Title: Lattice embeddings into the computably enumerable Turing degrees.

Abstract: This talk will attempt to explain in lattice-theoretic terms the status quo of
the problem of which finite lattices can be embedded into the computably
enumerable Turing degrees, concentrating on the lattice theory rather than the
computability-theoretic constructions.

In 2000, Lerman proved a characterization of the embeddable finite lattices
for join-semidistributive lattices, building on joint work with Lempp in the
mid-1990′s. (It is generally believed that once the join-semidistributive case
is solved, the full characterization is not too difficult.) Subsequent work by
Lempp, Lerman and Solomon produced only minor improvements and ended in a
stalemate on this problem which has been open for almost 50 years.

The problem with Lerman’s condition is that it is not known to be decidable
(and so in particular not expressible by first-order formula in the language
of lattices); rather, it requires the existence of finite sequences of
sequences of lattice elements subject to fairly delicate transition rules.

I will try to explain Lerman’s condition in detail, motivating it somewhat by
a hint of where each feature comes from in computability-theoretic terms.