation along any path in $\mathbb{C P}^1$ that avoids a finite set of points (namely the “critical points” of $P$ ).
Let $\widetilde{R S}$ be the ramified Riemann surface corresponding to the algebraic germ $\left[f_0, p_0\right]$ where $f_0(z)=\sqrt[4]{\sqrt{z}-1}$ and $p_0=4$
(a) Determine the branch/ramification points on $\widetilde{R S}$ and their images under the map $\pi: \widetilde{R S} \rightarrow \mathbb{C P}^1$. Draw a picture (like Figure 5.12 of Schlag’s book) to illustrate.
(b) Using the Riemann-Hurwitz formula, compute the genus of $\widetilde{R S}$.
Let $\Gamma$ be a Fuchsian group, i.e. a discrete group of $\mathrm{PSL}_2(\mathbb{R})$ that acts properly discontinuously on the upper half-plane $\mathbb{H}$.
(a) Show that the stabilizer of any point $p \in \mathbb{H}$ is a finite cyclic group. (From what we saw in class, this implies that $\mathbb{H} / \Gamma$ is a Riemann surface.
(b) Show that $\Gamma$ acts freely on $\mathbb{H}$ if and only if it is torsion-free.
Let $X$ be the hyperelliptic curve defined $f(x, y)=y^2-x^8+1$, and let $Y$ be the hyperelliptic curve defined by $g(z, w)=w^2-z^5+z$. Show that the function $f: X \rightarrow Y$ defined by
$$
f(x, y)=\left(x^2, x y\right)
$$
extends to a holomorphic map $F: X \rightarrow Y$ which is unramified.
Let $X$ be the elliptic curve defined by the zero set ${[x: y: z] \in$ $\left.\mathbb{C P}^2 \mid y^2 z-x z^2+x^3=0\right}$. Show that the differential form $\omega$ given in the affine chart $U_3 \cong \mathbb{C}^2$ by $\omega=d x / y$ defines a holomorphic differential on $X$.
MY-ASSIGNMENTEXPERT™可以为您提供SITES.GOOGLE MA307 RIEMANN SURFACE交换代数课程的代写代考和辅导服务！
