The Lusternik-Schnirelmann category of a topological space $X$ is a measure of the minimum number of open, contractible sets needed to cover $X$. Calculating this invariant for Lie groups is an open problem going back to Ganea in 1971 and little progress has been made since that time. In this talk, we show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group $G$ is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in $G$ corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of $G$. This is joint work with M. Hunziker.
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