The field of computational complexity and algebraic methods is experiencing significant developments, driven by innovative approaches to fundamental problems. Researchers are exploring new frontiers in p-adic numbers, linear recurrence sequences, and polynomial-time algorithms, leading to breakthroughs in our understanding of complex systems. Notably, recent studies have made progress in solving long-standing problems, such as the Skolem Problem, and have introduced new concepts, like spectral sparsification for Constraint Satisfaction Problems. These advances have far-reaching implications for cryptography, optimization, and computer science as a whole. Noteworthy papers include: Polynomial-time Tractable Problems over the p-adic Numbers, which solves a longstanding problem on the computational complexity of systems of linear equations over p-adic numbers. On the p-adic Skolem Problem, which presents algorithms for determining whether a given linear recurrence sequence has a p-adic zero. Rank Bounds and PIT for Σ^3 Π Σ Π^d circuits via a non-linear Edelstein-Kelly theorem, which proves a non-linear Edelstein-Kelly theorem for polynomials of constant degree and obtains constant rank bounds for depth-4 circuits. A Theory of Spectral CSP Sparsification, which introduces a notion of spectral energy for fractional assignments and defines a spectral sparsifier for Constraint Satisfaction Problems. On the Degree Automatability of Sum-of-Squares Proofs, which broadens the class of polynomial systems for which degree-d SoS proofs can be automated.