6 July 2018
ALGAR 2018: Sums of squares in fields
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. These will include tools from the algebraic theory of quadratic forms, in particular the Cassels-Pfister Theorem and upper bounds for the pythagoras numbers for certain extensions.
To obtain lower bounds for the pythagoras number of a field is generally an even more difficult problem. 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.
The main target group consists of Master students and PhD students in fundamental mathematics. More advanced mathematicians are also welcome to participate.
3 ECTS credits are awarded upon successful completion of the programme.
EUR 250: Regular registration
Reduced fee for students - €200
Reduced early registration fee for all categories (registration before 17 April) - 10% reduction
Students without funding may apply for support. For details please contact the organising committee (firstname.lastname@example.org).