Title: An Introduction to Configuration Spaces and some of their Applications

Abstract:

We discuss some foundational results on the topology and geometry of configuration spaces of ordered and unordered pairwise distinct points on a topological space, most particularly on a manifold. We illustrate some of the nicest ideas in the theory with basic examples and homology computations. Various classical applications will be reviewed. Finally if time permits, we explain how configuration spaces and labeled analogs can be used to model mapping spaces of continuous and rational maps. This leads to effective homological computations for both based and free loop spaces.



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Title: Homological Splitting Techniques in Algebraic Topology

Abstract:

A space is said to split stably if after ”suspension” it breaks down into the one-point union of simpler spaces. It is said to split homologically if it has the homology of such a union. Splitting a space X can be greatly useful in getting information about X or in comparing it to other spaces. In this talk we describe general methods to split filtered spaces stably and homologically in terms of the pieces of the filtration. This expands on works of Stylian Zanos and Fred Cohen. Examples and applications to geometry will be presented.



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