Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-i products; a family of coherent homotopies derived from the broken symmetry of Alexander-Whitney's chain approximation to the diagonal. Later, following a viewpoint developed by Adem, Steenrod defined his homonymous operations for all primes using the homology of symmetric groups. This viewpoint enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at 2. In recent years, thanks to the development of new applications of cohomology -- most notably in Applied Topology and Quantum Field Theory -- having a definition of Steenrod operations that can be effectively computed in specific examples has become a key issue. In this talk, I will review Steenrod's definition of the operations and describe an effective construction of them at every prime.

Let k be a field. Totaro studied the de Rham cohomology of algebraic stacks over k, and computed it for classifying stacks of linear algebraic k-groups in many cases. Combining previous work of Drury, May and Epstein, I define and study Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field k of characteristic p>0. These operations share many properties with their topological analogues, but there are also important differences. I then determine the Steenrod operations on the de Rham cohomology of linear algebraic k-groups computed by Totaro.