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The Algebraic Combinatorics Seminar on Quasigroups, manifolds, and the completion of partial Latin hypercubes.

Speaker: Charlotte Aten, University of Denver

In my recent work with Semin Yoo, we produced a generalization of a
construction of Herman and Pakianathan which assigns to each finite
noncommutative group a closed surface in a functorial manner. While Herman
and Pakianathan built \(2\)-manifolds from groups, we build \(n\)-manifolds
from \(n\)-quasigroups, the \(n\)-ary analogue of quasigroups. I will give
examples of these constructions, show some pictures, and indicate how all
homeomorphism classes of triangulable orientable manifolds arise from this
construction. We will see that a natural question about building compact
manifolds is equivalent to a completion problem for Latin hypercubes, the
\(n=2\) case of which is the Evans Conjecture.

Local pseudo-randomness and discrete curvature
Speaker: Paul Horn, University of Denver

Eigenvalues and eigenvectors of matrices associated with graphs give a rich
window into their structure.  Among the many important properties that
eigenvalues certify is how the `random like’ the distribution of the edges
is.  In recent years, inspired my analogies between graphs and Riemannian
manifolds – highlighted by the relationship between the Laplacian matrix of
a graph, and the Laplace-Beltrami operator of manifolds – there has been
significant effort into defining notions of curvature for graphs.  Graph
curvature gives a `local’ way of understanding `global’ graph properties,
and many of these properties are similar to those understandable via
eigenvalues.  In this talk, I’ll discuss some of these notions of
curvature, and show how they certify something about how randomly
neighborhoods of vertices are stitched together.  I’ll also try to briefly
mention some related questions and results on strongly regular graphs, so
hopefully the talk won’t be completely out of place in an algebraic
combinatorics seminar.