I’ve recently become somewhat interested in smooth manifolds and surfaces as a result of preparation for various interviews, amongst other reasons. The concept of a surface is very intuitive, and is a concept that a student of mathematics is likely to encounter very early in their mathematical careers, though a reasonable definition of a surface takes more effort. The result of this formalisation is a remarkably elegant theory, which eventually leads to ideas of smooth manifolds – arguably one of the best sounding mathematical objects – and generalised calculus.
It took quite some time before I had a “big picture” of what a surface is, and how this fits into the grander theory. Undoubtedly I am missing some of the major pieces to this puzzle, but it does show some of the elegance of the theory (at least for me).
Continue reading “Off on a tangent”
Zombies are a staple of modern popular culture, and appear in a variety of forms, including the traditional slow-moving, unintelligent zombie hordes and less common fast-moving – and perhaps intelligent – zombies. The common theme in media featuring zombies is the the zombie infection, which may affect either the living or the deceased, or both. It is generally agreed that this infection is passed from zombie to non-zombie by means of a scratch or bite, and infection always leads to a transformation into a zombie. (In some more modern interpretations, this transformation may be reversible.)
The spread of the zombie infection is an interesting problem to model mathematically. There are many factors to consider: the chances of a non-zombie becoming infected in an interaction with a zombie; the rate at which interactions between non-zombies and zombies occur; the spread the zombie horde from place to place; and the “critical mass” of the zombie horde at which point there is insufficient food for all zombies. We can model parts of this problem in isolation, with some simplifying assumptions.
Continue reading “The Zombie Apocalypse”