Abstract: In this introductory talk we survey the most useful and fashionable geometric representations of graphs, mainly in the plane.
We give a brief account on planar graphs, that is, graphs which can be drawn in the plane without crossing edges. Then we try to estimate the number of unavoidable crossings in any embedding of a graph in the plane.
Next, we represent graphs with straight line segments as edges, and study how results from graph theory change under certain geometric constraints.
Finally, we investigate how economically a graph can be embedded in a square grid.
Abstract: Take a sheet of paper, fold it flat, and make one complete straight cut. What shapes can the unfolded pieces make? The earliest reference to this idea is a 1721 Japanese puzzle book, in which a Japanese crest is made by this method. Houdini used folding and cutting for a magic trick before he became a famous escape artist. This talk discusses our work on the fold-and-cut problem: given a collection of line segments (i.e., a plane graph), find a flat folding of the plane so that precisely those line segments are folded to a common line, along which we can cut to make the desired shapes.