Will Johnson (Fudan University)

Around definable types in valued fields

Note time change: 11am

Haskell, Hrushovski, and Macpherson showed that the theory ACVF of algebraically closed valued fields has elimination of imaginaries after adding the so-called "geometric sorts" to the language. The same result holds in $p$-adically closed fields ($p$CF) by work of Hrushovski, Martin, and Rideau. In the case of ACVF, one way to prove this is to encode imaginaries using definable types, and then encode definable types in the geometric sorts. While $p$CF does not have "enough" definable types to encode imaginaries, the encoding of definable types carries over. Surprisingly, the geometric sorts are unnecessary: any definable type in $p$CF has a code in the home sort (the field sort). This fact and its proof have some unexpected applications to definable groups and definable topological spaces in $p$CF. For example, certain quotient groups are definable rather than interpretable, and there is a unified notion of "definable compactness" for definable topological spaces. Parts of this talk are joint work with Pablo Andújar Guerrero.