# Topological complexity, Euclidean embeddings of $RP(n)$, and the cohomology of configuration spaces of pairs of distinct points in $RP(n)$

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Peter Landweber, Rutgers University
Fine Hall 314

This will be a report on joint work with Jesus Gonzalez, about topics related to topological complexity (TC), introduced by Michael Farber in 2003 as a numerical measure of the complexity of robot motion planning problems. TC of real projective space $RP(n)$ coincides with the Euclidean immersion dimension of $RP(n)$ for $n$ different from 1, 3 and 7. For symmetric TC of $RP(n)$, there is a close relation to the Euclidean embedding dimension of $RP(n)$. Our current study of symmetric TC involves configuration spaces of pairs of distinct points in $RP(n)$ and has led to a calculation of their integral cohomology groups.