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Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The mathematics of Design Theory traces its roots back to antiquity, with Latin squares appearing in medieval Islamic manuscripts and in the 17th and 18th century works of Choi Seok-jeong, Jacques Ozanam and Leonhard Euler. The subject is known for its longstanding open problems, many of which took over a century to resolve: Euler's 36 Officers Problem was posed in the 18th century and solved by Tarry in 1900; the Existence Problem for Combinatorial Designs was posed in the 19th century and solved for large orders by Keevash in 2014. The impetus for this workshop comes from the dramatic advances that have been made in Design Theory during the last decade since this breakthrough. This period has seen an explosion of solutions for large orders to high profile classical problems in Design Theory, based on the key new idea that probabilistic arguments, often in conjunction with algebraic constructions, can yield powerful new approaches for finding these solutions for large orders. The time is therefore ripe for bringing together experts in both the new and classical methods of Design Theory to share ideas, make new collaborations, solve problems and identify important directions for future research.