The goal of the talk is to present my own personal view of design theory. This is likely to match the view of no one else on this planet, but I suspect it overlaps many common approaches. Starting from Gaussian quadrature in 1826, I will view designs as ways to spoof an observer, as toys for adversarial tasks in pseudorandomness. For example, if I want to convince an opponent that I am generating $k$-element subsets of a $v$-set uniformly at random, and I know that all my opponent can do is choose a fixed set $T$ of $t$ points and compute $B \cap T$ for all the samples $B$ that I generate, I see that I need only choose a $t-(v, k, \lambda)$ design and generate blocks of the design uniformly at random instead. For instance, for $v = 24$, $k = 8$, $t = 5$, this reduces each 20 coin tosses to ten.

The talk briefly reviews Gaussian quadrature and spherical $t$-designs before embarking on a tour of designs in association schemes, emphasizing a few joint projects with Doug Stinson. From the familiar - block designs and orthogonal arrays, vector space designs and designs of linear transformations - to more exotic objects such as Room squares, $\lambda$-transitive sets of permutations, ordered orthogonal arrays, and designs in sporadic $Q$-polynomial association schemes, I will mention bounds, constructions, and open problems that pique my interest. If time permits, I will discuss a connection to polynomial ideals of zero-dimensional varieties.

Bio: Bill Martin is a Professor of Mathematical Sciences and Computer Science at Worcester Polytechnic Institute. He serves on the editorial boards of BICA and JCD.