Abstract: Steenrod operations, which were introduced by Norman Steenrod in 1947, remain one of the most formidable tools in homotopy theory even today. The eighties witnessed the development of equivariant homotopy theory, which is a theory sensitive to symmetries of shapes. However, equivariant analog of Steenrod operations are practically unknown beyond the group of order 2.

In this talk, I will introduce an abstract theory of eulerian sequences which readily extends to equivariant homotopy theory. Our main result says that corresponding to every eulerian sequence there is a "Steenrod operation". This talk will discuss how Eulerian sequences recover all known Steenrod operations, as well produce new equivariant Steenrod operations for all finite groups. This work is joint with A. Waugh, F. Zou, and M. Zeng.