Infinite Limits

Hey! It's been a while since I last posted even though I mentioned that I would be more regular. That's my bad, I misjudged how busy I would be. Anyways.. enjoy this continuation of the last post! This is rather elementary, so please leave a comment if any of you would like a more detailed/longer explanation of the proof. An Intro to Perelman's Work Image Credit: CUNY - Mathematics To delve deeper into Grigori Perelman's proof of the Poincaré Conjecture, we need to explore several sophisticated concepts in differential geometry and geometric topology. Perelman's work, which built on Richard S. Hamilton's program involving the Ricci flow, is deeply rooted in the analysis of geometric structures on manifolds. Here, we'll attempt to shed light on some of the mathematical intricacies involved. The Poincaré Conjecture posits that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere S^3. A manifold is simply-connected if it lacks "holes,...

 Hey! It's been a minute since I wrote one of these, so I'm genuinely glad to be back. Happy New Year! I should be more regular now because I'm done with all university applications and I'm relatively free (as free as you can be by year 12). Today I'll be jabbering on a bit about the Poincaré Conjecture, one of the most special (solved) problems in mathematics. The Poincaré Conjecture Image Credit: Clay Mathematics Institute The Poincaré Conjecture is a fundamental problem in the branch of topology, best known as one of the 7 Millennium Prize Problems of the Clay Mathematics Institute. It was one of the most famous and long-standing unsolved problems in the history of mathematics until it was finally proven in 2003 by the Russian mathematician Grigori Perelman. Perelman's proof was a monumental achievement and earned him the Fields Medal and 1 million US Dollars (from the Clay Mathematics Institute) - both of which he declined (which I'll return to later). H...

The Mandelbrot Set is a set of complex numbers commonly used in art, due to its mathematical beauty. It was first defined by Robert W. Brooks   and Peter Matelksi in 1978, and the first visualisation was made by Benoit Mandelbrot in 1980.   Credit: American Scientist A lot of websites, videos, applets, and so on do a repeated zoom into the boundary of the Mandelbrot Set. But in mathematical terms, we define the Mandelbrot Set as the set of complex numbers which (under iteration) stays bounded within two. Here’s how it’s mathematically represented (don’t skip this, it’s easy math): f c (z) = z 2 + c, f c (z)<= 2 Let’s take an example here to understand this.   If c=-2,   f -2 (0) = 0 2 + (-2) = -2 (You start with z=0 as default) f -2 (-2) = (-2) 2 + (-2) = 4+ (-2) = 2 (You take the result of the previous function and input it as z in the next iteration) f- 2 (2) = (2) 2 + (-2) = 4+ (-2) = 2 We know that the result of the previous function (2), upon another itera...