Title:  Partial Difference Sets, Maximal Caps, and the Game of SET: How My Research is All Fun and Games After All

Abstract: A partial difference set (PDS) is a subset of a group with some very elegant combinatorial properties such that they can be used to construct strongly regular graphs and, in some cases, optimal error-correcting codes.  I have sought new constructions of PDSs for about 30 years now and still feel like it is a fun game. Meanwhile, maximal caps in affine space are sets of largest size that don’t contain an entire line. The Game of SET is a popular card game with deep mathematical connections.  About 15 years ago, I developed a PDS construction that included an example in an elementary abelian group of order 81.  About that same time, another researcher was showing that the nonzero points of affine space in AG(4,3) could be partitioned into 4 maximal caps.  Also around that time, I was getting consistently defeated in the game of SET by my 12-year old daughter and asking her mathematical questions related to the game. While attending a talk about two years ago, I realized that all of these things – the PDS constructions, maximal caps in AG(4,3), and the answer to one of the mathematical SET questions – were all the same.  Not just related, they were the same. In this talk, we will look at these correspondences and also consider some possible avenues for future exploration.