12/07/2012, 16:30 — 17:30 — Room P4.35, Mathematics Building Tibor Beke, University of Massachusetts
The sign pattern theorem and Brouwer's fixed point theorem
This work grew out of my attempt at concocting a proof of the
Brouwer fixed point theorem that is suitable for a first course in
topology. It should not involve algebraic topology and special
tricks like the no-retraction theorem, and should make the
statement itself plausible. Already in dimension two, Brouwer's
fixed point theorem is quite surprising and (visually) not very
compelling --- a contrast to the one-dimensional case where the
statement is equivalent to the intermediate value theorem that is
visually "obvious". We present a proof the Brouwer fixed point
theorem as a higher-dimensional generalization of the intermediate
value theorem. The proof itself is purely combinatorial and reduces
to the "sign pattern theorem" about (higher dimensional) matrices
containing two types of symbols, + and -. This talk should be
suitable (and hopefully, ideal) for undergraduate students.
