Combinatorics Seminar

When: Sunday, Jan. 16, 10am Where: Schreiber 309 Speaker: Wafiq Hibi, University of Haifa Title: The Beckman-Quarles theorem for rational spaces

Abstract:

Let R^d and Q^d denote the real and the rational d-dimensional spaces, respectively. For a real number b, a mapping f:X --> X, where X is either R^d or Q^d, is called b-distance preserving if dist(x,y)=b implies dist(f(x),f(y))=b for all x,y in X.

The Beckman-Quarles theorem states that every unit distance preserving mapping f:R^d --> R^d is an isometry, provided d is at least 2. A few papers have been written about rational analogues of this theorem, showing that for some values of d the property: "every unit distance preserving mapping f:Q^d --> Q^d, is an isometry" holds as well. I will discuss the following recent result: For every d satisfying d => 5, every unit-distance preserving mapping of Q^d to Q^d is an isometry.