The AKE Philosophy: from Arithmetic Geometry to Classification Theory – AKE PACT

Our main goal is to develop new methods in the model theory of henselian valued fields to answer questions stemming from both Classification Theory and Arithmetic Geometry. The guiding philosophy binding these applications together is the crucial role played by Ax--Kochen--Ershov-type theorems. We plan to mix tools from the algebraic side, including studying fields via Galois theory and field topologies, together with model-theoretic ingredients like embedding lemmas and the study of valued fields via their nonstandard methods.

We have two main objectives in mind in each of the areas of classification theory and arithmetic geometry, but there is ample scope for the research to grow broader over time. In our work on classification theory, the main motivation is the Shelah Conjecture which predicts that every infinite NIP field is either separably closed, real closed or admits a nontrivial henselian valuation. The finite-dimensional cases of this conjecture were shown in a series of spectacular papers by Johnson in 2020 in which he introduced a wealth of machinery. Combining this with our new approaches, we aim to prove the Henselianity Conjecture in characteristic 0 and develop characterization of inp-minimal fields. On the arithmetic-geometric side, we plan to axiomatize (existential) theories of henselian valued fields of positive characteristic, pushing the knowledge boundary both for perfect and imperfect fields. Moreover, we will prove a relative version of Hilbert's 10th problem in henselian valued fields of mixed characteristic. In a fifth objective, we aim to broaden our expertise and apply our recent generalization of the famous Ax-Kochen-Ershov Theorem to incorporate imperfect residue fields into the theory of motivic integration for complete discretely valued fields in mixed characteristic.