# Courses

**Specializations in Inverse Galois Theory.**Specialization is one the few general tools that provide a bridge between geometry and arithmetic. Given a geometric system involving some independent parameters, it makes it possible to specialize the parameters in the base field (the field Q of rationals for example) and still preserve the structure of the system. This is the so-called Hilbert property. The system structure can be for instance the Galois group of a field extension, where the Hilbert property follows from Hilbert's celebrated irreducibility theorem from 1892. The course is aimed at reviewing the specialization process and revisiting it towards some modern developments: some very recent results by Corvaja-Zannier, Bary-Soroker-Fehm and others on the Hilbert property of certain unirational varieties, others by Debes-König-Legrand-Neftin on the specialization set of covers of the line, finally some on different aspects of the Grunwald Problem including work by Harpaz-Wittenberg on the Brauer-Manin obstruction.**Number Theory in Function Fields.**This course aims to introduce various methods in the study of problems in analytic number theory in the setting of function fields with a focus on application to classical problems in analytic number theory. The first part will introduce a geometric approach, in which, one re-formulates the problem in a generic setting and then specializes using the Chebotarev density theorem. This enables one to reduce function field analogues of problems such as the Goldbach conjecture to a computation of a generic Galois group, and the course will present the state-of-the-art methods to compute such groups. The second part will introduce a probabilistic approach, in which one models the statistic of divisors (also referred to as the 'anatomy') of integers, or polynomials over a finite field, in terms of random permutations. This beautiful theory has many applications in number theory and in combinatoric group theory, and some of the applications to random polynomials will be presented in one of the other courses.**Random polynomials.**The study of random polynomials has a long history and it lies between probability-analysis and algebra-number theory. Recently there has been breakthroughs in the study. Rivin gave a nearly sharp upper bound on polynomials with Galois group smaller than the full symmetric group, finishing nearly a decade of study. BarySoroker-Kozma-Koukoulopoulos study random polynomials with restricted coefficients and showed that they are almost surely have big Galois group, making the first progress on a conjecture of Odlyzko-Poonen. The objective of the course is two folded. First we aim to equip the students with vast tool box in the study of random polynomials. Then, we introduce the above recent breakthroughs.