The dream is to define such a hierarchy of invariants for generic vector fields such that, whenever all the invariants of order less than or equal to k have zero value for a given field and there exists a nonzero invariant of order k+1 , this nonzero invariant provides a lower bound for the field energy." >From Arnol'd and Khesin, "TOPOLOGICAL METHODS IN HYDRODYNAMICS" Springer (1998), page 176. The first invariant in Arnol'd's hierarchy is the "helicity" of a vector field, which measures the extent to which the field lines wrap and coil around one another. It is related to the linking number of two closed curves, and when nonzero provides a lower bound for the field energy. It is expected that the second invariant in the hierarchy should be related to Milnor's mu-invariant for 3-component links. Most attempts to define this "second order helicity" use the Massey triple product interpretation of Milnor's mu-invariant, but apply successfully only to highly degenerate fields, such as those concentrated in toroidal tubes. Our goal here is to interpret Milnor's mu-invariant for a 3-component link as an "ambiguous Hopf invariant" of a related map of the 3-torus to the 2-sphere, and then in the case that the three pairwise linking numbers are zero and the ambiguity disappears, to express it as an integral. We do not yet know whether this will successfully lead to a second invariant of vector fields in Arnol'd's hierarchy. In the first talk on Tuesday March 25, Herman Gluck will provide background information, and give an overview of the proofs of the three main theorems. The following week, Paul Melvin will discuss Milnor's mu-invariant for 3-component links. In subsequent talks, Rafal Komendarczyk will discuss, among other things, an approach to the first main theorem via framed cobordism of framed links, and Clay Shonkwiler will give a proof of it via groups of string links. Then Dennis DeTurck will give proofs of the remaining theorems, and Shea Vela-Vick will give an overview of the competition, discussing Massey triple products and the resulting integrals.