Speaker: Amin Bahmanian (Illinois State University)
Title. High-dimensional Combinatorics
Abstract. In light of recent interest in higher-dimensional analogues of various combinatorial objects such as permutations, Latin squares, Hadamard matrices, association schemes, and symmetric block designs, we consider the following problem. Given a triple of non-negative integers $(v,k,\lambda)$, can we fill the $v^3$ cells of a $v\times v\times v$ cube with $\{0,1\}$ in such a way that each layer parallel to each face contains $k$ ones, and that for every two parallel layers there are exactly $\lambda$ positions where they have matching ones? We construct infinite families of these objects using difference sets, symmetric designs, doubly regular tournaments, Hadamard matrices, Latin cubes, and association schemes on triples.
This is joint work with Vedran Kr\v{c}adinac, Lucija Reli\'{c}, and Sho Suda.