"Steenrod homotopy" is synonymous with "strong shape". Polish (=separable metrizable complete) uniform spaces seem to be a reasonable domain of study on their own, though one might be able to generalize relevant constructions mentioned below to arbitrary complete uniform spaces should there arise such a need.
In 1950s, J.R. Isbell introduced finite dimensional uniform polyhedra, which he showed are uniform ANRs, and proved that every residually finite-dimensional (that is, admitting arbitrarily fine uniform coverings with finite dimensional nerves) Polish uniform space is the limit of an inverse sequence of finite dimensional uniform polyhedra. Based on his work, Segal, Spiez and Günther constructed a semi-uniform Steenrod homotopy category of resudually finite-dimensional Polish uniform spaces in 1994, whose restriction to finite-dimensional uniform polyhedra is the semi-uniform homotopy category. (Semi-uniform homotopy is a possibly non-uniformly-continuous homotopy through uniformly continuous maps.) A Steenrod homotopy category of residually finite dimensional Polish uniform spaces (whose restriction to finite dimensional uniform polyhedra is the usual, non-uniform, homotopy category) was constructed in a 2009 paper by the speaker, where it was applied to give simple proofs of the famous Krasinkiewicz-Minc and Krasinkiewicz-Geoghegan theorems in 0-dimensional Steenrod homotopy of compacta, and to prove that a continuum is Steenrod connected (=pointed 1-movable) if and only if every its uniform covering space has only countably many uniform connected components.
In this talk, I would like to present an extension of the Steenrod homotopy category to all (i.e., not necessarily residually finite dimensional) Polish uniform spaces and to discuss the application of this extension to getting a geometric grasp of functional spaces (such as iterated loop spaces of compacta) and classifying spaces of topological groups (such as the group of p-adic integers). In slightly more specific terms, the latter yields a substantial progress on the Hilbert-Smith conjecture (which I announced at the CAT'09 conference in Warszawa last summer), whereas the former should apply to prove (hopefully not only in the UV_1 case) a higher-dimensional generalization of the aforementioned Krasinkiewicz-Minc theorem.
The construction of the infinite-dimensional Steenrod homotopy is based on a solution to Isbell's "Research Problem B_2" (informally stated) that I see as asking to define infinite-dimensional uniform polyhedra in such a way that they would be uniform ANRs, and every Polish uniform space would be the limit of an inverse sequence of uniform polyhedra. This definition of infinite dimensional uniform polyhedra in turn involves a rethink of much of PL topology, which will have to be the main focus of the talk.