Circles of Apollonius... and magnetism!
These two concepts go together in a very natural way. Prepare for a breathtaking real-world application of Euclidean geometry!
Suppose you have two identical long wires side by side, parallel to each other and connected to each other at one end, and a current is flowing through one wire and back through the other in the other direction. As in this video, each wire generates a magnetic field, and the magnetic field forces the two wires towards each other. The question is, just before the wires start to move towards each other, what does the magnetic field look like?
Characters of the symmetric group
Last week, I sat down to compute the character table of $S_5$ for the first time in several years.
My first instinct, following what I learned five years ago, was to use the standard tricks for computing character tables: compute the characters of some easily constructed representations, use the fact that the sum of the squares of the dimensions of the irreducible characters is $|S_5|=120$, and use the orthogonality of the rows and columns of the character table to finish it off.
This, however, is rather tedious for a group as large as $S_5$. But I had recently learned that the irreducible representations of the symmetric group are completely classified, and can be constructed using group actions on standard Young tableaux. Was there a way to use this theory to compute each entry of the character table directly?
Introduction
Welcome! The purpose of this blog is to record some of the particularly beautiful mathematical ideas I have seen or invented, and share them with you.
The process of doing mathematics is much like a quest to uncover mathematical truths. Sometimes, such a truth may be valid but uninteresting, just another pebble or grain of sand along the beach. But other times, you will uncover a gemstone - a particularly aesthetic, beautiful, or useful truth hiding in the vast sandpiles of information.
This blog is devoted to the gemstones of my mathematical investigations. Enjoy!