Algebraic and Homological Aspects of Hermitian $K$-Theory

In 1970, S. P. Novikov proposed a systematization of algebraic results of the surgery theory based on the Hamiltonian formalism over rings with involution. His results have had a significant impact on the development of Hermitian analogs of algebraic $K$-theory. This article was written at S. P. Novikov's suggestion and aims to present the current state of research at the interface between the problems of manifold theory and Hermitian $K$-theory of rings with involution.