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Colloquium, fall 2005

Day: Thursday, November 17th Time: 4:00pm Place: BH 401 Title: Discrete Point X-Rays Speaker: Richard Gardner Abstract: Around 1994, Larry Shepp introduced the term discrete tomography. Here the focus is on determining finite subsets of the n-dimensional integer lattice Z^n by means of their discrete parallel X-rays.  A discrete parallel X-ray of a finite subset F of Z^n in the direction of a vector v in Z^n gives the number of points in F lying on each line parallel to v.  The points in F can model the atoms in a crystal, and there is a genuine application of discrete tomography in high resolution transmission electron microscopy.   This talk will briefly survey some basic results in discrete tomography, and then introduce the discrete point X-ray of a finite subset F of Z^n at a point p in Z^n, which gives the number of points in F lying on each line through p.  It turns out that the theory of discrete point X-rays is more difficult than the parallel case, requiring some old and some very new results in number theory. The work was carried out jointly with Paolo Dulio of the Politecnico di Milano and Carla Peri of the Universit\`{a} Cattolica in Milan.   Almost everything will be illustrated by pictures!