2007-2008: Outstanding Scholarly
Achievement Award

AWARD CEREMONY
Thursday, May 15, 2008
4-5 p.m. Lecture, Orchard Suite II & III
5-6 p.m. Reception, Lobby Suite

Generalizations of the Brouwer Fixed Point Theorem
Dr. Marcus March
Professor, Mathematics/
Statistics Department

Although almost 100 years old, new uses of L.E.J. Brouwer’s fixed point theorem are still being discovered.  His theorem is of interest and utility to mathematicians, physical scientists, and social scientists alike.  In this presentation, I will discuss Brouwer’s theorem and my two generalizations of his result.  We will begin with some basic examples of spaces with and without the fixed point property and proceed to more complicated abstract examples.  All examples will be discussed from a visual perspective.

A transformation  f  on a space  X  has a fixed point if  f  leaves some point  x  fixed.  That is, for some  x in  X ,  f(x)=x.  A space  X  has the fixed point property if every continuous transformation on  X  has a fixed point.  Brouwer’s theorem states that each  n-dimensional “cube” has the fixed point property.  A  1-dimensional cube is the interval  [0,1]  on the number line.  A  2-dimensional cube is a square plus its interior.  A  3-dimensional cube is a box plus its interior.  For  n > 3, an  n-dimensional cube is harder to visualize, but is, nonetheless, a higher dimensional version of the  3-cube.  The  n-dimensional version of Brouwer’s theorem was first proved in 1910.  In applications, fixed point results are primarily used to establish existence theorems.  A few examples of existence theorems that follow from Brouwer’s result are

·  existence of solutions of ordinary differential equations

·  existence of Nash equilibria in economics

·  existence of equilibria in game theory

·  existence of best approximations in optimization theory

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