Explore the intricacies of mirror symmetry in the context of hyperkähler manifolds in this comprehensive lecture. Delve into advanced mathematical concepts as the speaker unravels the complexities of these geometric structures and their symmetrical properties. Gain insights into the fascinating interplay between algebraic geometry and theoretical physics, and discover how mirror symmetry provides a powerful framework for understanding hyperkähler manifolds. Enhance your knowledge of this specialized field through a detailed examination of key theories, proofs, and applications presented by an expert in the subject.