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The talk will begin with some background on different kinds of for- mulas and what it means to define, or to interpret a structure A in a structure B. An “effective” interpretation allows us to effectively recover a copy of A from an arbitrary copy of B. Montalb ́an defined a very general notion of effec- tive interpretation, in which the elements of A are represented by tuples from B of arbitrary arity, and the formulas that define the interpretation also have no arity. R. Miller introduced a notion of “computable functor.” Harrison-Trainor, Melnikov, Miller, and Montalb ́an showed that the existence of an effective in- terpretation is equivalent to the existence of a computable functor. Maltsev showed that a field can be defined in its Heisenberg group, using existential formulas with a pair of parameters. There is no definition without parameters. I will describe joint work with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Russell Miller, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar, in which showed that there are uniform formulas (the same for all fields) that provide an effective interpretation of the field in its Heisenberg group. Our first proof involved finding a computable functor and applying the result of Harrison-Trainor, Melnikov, Miller, and Montalb ́an. We then managed to find an explicit interpretation, using finitary existential formulas.
Interpreting A Field In Its Heisenberg Group