1. Solve mathematical problems by computer.
2. Use mathematical tools in solving problems.
3. Write own mathematical software.
4. Define new mathematical structures.
5. Compare with other colleagues in how fast can the stated problem be solved.

4 hours of lectures a week. The lecturer is going to follow the lecture materials.

Each student will get a project in one of the three major areas: discrete mathematics, probability or mathematical modelling. The project includes analysis of a problem, studying as well as solving it using the computer.

1. Introduction to the main topics and challenges. Student project presentations. Generation of all subsets of a given set. Different lexicographical orderings.
2. Generation of all k-subsets, all functions, injective and surjective functions. Adding constraints and software solutions.
3. Permutations. The successor function. Permutations without fixed elements. Integer partitions.
4. Bell's numbers. Stirling numbers. Software solutions. Combinatorial designs.
5. Introduction to mathematical modelling. Weak formulation of a mathematical model. Introduction to the finite element method.
6. Implementation of the finite element method for the 1D problem. Methods of solving linear systems.
7. Finite element method for a two-dimensional Poisson problem. GMSH and non-trivial domains.
8. Midterm exam
9. Weak formulation and finite element solvers for several models from practice.
10. Parameter tables for designs. Construction strategies. Difference sets. Introduction to Monte Carlo simulations.
11. Introduction to Markov chains. Transition probability matrix. Chapman-Kolmogorov Equations.
12. Simulation of Markov chains. Classification of States.
13. Limiting distribution. Stationary distribution. Ergodic theorem.
14. Project work and presentation.
15. Final exam