Automorphisms on algebraic varieties: K3 surfaces, hyperkähler manifolds, and applications on Ulrich bundles

One of the main tools to study the geometry of complex algebraic varieties is the group of automorphisms. The first part of this thesis concerns the study of symplectic automorphisms of finite order on K3 surfaces, and birational symplectic maps of finite order on projective hyperkähler manifolds which are deformation equivalent to the Hilbert scheme of K3 surfaces. In the second part of this thesis, the automorphism groups of rational homogeneous spaces are used to study Ulrich bundles in smooth projective varieties.