Combinatorics Seminar

When: Sunday, May 21, 10am

Where: Schreiber 309

Speaker: Peter Horak, University of Washington, Tacoma

Title: Tiling Z^n by Translates


We will focus on tiling of Z^n by translates of a finite set. Although this is a very special type of a tiling, it provides the simplest known counterexample to part (b) of the 18th problem of Hilbert: If congruent copies of a polyhedron P tile R^3 is there a group of motions so that copies of P under this group tile R^3 ? We will show that a polynomial method, Fourier analysis and algebraic geometry provide powerful tools in this area. Applications of these methods to various problems and conjectures, Golomb-Welch 1969, Lagarias-Wang 1996, will be discussed. For example, the approach will be illustrated by a short proof of the fact that each tiling of Z^n by a set of a prime size has to be periodic.

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