In a nutshell


Lecture series on sums of squares in fields and on diverse tools from algebra and arithmetic used in their study.


Stadscampus, University of Antwerp, Belgium


2 - 6 July 2018


The target group consists of Master students and PhD students in fundamental mathematics. More advanced mathematicians are also welcome to participate.

Course description

The second edition of the summer school ALGAR is dedicated to the study of sums of squares in fields. In four lecture series and a few special talks we introduce the audience to the necessary algebraic, arithmetic and geometric methods  that yield to beautiful and deep results on the length of sums of squares in certain types of fields.

Sums of squares are studied since antiquity, starting from the determination of pythagorean triples of integers, Bramagupta’s composition formula for sums of two squares and Lagrange’s four squares theorem. We focus on the study of the so-called pythagoras number of a commutative ring, defined as the smallest number p such that every sum of squares is equal to a sum of p squares. 

We introduce and explore the sophisticated methods involved in the study of pythagoras numbers of fields. We aim in particular to exhibit the diverse methods and tools from algebraic geometry that apply for determining the lengths of sums of squares in function fields of real surfaces and of curves over number fields.


3 ECTS credits are awarded upon successful completion of the programme.

Registration fee


The fee includes course material, coffee breaks, a reception and a conference dinner. Does not include accommodation and meals.

Application details

Online through Mobility Online. The application deadline is: 4 June