Sztochasztika

1. Ordinary Differential Equations:
Existence, uniqueness, continuous dependence, and the underlying fixed-point theorems. Continuous and discrete-time dynamical systems. Limit sets. Linear equations. Local theory around hyperbolic equilibrium points and periodic orbits, stable and unstable manifolds. Stability and Lyapunov functions.
References:
V.I. Arnold, Ordinary Differential Equations, The MIT Press, Cambridge, Massachusetts, and London, England, 1998. (Chapters 1-4)
D.K. Arrowsmith & C.M. Place, An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 1990. (Chapters 1-4)

2.Dynamical Systems:
Structural stability. Grobman-Hartman lemma. Basic types of local bifurcations. Center manifolds. Attractor-repeller pairs, Morse decomposition, Conley recurrence, Morse-Smale systems. Smale horseshoe. Homoclinic orbits, Birkhoff-Smale theorem.
References:
C. Robinson, Dynamical Systems, CRC Press, Boca Raton, 1995. (Chapters 4; 5; 7.4.1-7.4.2; 7.12; 9.1)
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin, 1995. (Chapters 4.1-4.4)

3. Chaos and Ergodic Theory:
Symbolic/shift dynamics. Interval mappings. Invariant measures in discrete and continuous-time dynamical systems. Ergodic measures. Topological entropy. Hausdorff dimension. Iterated function systems, fractals.
References:
P. Walters, An Introduction to Ergodic Theory, Springer, Berlin, 1982. (Chapters 1; 4)
K. Falconer, Techniques in Fractal Geometry, Wiley, New York, 1997. (Chapter 2)

4. Numerical Dynamics:
Comparison of original and discretized dynamics concerning attractors, invariant manifolds, and interval arithmetic, fixed-point index. Chaos of the Smale horseshoe type. Relationship between structure and convergence estimates, symplectic discretizations.
References:
M. Stuart & A.R. Humphries, Dynamical Systems and Numerical Dynamics, Cambridge University Press, Cambridge, 1996. (Chapters 4; 6-7)

5. Functional Differential Equations:
Basic properties of the solution operator: existence, uniqueness, continuous dependence, compactness. Linear equations. Elements of the qualitative theory: stability, Lyapunov functions, stable and unstable manifolds around hyperbolic equilibria.
References:
K. Hale, Functional Differential Equations, Springer, Berlin, 1971. (Chapters 1-5; 8-13; 19-20; 22; 24-26)

6. Dynamical Models in Biology:
Population dynamics: Kolmogorov systems, age-structured and spatial ecological systems. Epidemic models. Evolution theory, population genetics, game-theoretic models.
References:
J. Hofbauer & K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. (Chapters 12-13; 18-19)
M. Farkas, Dynamic Models in Biology, AP, New York, 2001. (Chapters 1-2; 5)

7. Control Theory:
Basic properties of linear control systems: controllability, observability, stabilizability, realization. Optimal controls, Pontryagin's maximum principle. Dynamic programming, the Hamilton-Jacobi-Bellman equation.
References:
A. Strauss, An Introduction to Optimal Control Theory, Springer, Berlin, 1982.
E.D. Sontag, Mathematical Control Theory, Springer, Berlin, 1998. (Chapter 8)

8. Numerical Methods for ODEs:
Approximate solutions of initial value problems, convergence, stability, Lax equivalence. Single and multistep methods. Numerical solutions of two-point boundary value problems: shooting, finite difference. Galerkin and finite element methods.
References:
A. Quarteroni, R. Sacco & F. Saleri, Numerical Mathematics, Springer, Berlin, 2000. (Chapters 11-12)

9. Classical Partial Differential Equations:
Method of characteristics. Basic concepts of Laplace/Poisson, heat, and wave equations. Harmonic functions. Weyl's lemma. Fourier method.
References:
F. John, Partial Differential Equations, Springer, Berlin, 1971. (Chapters I-IV)
L.C. Evans, Partial Differential Equations, AMS, Providence, 1998. (Chapter 2)

10. Sobolev Spaces:
Sobolev spaces, embedding theorems, boundary trace, singular integrals. Distributions, fundamental solutions via convolution, Fourier analysis. Operator semigroups and their infinitesimal generators, perturbation and approximation theorems.
References:
L.C. Evans, Partial Differential Equations, AMS, Providence, 1998. (Chapter 5)
M. Reed & B. Simon, Methods of Modern Mathematical Physics II., AP, New York, 1975. (Chapters IX.1-2, 5-6)
J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985. (Chapters 1-7)

11. Linear Elliptic Equations:
Weak solutions. Lax-Milgram lemma, variational formulation, eigenvalue and eigenfunction theory, energy inequalities, interior and boundary elliptic regularity, maximum principles.
References:
L.C. Evans, Partial Differential Equations, AMS, Providence, 1998. (Chapter 6)

12. Linear Parabolic and Hyperbolic Equations:
Weak solutions of linear parabolic equations, energy inequalities, regularity theorems, maximum principles. Linear hyperbolic equations: weak solutions, energy inequalities, propagation of disturbances.
References:
L.C. Evans, Partial Differential Equations, AMS, Providence, 1998. (Chapters 7.1-7.2)

13. Reaction-Diffusion Equations:
Existence and uniqueness for quasilinear equations on Banach space pairs. Regularity theorems. Qualitative theory elements: stability, stable and unstable manifolds, traveling waves.
References:
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981. (Chapters 1.4-1.5; 3.1-3.6; 5.1-5.4)
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, Berlin, 1983. (Chapters 15-16)

14. Nonlinear Hyperbolic Equations:
Hyperbolic conservation laws. Burgers equation, shock waves, Riemann problem. Hopf-Lax formula, Oleinik's theorem. Compensated compactness, viscous approximation, Tartar and Murat theorems, DiPerna theory.
References:
L.C. Evans, Partial Differential Equations, AMS, Providence, 1998. (Chapters 3.3-3.4)
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, Berlin, 1983. (Chapters 15-16)
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer, Berlin, 1997.

15. Numerical Methods for PDEs:
Projection Methods: Galerkin, finite element. Functional-analytic foundation of finite element methods for elliptic equations. Finite difference methods for Laplace, heat, and wave equations, stability conditions.
References:
A. Quarteroni, R. Sacco & F. Saleri, Numerical Mathematics, Springer, Berlin, 2000. (Chapter 13)
D. Braess, Finite Elements, Cambridge University Press, Cambridge, 1997. (Chapters 1-2)

16. Large Linear Systems:
Direct methods for solving block-structured and general sparse matrix equations. Iterative methods, gradient and conjugate gradient methods. Relaxation procedures. Preconditioning.
References:
A. Quarteroni, R. Sacco & F. Saleri, Numerical Mathematics, Springer, Berlin, 2000. (Chapters 3-4)