Sample Abstract
The Brouwer Fixed Point Theorem is a result in
topology that has some startling and entertaining
consequences. For instance, it implies that if you take
a map of Maryland and spread it out on your desk, there
is a point on the map that lies directly over the part
of Maryland it represents. The speaker will attempt to
demonstrate a (rather implausible) method of applying
the theorem to convert crumpled paper into cash.
To follow this talk, the listener should have some notion of what it means for a function to be continuous.
Description
The talk is presented as a wager. I take a sheet of paper,
set it on a desktop, mark its position, and have an audience member
crumple the sheet and place it within the marks. I then propose
the following wager: I will bet any member of the audience that
there is a point on the page that lies directly over its pre-crumple
position. For the remainder of the talk, I convince the audience
that I may collect on this bet.
This talk was inspired by a lecture given by Francis Su (creator of the Math Fun Facts web page) to the Harvard Graduate Student Seminar. His subject was applications of Sperner's Lemma, one of which is a straightforward proof of the Brouwer Fixed Point Theorem.
Level
Undergraduate.
While all of the continuity arguments are supplemented with large doses of intuition, a listener who has not at least seen continuity of real, one-variable functions before will struggle with this talk. Knowledge of the epsilon-delta definition of continuity is useful but not necessary. There is also some uniform continuity hand-waving towards the end of the talk, my assumption being that much of the audience will not understand uniform continuity.
Mechanics
I need a desktop (though the floor might do)
and an overhead projector. Ideally, I should have a
co-conspirator to take the sucker bet.
Appearances
Transparencies for Brown SUMS Talk (HTML)