In 2025, the ALGAR summer school will be dedicated to linear algebraic groups and algebras with involution. We will study linear algebraic groups and explore their connections to quadratic forms and central simple algebras with an involution. On this basis, we will discuss cohomological invariants and see a presentation of Weil’s characterization of classical semi-simple groups of adjoint type as automorphism groups of algebras with involution. We will further look into the rationality problem for certain group varieties and the related notion of R-equivalence, introduced by Y. Manin.

The main part of the summer school takes place in the week from 25 to 29 August. There is some familiarity with the basic notions of quadratic forms, Galois cohomology and linear algebraic groups will be assumed.  It will consist of lectures accompanied by exercise sessions and a couple of research talks. Participants who want an initiation into the topics of the main week, are warmly invited to the introductory days from 20 to 22 August. The confirmed speakers are:

• ​Nicolas Garrel (University of Antwerp, Belgium)
• ​Philippe Gille (Université de Lyon, France)
• ​Diego Izquierdo (Institut Polytechnique de Paris, France)
• ​M. Archita (University of Antwerp, Belgium)