  The Orlov spectrum is a new invariant of a triangulated category. It was
introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R.
Rouquier. The supremum of the Orlov spectrum of a triangulated category is
called the ultimate dimension. In this work, we study Orlov spectra of
triangulated categories arising in mirror symmetry. We introduce the notion of
gaps and outline their geometric significance. We provide the first large class
of examples where the ultimate dimension is finite: categories of singularities
associated to isolated hypersurface singularities. Similarly, given any nonzero
object in the bounded derived category of coherent sheaves on a smooth
Calabi-Yau hypersurface, we produce a new generator by closing the object under
a certain monodromy action and uniformly bound this new generator's generation
time. In addition, we provide new upper bounds on the generation times of
exceptional collections and connect generation time to braid group actions to
provide a lower bound on the ultimate dimension of the derived Fukaya category
of a symplectic surface of genus greater than one.
