Infinite Limits

 Hi! Welcome to this 3-part series on the mathematics behind quantitative finance! Took a while to return, but I'm done with IB exams now (well... since mid-May, really. But I've been on holiday or busy on and off since). Anyways, this part of the series will cover: Introduction to Quantitative Finance   Fundamental Mathematics in Quantitative Finance   Financial Instruments and Markets    Rest in Peace to  Jim Simons , whom this series is dedicated to. Jim Simons, founder of Renaissance Technologies and Father of Modern Quantitative Finance Introduction to Quantitative Finance Overview of Quantitative Finance Quantitative finance is a field that uses mathematical models, statistical techniques, and computational tools to understand and solve financial problems. It plays a crucial role in the financial industry by providing methods for pricing complex securities, managing risk, and optimizing portfolios. Quantitative finance integrates a...

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...

In the intricate world of fluid dynamics, where the movement of liquids and gases forms the backdrop of countless natural phenomena, the unsolved Clay Institute Millenium Prize Problem of the Navier-Stokes equations stands as a beacon of understanding. These mathematical expressions encapsulate the intricate forces, pressures, and velocities that orchestrate the motion of fluids. George Gabriel Stokes Image Credit: Clay Mathematics Institute Fluid dynamics is a captivating branch of physics and mathematics that delves into the behaviour of fluids in motion. Whether it's the ocean currents, the streams, or the turbulence in a boiling pot, understanding the principles governing fluid motion requires strong mathematical concepts and physics insights. The Navier-Stokes equations, named in honour of the French engineer Claude-Louis Navier and the Irish mathematician George Gabriel Stokes, are the fundamental equations that explain fluid dynamics. In their general form, they can be expre...