Infinite Galois Theory in the Context of Hilbert's Tenth Problem

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At the International Congress of Mathematicians in 1900, David Hilbert presented a list of 23 problems he deemed the most important questions in mathematics for the next century. Hilbert's Tenth Problem asks for an algorithm that decides for any given Diophantine equation if it has a solution in the integers or not. In particular, this algorithm would also be able to decide for any such equation if it has a solution in the rationals. Hilbert's Tenth Problem remained open until 1970 when Yuri Matiyasevich finally proved that such an algorithm could not exist. The methods introduced in his paper, however, only apply in the case of integer solutions, leaving the problem over the rationals open. In his original paper, Matiyasevich claims that Hilbert, who believed in the existence of an algorithm, would thus prob-ably not be satisfied with the current solution that only applies to the integers. He also points out that, at the time of writing in 1970, "progress in [the rational] case has been rather meagre." In recent years, progress has been made towards solving Hilbert's Tenth Problem over the rationals, in particular by Professor Koenigsmann's research group. In my research, I want to carry on the progress made so far and employ new techniques towards attacking the problem. In particu-lar, I want to draw from recent results in the areas of valuation theory, model theory, and infinite Galois theory, which are new and promising areas when it comes to applying them to Hilbert's Tenth Problem. Hilbert's Tenth Problem can naturally be translated to the language of model theory, amounting to the question of whether the existential first-order theory of the rational numbers is decidable or not. Moreover, methods from model theory have invoked more general valuation the-ory. This has led to a number of interesting new results and conjectures, among them the Elemen-tary Galois Conjecture (EGC). This conjecture suggests a deep connection between the absolute Galois group and the existence of certain henselian valuations, which would have far-reaching consequences. One such consequence provides evidence that the absolute Galois group over the rational numbers encodes sufficient information to answer Hilbert's Tenth Problem. At the mo-ment, however, this group is far from being fully understood, so my research will comprise a fur-ther investigation of this object in the context of Hilbert's Tenth Problem. The Elementary Galois Conjecture is also closely related to a famous, though not yet well understood conjecture, Grothendieck's Section Conjecture in his program called "anabelian geometry" from 1983. This project falls within the EPSRC Logic and Combinatorics research area.