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Riemann Mapping Theorem – The Crux of Complex Analysis

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Today I’m going to put some pieces of information on Riemann Mapping Theorem from across the web which I have been studying for a while. This post assumes a fundamental understanding of Conformal Mappings used in Complex Analysis. If not, this picture below should serve as an end-to-end introduction.

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Another one:

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Another one which makes it clearer:

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I’m in no mood to cover Möbius Transformations so I’ll directly move to RMT.

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So now let’s come back to RMT.

The Riemann mapping theorem states that if U is a non-empty simply connected (it just means that the domain U has no holes and is wholly connected within itself) open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk

D = \{z\in \mathbf{C} : |z| < 1\}.

The illustrations below further widened my outlook on RMT and its immense usefulness.

This slideshow requires JavaScript.

I thereby conclude today’s post. I’ll come back with a new post sometime later in the week where I’ll cover further ramifications of a famous lemma which will kick start our discussion in a wholly new direction.

 

 

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Mandelbrot Locally Connected (MLC) Conjecture

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** This is the revival post for my blog so I’ll keep it crisp and clear for now. **

My latest readings have kept me busy with the beautiful intricacies of Julia Sets, Fatou Sets and Mandelbrot Sets. In this post, I’m simply reconstructing the explanation of a contributor on MathOverflow giving reasons as to how Mandelbrot Set’s local connectedness has deep rooted implications for topics in Complex Dynamics and beyond.

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That’s it from my side for today. This is a sort of a primer for the theme I’m going to cover in my next few posts which will basically be a set of finely curated materials across the web and may or may not be my original work.

Courtesy: Jeremy Kahn

 

 

Dear Barkha Dutt: The Buck Stops Where?

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KAFILA - 10 years of a common journey

On the of latest edition, (telecast a few hours ago, on the evening of the 15th of January, 2013) of ‘The Buck Stops Here’, (a flagship news show on NDTV anchored by Barkha Dutt) – ‘India-Pakistan:Another Tipping Point‘, Admiral (Retd.) Ramdas, former chief of the Indian Navy said he knew that Indian forces have beheaded Pakistani soldiers in the past. Gen. (Retd.) V.P. Malik, former chief of the Indian Army contradicted him, and said this had never happened. Barkha Dutt was silent on this matter.

Below is a summary of some highlights of the discussion.

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Aap Ki Dilli – Dil Se !

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Aap Ki Dilli – Dil Se !

Meera's Musings

Today, Feb 10th, 2015 has been a very special and unforgettable day !

The stunning sweep of the Aam Aadmi Party’s victory, with 67 out of 70 seats in the Delhi Vidhan Sabha, has come at the conclusion of a long year, with many points at which the party had been written off by detractors.

As I look back on the past year, I believe these are the 5 most important reasons that Delhi voted so resoundingly for AAP.

1. The hard work and credibility of AAP’s candidates : Sir jee – AAP ki sarkar !

Candidates who had been elected as MLA’s in the 2013 election, conducted several Mohalla sabhas and meetings with Residents’ Welfare Associations (RWA’s) to ascertain the most pressing needs of their constituency. They then proceeded to expend their entire MLA LAD (Local Area Development) Funds, in line with these priorities. By the time the…

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Thinking of suicide? Read me

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Life Blog Series wants to dedicate a special piece written by “takingthemaskoff” which is the least I can do to promote its reach. Without delving into the intricacies surrounding the topic he has so heart wrenching-ly penned down, I would suggest you to go forward and give it a honest 5 minutes read.

takingthemaskoff

“You see the giant and the shepherd in the valley and Elah and your eye is drawn to the man with the sword and shield and the glittering armor. But so much of what is beautiful and valuable in the world comes from the shepherd, who has more strength and purpose than we can ever imagine.”

 -Malcolm Gladwell

 

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I survived a suicide attempt, spent years in rehab centers, jails, psych hospitals. Now I have worked as a supervisor at these type of facilities.

However my friend, he did not. This is what suicide looks like. This is him after hanging himself, right before he died.

 

The difference is nothing. He grew up in a dysfunctional home, where the norm was drug use, and physical, emotional, and psychological abuse. They didn’t have money, we did. He went to jail, he stayed. I went to jail, I got bailed out…

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A Black Hole Visits Baltimore

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I recommend this Black Hole to visit my University in India

Frontier Fields

[NOTE: This post is the fourth in a four-part series. Previous posts are: 1) Einstein’s Crazy Idea, 2) Visual “Proof” of Gravitational Lensing, and 3) Gravitational Lensing in Action.]

For the final part of this series of blog posts, let’s bring things back to Earth. The demonstration of a physical process will always seem a bit arcane when using unfamiliar objects as the example. Most folks don’t have a working relationship with galaxies, let alone the strange varieties one gets in the distant universe. Instead of taking the viewer into the universe, it can be more intuitive to bring the cosmic phenomenon closer to home.

Suppose that, say, a black hole decided to take a short vacation. Perhaps it got tired of the enormous responsibilities of being such a tremendous distortion of space-time. It needed a weekend off to cool its jets (absurdly geeky pun intended – sorry)…

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