H-minimality

My goal, in this talk, is to explain a new notion of minimality for (characteristic zero) Henselian fields, which generalizes C-minimality, P-minimality and V-minimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, h-minimality, can be defined, analogously to other minimality notions, by asking that 1-types, over algebraically closed sets, are entirely determined by their reduct to some sublanguage – in that case the pure language of valued fields. However, contrary to what happens with other minimality notions, particular care has to be taken with regards to the parameters. In fact, we define a family of notions: l-h-min for l a natural number or omega. My second goal in this talk will be to explain the various geometric properties that follow form h-minimality, among which the well-known Jacobian property, but also higher degree and higher dimensional versions of that property.