September 23
Russell Miller, CUNY
Interpreting a field in its Heisenberg group

The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1's along the diagonal and 0's below it. This group is obviously interpretable (indeed definable) in the field F. Mal'cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal'cev was unable to produce an interpretation without parameters.

After introducing the notions of a computable functor and an effective interpretation, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal'cev's result actually gives a definition of F in G(F), there is no parameter-free definition of F there.

This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.