Anabelian Geometry and Elementary Equivalence of Fields

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ABSTRACT for Award DMS-0401056 of PopThe present research project concerns modern Galois Theory,more specifically birational Anabelian Geometry. The mainproblem I will be investigating is trying to recover thebirational class of a variety (over an algebraically closedbase field) from the pro-l Galois theory of its function field.Related to this, there are several other questions, like theone by Ihara/Oda-Matsumoto concerning a geometric/combinatoricdescription of the Galois group of the field of rational numbers.The above question is also related to describing rational pointsof varieties via the so called Section Conjecture. This kind ofquestions were initiated -in the arithmetic situation- in someremarkable (unpublished) manuscripts by Grothendieck. But itappears that in higher dimensions, one has such "anabelianphenomena" even in the total absence of arithmetic, i.e., overan algebraically closed base field.The Galois Theory makes a bridge between two very differentmathematical aspects, namely some basic algebraic objects, likefields, or more general spaces (varieties) on the one side,and the way one solves algebraic equations, or constructscovers of the spaces in discussion, on the other side. Now the(birational) Anabelian Geometry asserts that in the case the fieldone works over is "primitive enough", respectively the geometryof the space one constructs covers of, is "complicated enough", thetotality of the "recipes" of solving all the algebraic equations,respectively of constructing all the covers, encodes the field,respectively the space under discussion. This opens a completelynew perspective in approaching some very fundamental mathematicalquestions. The present research project addresses some of the basicproblems in this mathematical field of research.