periodic difference operator, Ergodic theorems, spectral theory, Markov operators, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Riemann-Roch theorem, m- gonal Toda systems, Period matrices, variation of Hodge structure; degenerations, Hill's operator, Families, moduli of curves (analytic), Local and nonlocal bifurcation theory for dynamical systems, Theta functions and abelian varieties, Korteweg-de Vries equation, Elliptic curves, Riemann-Roch theorems, Jacobi variety, Jacobians, Prym varieties, periodic Toda system, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Holomorphic maps on manifolds, almost periodic difference operator, hyperelliptic curves with branch points, Its-Matveev-Baker formula, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Linear difference operators, symplectic structures, Divisors, linear systems, invertible sheaves, Difference algebra