Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete intersection curve. What is its topological genus? but I don’t know how can I calculate […]

Consider a polynomial $P(z) = z^4 \in \mathbb{C}[z]$. Set-theoretically $P(z)$ has one root equal to zero. From algebraic point of view it has four roots: root zero has multiplicity four. Also we can’t draw a curve in $\mathbb{C}$ around one of such roots but not around the others. Now consider a Riemann surface $X$ given […]

Edit: I already received a good answer to my second question. I’d be interested in a hint about the first one, as well. Thanks in advance! I’m interested in compact Riemann surfaces and their homology. In this question, Kundor proposes a nice drawing of the connected sum of tori, saying that it is clearer than […]

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn’t find it in Diamond and Shurman, and neither could I find an explicit formula with a simple google search. Certainly, there should be some explicit expression, no?

For a long time (a couple of years) I’m following the Q&A’s about “half-iterate of $\exp(x)$” etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder’s method, but also a $\mathbb R \to \mathbb R$ for fractional heights $h$ due to Hellmuth Kneser. I would like to understand the latter method of […]

Fix a positive integer $n$. Let $f:\mathbf{H}\longrightarrow \mathbf{C}$ be a modular function with respect to the group $\Gamma(n)$. Is the derivative $$\frac{df}{d\tau}:\mathbf{H}\longrightarrow \mathbf{C}$$ also a modular function with respect to $\Gamma(n)$? I think it’s clear that $df/d\tau$ is meromorphic on $\mathbf{H}$ and that it is meromorphic at the cusp. I just don’t know why it […]

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.

So, I’ve seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called ’tiling method’ as in section five of this paper (http://arxiv.org/pdf/1007.5467.pdf). Basically, you end up getting: $$ \begin{equation} K_M(\textbf{x}, \textbf{y}, t) = \sum\limits_{g \in G} K_\tilde{M}( \tilde{\textbf{x}}, g \cdot […]

I got stuck reading the proof of the following theorem: Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere. Proof: Let $g_{S^3}$ denote the round metric on $S^2$ (such that the area is $4\pi$). By the uniformization theorem there exists a map $\phi: S^2 […]

Intereting Posts

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Spectrum of finite $k$-algebras
What exactly is a Kähler Manifold?
How to solve for a variable that is only in exponents?
Lipschitz Smoothness, Strong Convexity and the Hessian
On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?
How to find large prime factors without using computer?
Transforming integral equation to differential equation
Zeroes of a Particular Function
Isometries of the sphere $\mathbb{S}^{n}$
Clarifying definition of outward unit normal
How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?
Circulant determinants
A push-out of a pull-back
What are functions used for?