# Mathematics, studia stacjonarne II stopnia

**In power from:** 27 Mar 2023

## Registration criteria

**Admission procedure**

Candidates applying for admission to a second-cycle study programme must be holders of a diploma of higher education.

The selection of candidates is conducted based on:

- a **diploma grade ** in the case of graduates of the same or related field of study;

**or**

- **an interview **covering major subjects appropriate for the undergraduate degree programme in Mathematics in the case of candidates with a degree in other fields of study.

The following grading scale is applied: **very good (5), good plus (4+), good (4), satisfactory plus (3+), satisfactory (3), unsatisfactory (a** **failing grade) (2). **The candidate who gets a failing grade in the interview cannot be admitted.

**Announcement of results**

The results of the recruitment will be available in the IRK account of each registered candidate.

**Required documents**

Having registered in the IRK system, the candidate is obliged to submit all the required paper-based documents. Failure to do so will be understood as a resignation from applying for the study programme, despite a valid registration and paying the required application fee. The documents need to be submitted to Centralny Punkt ObsÅ‚ugi Kandydata (the Central Service Point for Candidates) - time and date as specified on the IRK website.

**A list of documents required can be downloaded here**

**Addidional information for candidates**

**A tuition fee is required.**

**Example interview topics **

1. Propositional calculus. Functors and quantifiers. Basic laws of propositional calculus.

2. Algebra of sets. Basic operations on sets and their properties. Relations and their properties; an equivalence relation, an order relation; functions as relations.

3. Real numbers and their properties. Subsets of reals, bounded sets, limits of sets.

4. Complex numbers and their properties. Algebraic, trigonometric and exponential representations of complex numbers. Power and square root in the set of reals.

5. Numerical sequences. Limit of a sequence. Properties of convergent sequences and their examples.

6. Numeric series. The concept of convergence, convergence criteria, geometric and harmonic series.

7. Elementary functions. Examples, properties, graphs of basic functions.

8. A limit and continuity of functions. Properties of limits, basic theorems about continuous functions.

9. Differentiable functions. Derivative of a function and its basic properties. Derivatives of elementary functions. Basic theorems of a differential calculus. Taylor’s scheme. The use of differential calculus for function studies.

10. Indefinite and definite integrals. Definition and properties of integrals. Basic integration methods. Newton's Leibniz scheme. Applications of integrals.

11. Sequences and function series. Point convergence and uniform convergence. Convergence criteria. Power series. Examples of expanding functions into a power series.

12. Differential equations. The concept of ordinary differential equation. General and detailed solution, geometric interpretation. Basic examples of differential equations.

13. Matrices and determinants. Operations on matrices. Inverse matrix. Properties of determinants. A matrix raw.

14. Linear spaces. Definition and basic examples of linear spaces. Basis and space dimension.

15. Linear transformations. Definition and examples of transformations. Matrix of a linear transformation.

16. Systems of linear equations. Methods of solving systems of linear equations. Kronecker – Capelli Theorem. Cramer’s systems.

17. Groups. The concept and examples of groups. Abel’s and cyclic groups. A group of transformations. The order of a group; layers and quotients groups. Groups’ homomorphism.

18. Polynomials. Polynomial roots. Decomposition of polynomials. Bezout Theorem. The fundamental theorem of algebra.

19. Metric spaces. Definition and examples of metric spaces. A concept of a ball in a metric space. Open and closed sets. Complete, compact and consistent spaces. Homeomorphisms.

20. Elements of combinatorics. Permutations, variations and combinations. Definitions and examples.

21. Foundations of probability theory. A concept of probability. Operations on events. Properties. Conditional and total probability. Independence of events. Bernoulli scheme. Random variables. Examples of discrete and continuous distributions. Expected value and variance of random variable.