Abstract: Hecke operators were introduced by Erich Hecke in the 1930s.
They can be thought of as multivalued functions on the space of
lattices in the plane. They consequently act on certain functions on
this space — modular forms. In this talk I will give an accessible
introduction to Hecke operators and explain how they also act on the
free abelian group generated by the conjugacy classes in the modular
group SL_2(Z). Surprisingly, this leads to a non-commutative
generalization of the Hecke algebra. If there is sufficient time, I
will discuss how the Hecke action on loops leads to a Hecke action on
those “iterated Shimura integrals” that are constant on conjugacy
classes of SL_2(Z) and explain the hope that this Hecke action will
aid in the computation of their periods.