How do I modify the alt+[key] special character mappings? So pressing alt+[any key] gives you a special character. I am wondering if there is a way to change what special character is outputted for every alt+[key] keystroke. For example, I would like to map alt+t to the theta symbol, map alt+d to the degree symbol, and map alt+s to the square root symbol.
The simplest way to do what you need is probably to just set up text replacements in: System Preferences > Keyboard > Text > Replace With

If you really want to change key mappings, you can use Ukelele, or possibly Karabiner.