MODNET
Research Training Network in Model Theory
 Research > Task II: Model theory of fields and applications Task II: Model theory of fields and applications Description This topic is at the heart of our project, and is the one where the connections between model theory and other branches of mathematics are most striking. The use of techniques coming from geometric stability has led to several applications outside of model theory. While most of the questions listed below have a model-theoretic inspiration, their solution will mostly involve proving algebraic results. The topic can roughly be divided into two parts: Model-theoretic study of various fields, in particular with operators such as derivations, automorphisms, the λ-functions of separably closed fields, pseudo-analytic functions; Study of various objects from algebraic geometry and uniformity results. List of specific problems     II.1: Model theory of fields and applications to algebraic geometry a) Investigate groups definable in SCF: in particular, whether definable simplicity implies simplicity b) Develop model theory (e.g., find model companion, prove simplicity, describe imaginaries, definable groups, minimal types, nonmodular types, induced structure on definable sets, etc.) of fields expanded by one or several operators, e.g.: algebraically closed fields with (commuting) derivations and/or automorphisms, separably closed fields with stacks of Hasse derivations, derivations of the Frobenius, non-standardFrobenius. c) Extend model theory of Buium's D-varietiesto difference varieties (ACFA) and Hasse D-varieties (SCF). d) Investigate D-groups in differential Galois theory (non-linear case), making the connection with the Malgrange construction of the Galois groupoid of a foliation. Calculate the groups for classical equations (eg Painlevé equations). Sharpen Hrushovski's effective determination of the Galois group of a linear differential equation (improve the bound on the order of tensor powers). Similar questions for difference Galois groups (e.g., of q-difference equations). Connections with the difference Galois theory developped by Singer and Van der Put. e) Apply model theory to the study of  special points  of abelian and Shimura varieties, in the style of the Manin-Mumford theorem and André-Oort Conjecture. f) Continue the study of  difference algebraic geometry : difference schemes, varieties, blow-ups, (co-)homology groups etc.     II.2: PAC fields and finite fields a) Prove Zilber trichotomy for reducts of pseudofinite fields, if necessary assuming omega-categoricity. b) Describe imaginaries in PAC fields, and use this to study groups interpretable in omega-free PAC fields.     II.3: Fields with Pseudo-exponentiation, and analytic expansions of the complex numbers. a) Prove pseudo-exponentiation gives an analytic Zariski structure; likewise expansions by power functions. Find examples from quantum tori and non-commutative geometry. Examples from Hrushovski amalgamation. b) Refute Zilber s conjecture that complex pseudo-exponentiation is exponentiation     II.4: Motivic integration Develop a theory of motivic integration with parameters and a sensible theory of constructible motivic functions     II.5: Weil Cohomology a) Examine non-standard Weil cohomologies. b)  Connections of non-standard Frobenius with conjectured semisimplicity of action of Frobenius on Weil cohomology. c) Determine in which cases the p-th cohomological dimension of a field is an elementary invariant of the field. Progress Year 1, Year 2, Year 3, Year 4.

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