The present research project concerns modern Galois Theory, more specifically birational Anabelian Geometry. The main problem I will be investigating is trying to recover the birational class of a variety (over an algebraically closed base field) from the pro-l Galois theory of its function field. Related to this, there are several other questions, like the one by Ihara/Oda-Matsumoto concerning a geometric/combinatoric description of the Galois group of the field of rational numbers. The above question is also related to describing rational points of varieties via the so called Section Conjecture. This kind of questions were initiated -in the arithmetic situation- in some remarkable (unpublished) manuscripts by Grothendieck. But it appears that in higher dimensions, one has such "anabelian phenomena" even in the total absence of arithmetic, i.e., over an algebraically closed base field.
The Galois Theory makes a bridge between two very different mathematical aspects, namely some basic algebraic objects, like fields, or more general spaces (varieties) on the one side, and the way one solves algebraic equations, or constructs covers of the spaces in discussion, on the other side. Now the (birational) Anabelian Geometry asserts that in the case the field one works over is "primitive enough", respectively the geometry of the space one constructs covers of, is "complicated enough", the totality 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 completely new perspective in approaching some very fundamental mathematical questions. The present research project addresses some of the basic problems in this mathematical field of research.
