Explore the intersection of isogenies and lattices in cryptography through this comprehensive lecture. Delve into topics such as twin smooth integers, polynomials, and delayed encryption. Examine the applications of these concepts in sealed bid auctions and encryption algorithms. Investigate lattice-based cryptography, including the Reverse Minkowski Theorem and new lattice sampling techniques. Learn about the Hidden Number Problem, ideal lattices, and arithmetic curves. Gain insights into the current state of the art in isogeny-based and lattice-based cryptography, and understand their potential implications for future cryptographic systems.