Title: Configuration Spaces of Graphs

Abstract: The i-th rational cohomology of the n-th ordered configuration space of a (nice enough) topological manifold (for any fixed i and n) satisfies representation stability, a concept introduced by Benson Farb and Thomas Church which generalizes homological stability. This implies, for example, that given any such manifold we can calculate this cohomology for all n >> 0 simultaneously by a finite calculation. Furthermore, the dimension of this i-th cohomology is eventually a polynomial in n, the number of particles.

If we instead of manifolds look at graphs, the situation is more complicated: the dimension of the corresponding cohomology for a graph grows much faster than polynomially in some degrees, even in the simplest cases. To investigate this cohomology, it is useful to construct a deformation retraction of these configuration spaces with a CW structure. In this talk, we will do this explicitly for any locally finite graph. This allows us to compute the cohomology explicitly in a few cases, but the general case is still unknown. If the graph we are considering is finite, then this CW complex will also be finite, which allowed us to use it for calculations with a computer; if there is time we will present some of them.