In this course we will cover stable infinity categories and  Poincare infinity categories. The latter are stable infinite categories equipped with a non-degenerate quadratic functor. They are used to describe algebraic cobordism categories and the process of algebraic surgery.

We will explain the above  in this course with the ultimate goal to use these tools to understand Grothendieck-Witt and Witt groups. In particular we will compute these for certain Dedekind rings (like the integers Z).  The whole course is based on results obtained in joint work with Baptiste Calmes, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit,Markus Land,Kristian Moi, Denis Nardin and Wolfgang Steimle.