Invariable generation in finite groups with applications to algorithmic number theory

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This research proposal lies at the interface of two areas of pure mathematics: algebra and number theory. More specifically, the research seeks to build on a fascinating link which has emerged between a problem in the theory of "groups" (the algebraic structures which capture and allow us to study symmetries in nature) and one of the most famous unsolved problems in mathematics: the Inverse Galois Problem.Galois theory was discovered by the French mathematician Evariste Galois in the nineteenth century as a tool to study (integer) polynomial equations of degree greater than 4, and when they can be solved by radicals. To a set of roots of such a polynomial, Galois associated an algebraic structure which we now call a Galois group. This structure is a set together with a binary operation which preserves the symmetries in the roots of the polynomial, and studying this operation allows us to deduce properties of the roots. In this way, Galois developed a theory whereby one can translate questions about a very complicated polynomial equation to questions about its Galois group, which is often easier and more concise to study. In recent years, the theory, together with group theory and number theory in general, has shifted from being a purely academic endeavour to making significant contributions to cryptography, e-commerce and financial security. Galois groups are special examples of the "groups" we mentioned in the first paragraph, and have finite size. Thus, all Galois groups are finite groups, but what about the other way around? Is every finite group the Galois group of some integer polynomial? This is called the "Inverse Galois Problem" (IGP), and a complete solution has evaded mathematicians for almost 200 years. A groups which is the Galois group of an integer polynomial is said to "satisfy IGP". Some specific cases have been dealt with (the group of all symmetries of a finite set of size n - the symmetric group of degree n - is known to satisfy the IGP, for example), but even some relatively "small" groups remain elusive. In 2008, the number theorists Jouve, Kowlaski and Zywina announced a new technique to study the IGP in the Weyl groups of simple algebraic groups - an important class of groups in geometry. This spawned a renewed optimism for the IGP, and led to further developments of the technique by Lucchini and Tracey (using powerful group theoretic techniques) and the mathematicians Eberhard, Ford and Green (using powerful techniques from probability theory and combinatorics). This research proposal seeks to build on these techniques by combining the group theoretic, probabilistic, and combinatorial approaches mentioned above. The specific problems we propose range from answering important questions concerning these techniques in the finite simple groups (the "building blocks" of finite groups), to answering some long-standing questions posed by B.L. van der Waerden and J.P. Serre. As a final ambitious problem, we seek to solve the IGP in the case when G is the Mathieu group M23 - one of the most famous and important finite groups in which it is unknown whether or not the IGP is satisfied.