A major open problem in the model theory of valued fields is to gain an understanding of the first-order theory of the power series field F((t)), where F denotes a finite field. For sufficiently "nice" henselian valued fields, the Ax-Kochen/Ershov philosophy allows to reduce questions of elementary equivalence and elementary embeddings to the analogous questions about the value group and residue field (or related structures). In my talk, I will present a new such principle which applies in particular to a large class of algebraic extensions of F((t)), albeit not to F((t)) itself. The talk is based on joint work with Konstantinos Kartas and Jonas van der Schaaf.

The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are often inter-definable with their open core. Structures such as M = (ℝ,<, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk, we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination (every definable set is constructible), and its definable functions are generically continuous.

I will discuss several interesting examples of classes of structures for which there is a sensible first-order theory of "almost all" structures in the class, for certain notions of "almost all". These examples include the classical theory of almost all finite graphs due to Glebskij-Kogan-Liogon'kij-Talanov and Fagin (and many more examples from finite model theory), as well as more recent examples from the model theory of infinite fields: the theory of almost all algebraic extensions and the universal/existential theory of almost all completions of a global field (both joint work with Arno Fehm). Interestingly, such asymptotic theories are sometimes quite well-behaved even when the base theories are not.