Project objectives:  The research objectives of the project concern problems in four closely interrelated themes
currently forming the main research topics of the different teams: the inverse Galois problem, geometric Galois
theory, number theory and differential Galois theory.  The main aim of the project is to improve current
approaches to several related open problems in these themes, via the integration of explicit methods.  Some of
the precise research objectives are listed below. 

(1) In the inverse Galois problem: explicit realisation of all unitary groups over prime fields as Galois groups,
      completing the realisation of the full set of classical finite groups; also, realisation of new finite groups as
      Galois groups via geometric methods (braid groups).

(2) In geometric Galois theory: determination of the Grothendieck-Teichmüller group GT, application
      to proving specific results on the absolute Galois group Gal(Q) of the rationals and determination of
      the precise relation between GT and Gal(Q); also, combinatorial determination of the Galois action
      on dessins d'enfants; also, study of the finite quotients of fundamental groups of curves in characteristic p;
      also, computations in GT aiming at the Shafarevitch conjecture on Gal(Q).

(3) In number theory: development of effective computational methods in class field theory; also,
      fast methods for decomposition of integers into prime factors; also, algorithms for computing
      with elliptic curves, and Fermat-like problems (finding integral solutions for Diophantine equations).

(4) In differential Galois theory: effective computational methods for general families of equations; also,
     development of differential Galois theory over p-adic fields and finite fields; also, the differential inverse        
     Galois problem (asking whether any linear algebraic group is a differential Galois group).