One can attempt to study a (pointed) space X by probing it with (pointed) maps from other (pointed) spaces Y. It turns out that probing with maps from Y = S^n provides a lot of information about X. In particular, if we relax "maps" to "maps up to homotopy", the collection of such maps form the nth homotopy group of X. These are strong invariants in the sense that a map between connected CW-complexes is an equivalence if it induces isomorphisms on nth homotopy groups for each n. However, when we pass to "maps up to homotopy" we are destroying interesting information about maps between Y and X. In this talk we will study the space of maps from spheres S^n to "nice" spaces X, how this relates to the study of homotopy groups and homology groups of X, and, time permitting, how the study of such spaces naturally lead to the first theorem in stable homotopy theory, the Freudenthal suspension theorem.
Zoom Link: https://washington.zoom.us/j/92849568892