Abstract: A semigroup S is called n-collapsing if, for every a_1,....., a_n in S, there exist functions f \neq g (depending on a_1,....., a_n), such that:

a_{f(1)} \cdots a_{f(n)} = a_{g(1)} \cdots a_{g(n)};

it is called collapsing if it is n-collapsing, for some n. More specifically, S is called n-rewritable if f and g can be taken to be permutations; S is called rewritable if it is n-rewritable for some n.

Semple and Shalev extended Zelmanov's solution of the restricted Burnside problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent.

In this talk, we consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we will see that the following conditions are equivalent, for all unital algebras A over an infinite field:

(1) The multiplicative semigroup of A is collapsing. (2) A satisfies a multiplicative semigroup identity. (3) A satisfies an Engel identity.

We deduce that, if the multiplicative semigroup of A is rewritable, then A must be commutative

Abstract: The Steenrod algebra of (stable) primary operations in mod p cohomology has a rich and fruitful history in homotopy theory, notably with the Adams spectral sequence. Secondary cohomology operations detect additional information not seen by primary operations. We will introduce secondary operations and discuss some of their properties. Then we will present sample calculations and applications from classical homotopy theory, such as the Peterson-Stein formulas and some homotopy groups of spheres.