Homological Methods, Singularities, and Numerical Invariants

The contents of this dissertation are various in nature and background. A connection between them is a common effort to measure the singularities of a commutative Noetherian ring using homological tools, and associated numerical invariants. First, we study the index of a Gorenstein local ring, a numerical invariant that is defined in terms of Auslander's delta invariant. In particular, we find a counterexample to a conjecture of Songqing Ding relating the index and the minimal Löewy length of an Artinian reduction of the ring. Successively, we focus on two numerical invariants for rings of prime characteristic, the F-pure threshold and the diagonal F-threshold. We relate them with a third number, the a-invariant, proving most of a conjecture made by Hirose, Watanabe, and Yoshida. Finally, we study Golod rings. We present an example of a quotient of a polynomial ring over a field by a product of monomial ideals that is not Golod. This answers, in negative, a question of Welker.