The current research in the field is significantly advancing the understanding of computational complexity, particularly in the areas of polynomial identity testing, hazard-free formula complexity, and hardness amplification. Innovations in polynomial identity testing for non-commutative circuits have led to efficient algorithms for higher-depth circuits, addressing a long-standing open problem. In hazard-free formula complexity, new insights have broken the monotone barrier for non-unate functions and provided tighter bounds for random Boolean functions. Hardness amplification research is exploring rare-case hardness against various adversaries, constructing functions that are exceptionally difficult to compute on a majority of instances. Additionally, group theory techniques are being employed to demonstrate that counting problems on graphs are nearly as hard in a subset of instances as they are in all instances. These developments collectively push the boundaries of what is computationally feasible and deepen our theoretical understanding of computational limits.