The fourth edition of the ALGAR summer school is dedicated to valuation theory and its use in arithmetic. Five lecture series and some special talks will give a multifaceted introduction to the role of valuations at the crossway of quadratic form theory and model theory. This includes the role of local-global principles in recent results on definability of certain integrally closed domains in their field of fractions.
Date: 6 - 10 July 2020
Update 27 March: Due to the recent developments of the covid-19 pandemic, the physical version of the summer school was cancelled. We have however decided to develop an online alternative.
The summer school will be hosted via Blackboard Collaborate Ultra, the collaborative online learning platform of the University of Antwerp. Participants need to have a stable internet connection, a headset, a computer and a webcam (Google Chrome Browser is highly recommended). The summer school offers a variety of teaching methods. Most sessions will be held “live” as to enable interaction and debate. Some theoretical or introductory sessions may be recorded and shared beforehand.
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.
The summer school will start on Monday the 6th of July in the morning and will end on Friday the 10th of July in the afternoon.
The schedule can be found here (pdf - 98 kB). A final version of the schedule will be released soon.
Master students and PhD students in fundamental mathematics. More advanced mathematicians are also welcome to participate.
3 ECTS credits are awarded upon succesful completion of the programme.
Participants will be able to get in touch with peers attending other summer schools at the Antwerp Summer University. A visit to the beautiful city hall, a networking reception, a guided city walk, a quiz night, a football game and a day-trip to another Belgian city such as Bruges or Brussels are only some examples of these activities.
All activities of the social programme are offered free of charge, in some cases participants will be asked for a deposit which will be reimbursed upon participation to the activity.