Title: Around tame fields: Fq((Q)), eliminations, transfer, and beyond
Abstract: I will report on recent advances in and around the theory of
tame valued fields. Beginning with the relevant Ax-Kochen-Ershov
principles (now in arbitrary imperfection degree), I will explain recent
joint work with Boissonneau in which we prove elimination results in
theories of separably tame fields with finite residue fields, showing
that each formula is equivalent (modulo such theories) to an "E-1"
formula, the precise forms of which I will describe. These results,
though not the methods, are related (and partially generalize) to work
of Lisinski who deduced the decidability of (F_q((Q)),t) -- among others
-- from an argument of Kedlaya; I will describe that part of the story
as well. Finally, I will describe related work of Soto Moreno, who has
refined the NIP transfer arguments (of Delon, Bélair, Jahnke--Simon, and
others) in the more specialised setting of (separably) algebraically
maximal fields.