This talk will be in the commutative Noetherian local setting. If you give me a ring R of Krull dimension two, tell me that the corresponding geometric object is smooth and ask me to compute Ext^5(M,N) for two finitely generated R-modules M and N, it would take me less than two seconds to answer: it is zero. Things only get interesting if you remove the smoothness hypothesis. Then I can talk about nonzero arbitrarily higher ext modules and their annihilators. In this talk, I will tell you three related but separate theorems regarding these annihilators. The first one is joint with Akdenizli-Aytekin-Çetin and it is about a topology on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. The second one is joint with Ryo Takahashi and gives upper bounds on the dimension of singularity categories. Finally, the last one is about Frobenius structures on the category of MCM modules and is joint with Benjamin Briggs.