Mathematics 165

Combinatorics and Designs (127797)

Noam D. Elkies

2025 Spring (4 Credits)

Schedule: TR 1200 PM - 0115 PM

Instructor Permissions: None

Enrollment Cap: n/a

In combinatorics and elsewhere one often encounters a “design”,or a collection of subsets of some finite set S whose elements areevenly distributed in a suitable sense; for instance the collection ofedges of a regular graph (each of whose vertices is contained in the samenumber of edges) or the collection of lines of a finite projective plane(any two of whose points are contained in a unique line).Of particular interest are designs symmetric under a large group ofpermutations of S. The consideration of specific classical designs andtheir symmetries will lead us to the general study of designs andpermutation groups. We conclude with the construction and detailedanalysis of the remarkable designs associated with Mathieu’s sporadicgroups of permutations of 12- and 24- element sets.

Recommended Prep:

The ability to write proofs and some knowledge of linear algebra will be needed.

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