Antwerp Summer School in Algebra and Arithmetic
In 2022, the ALGAR summer school aims to present ideas and techniques from constructive mathematics in relation to open problems in quadratic form theory. Some familiarity with the basics of quadratic form theory will be assumed, but no previous exposure to mathematical logic and constructive mathematics will be required. The lectures will be accompanied by exercise sessions and a couple of research talks.
Target group: Master students and PhD students in pure mathematics. Also more experienced mathematicians are welcome to participate.
The theory of quadratic forms and central simple algebras over general fields is based to a large extent on elementary arguments. Nevertheless, many great theorems in this area are full of mysteries. Often they tell us the existence of some representation for certain objects, but the proof does not provide us with such a representation.
Constructive mathematics is an active branch of research concerned with the systematic search for computational proofs. Here the Law of the Excluded Middle and the Axiom of Choice have to be avoided. From a computational proof, one can usually extract extra information. For example, if the statement is about some way to represent a certain object, then a constructive proof will show us how to find such a representation and give us a bound on the number of parameters involved in the description.
In the summer school, general background and specific techniques from constructive algebra will be explained and linked to concrete open quantitative problems in quadratic form theory.
The programme will include a glimpse into topos theory and its role in mathematical logic. It will be tailored to young researchers in quadratic form theory and it will open perspectives for future research.
3 ECTS credits are awarded upon successful completion of the programme.