MODAG (291111)

  https://cordis.europa.eu/project/id/291111

  FP7 (2007-2013)

  Model Theory and asymptotic geometry

  ERC Advanced Grant - Mathematical foundations (ERC-AG-PE1)

  mathematical logic  ·  arithmetics  ·  geometry  ·  combinatorics  ·  algebraic geometry

  2012-01-01 Start Date (YY-MM-DD)

  2017-09-30 End Date (YY-MM-DD)

  € 1,393,499 Total Cost

  Description

The principal methods of model theory, in its connections to algebraic geometry, have been quantifier elimination (e.g. Tarski's theorem) and structural stability (used here in a wide sense, including simplicity, NIP and structural o-minimality.) We propose to move beyond current limitations on both fronts, by means of three interrelated projects. (1) A study of definable sets in global fields. A successful quantifier elimination result in this setting would extend the reach of model theory to wide areas of number theory and geometry that have not been accessible before, including points of small height in number theory, and the Gromov-Witten invariants of a variety in geometry. (2) A study of limits of o-minimal metric structures as quotients of non-archimedean structures, extending similar measure and group-theoretic work that has led to a resolution of Pillay's conjectures in the o-minimal setting, and leading towards a model theory of Calabi-Yau degenerations. (3) Model theoretic asymptotic limits lead to measure and dimension theories, with associated dependence theories, that resemble known structures from stability theory but do not lie within the stable realm or its current extensions. Preliminary stability-theoretic considerations have already led to significant applications in combinatorics. We propose creating a structural stability theory based on pseudo-finite dimension, expected to create a long-term bridge between model theory and additive combinatorics.

  Complicit Organisations

1 Israeli organisation participates in MODAG.