Valuations are a central tool in the study of the arithmetic properties of fields.
Discrete valuations occur in number theory and algebraic geometry, for example in the context of local-global principles for properties of quadratic forms, central simple algebras or related objects in Galois cohomology.
In recent years valuations and their interplay with quadratic forms and central simple algebras - e.g. in the form of a local-global principle - have proven to be critical in finding first-order definitions of natural subsets of fields from arithmetic. In the field of rational numbers, for example, it has been shown that the set of squares, the ring of integers, and more generally any finitely generated subring, have a universal first-order definition, relying heavily on the study of quaternion algebras through valuations. These definability results, in turn, have applications to longstanding questions from logic and model theory.
In our summer school we want to give an overview on the methods that are involved in this type of results. This will contain the discussion of certain well-known theorems on valuations and algebraic structures in a classical number-theoretic context, such as the Albert-Brauer-Hasse-Noether theorem (for central simple algebras), the related Hasse-Minkowski theorem for quadratic forms, as well as statements describing the ramification of such structures, like Hilbert's Reciprocity theorem for quaternion algebras.
For participants coming from a more model-theoretic background, this will offer a systematic introduction to certain algebraic techniques.
At the same time, participants coming from a more arithmetic background are offered an introduction to the challenging open problems in the area of definability and related to Hilbert’s 10th problem.
This summer school is particularly directed to PhD students, but it is open to master students and to more experienced researchers as well.