Final Report Summary - MODAG (Model Theory and asymptotic geometry)
Gödel's theorem shows the impossibility of axiomatizing all of mathematics; any attempt to do so will run into incompleteness phenomena. On the other hand, model theory is made meaningful to the foundations of mathematics in showing that some restricted, but rich and wide areas of mathematics can be axiomatized, and that this axiomatization can be useful both for a general understanding of the field, and for obtaining specific results within it. The earliest and best known example is that of real algebraic geometry, axiomatized by Tarski via the theory of real closed fields. In the decades since Tarski's theorem, the same has been achieved for a variety of other important geometries, notably over other completions of the rational numbers (local fields.)
However, number theory and algebraic geometry recognize a distinction between local and global phenomena. Solving a polynomial equation in integers implies a solutions in the real numbers and in each other completion (local solutions), but this is not enough; there is an additional deep structure that must be taken into account, governing the way the local data can be tied together. Similarly, a germ of a function near each point of a manifold, with prescribed poles, cannot always be glued together to a global solution; this leads to cohomological questions. No aspect of this global geometry was previously known to be amenable to axiomatization; only undecidability results were available.
This project has made considerable progress towards an axiomatic understanding of global geometry. A universal theory was shown to exist, reflecting global constraints; investigations of formulas in this language leads immediately to deep geometric questions that have not previously been encountered by logicians. It was proved that the theory axiomatizes all universal laws that govern algebraic curves on algebraic varieties in this language, i.e. formulas that hold universally in the field of algebraic functions.
Given a universal theory of this kind, one wishes to understand more general formulas, going beyond universal quantifiers. In technical language, the next step is to determine a model companion to the universal theory; in algebra, this corresponds to the move from the axiomatization of additional and multiplication in the theory of fields, to the fundamental theorem of algebra and the Hilbert Nullstellensatz. The existence of a model companion is known to depend on two questions: a certain finiteness principle (analogous to Noetherianity) and an amalgamation principle allowing two models to co-exist. On the former question, work is ongoing. Amalgamation has been fully understood. In another direction, stability has been proved for a wide class of formulas in this theory; opening the way to the use of some of the deeper tools of model theory in this setting.
Other achievements include the creation of structural model theoretic tools for valued fields, for dependent theories, for the model theory of asymptotic geometry and for relations with asymptotic finite combinatorics.
However, number theory and algebraic geometry recognize a distinction between local and global phenomena. Solving a polynomial equation in integers implies a solutions in the real numbers and in each other completion (local solutions), but this is not enough; there is an additional deep structure that must be taken into account, governing the way the local data can be tied together. Similarly, a germ of a function near each point of a manifold, with prescribed poles, cannot always be glued together to a global solution; this leads to cohomological questions. No aspect of this global geometry was previously known to be amenable to axiomatization; only undecidability results were available.
This project has made considerable progress towards an axiomatic understanding of global geometry. A universal theory was shown to exist, reflecting global constraints; investigations of formulas in this language leads immediately to deep geometric questions that have not previously been encountered by logicians. It was proved that the theory axiomatizes all universal laws that govern algebraic curves on algebraic varieties in this language, i.e. formulas that hold universally in the field of algebraic functions.
Given a universal theory of this kind, one wishes to understand more general formulas, going beyond universal quantifiers. In technical language, the next step is to determine a model companion to the universal theory; in algebra, this corresponds to the move from the axiomatization of additional and multiplication in the theory of fields, to the fundamental theorem of algebra and the Hilbert Nullstellensatz. The existence of a model companion is known to depend on two questions: a certain finiteness principle (analogous to Noetherianity) and an amalgamation principle allowing two models to co-exist. On the former question, work is ongoing. Amalgamation has been fully understood. In another direction, stability has been proved for a wide class of formulas in this theory; opening the way to the use of some of the deeper tools of model theory in this setting.
Other achievements include the creation of structural model theoretic tools for valued fields, for dependent theories, for the model theory of asymptotic geometry and for relations with asymptotic finite combinatorics.