**Name and the Goals of the Study Programme**

Master study programme of Mathematics (MA) represents master studies in the field of pure (theoretical) mathematics at the University of Novi Sad, which is conducted at the Faculty of Sciences.

**Professional Title, Academic, or Scientific Title**

Upon completion of the studies, a student acquires a title of Master in Mathematics.

**Admission Conditions**

There are 9 compulsory courses and the Master thesis, which are defined by the curriculum of the programme. Elective courses are divided into three groups. The first group consists of two courses from the field of algebra and student has to choose one. The second group consists of 17 elective courses with a choice of at least 4, where a student, by making a choice, profiles him/herself in one of the areas of mathematics. The third group comprises 5 methodical-psychological-pedagogical courses. The purpose of this group is to enable a student who finished M3 or M4 bachelor study programme, to collect 30 ECTS from that group of courses and 6 ECTS more from the School practice, thus qualifying him/her for a job of the schoolteacher.

**The Structure of the Study Programme**

By choosing this programme, student orients himself toward extensive study of sophisticated theoretical results (and by that to direct him/her towards the profession of a mathematician-researcher). Student can extend his/her knowledge by several highly-specialized courses, some of which are inherited from the master study programme MB: *Applied mathematics*. These elective courses are independent by its contents, and therefore, there are no extra requirements for attending them, exception being the year of study to ensure logical continuation of knowledge obtained in compulsory courses. Each course has an adequate amount of ECTS credits, and the total number of classes.

**The Time Allotted for the Realization of Particular Study Forms**

It lasts for 2 years and has a total credit value of 120 ECTS.

**Prerequisites for the Registration for Particular Courses or Group of Courses**

Teaching method is usually frontal instructions, with the adequate use of classical and modern visuals. Due to the specific nature of Mathematics as a science, the tutorials have a theoretical character, where students exercise the presented theoretical principles and analyze typical problems and their solutions, which further enables them to independently apply the acquired techniques. Several courses (mainly in methodology of mathematics or numerical mathematics) rely on tutorials with the aid of computers, or they are of some other type (for example teaching practice), or comprise students’ independent work.

The purpose of the study programme of Mathematics stands for education of mathematician-researchers and their further advancement in scientific-research work. This also involves creating the young scientists at universities, scientific institutes and other institutions and facilities, where realization and modelling of practical problems involves using advanced mathematical structures. As a secondary purpose, achieved by a group of specific courses, there is education of teachers capable for using modern methodological principles, techniques in preparation and teaching mathematics classes in primary and secondary schools.

The goal of the master study programme of Mathematics is obtaining more advanced, but still basic knowledge of all the major sub-disciplines out of the area of theoretical mathematics, including in particular: Calculus (with applications to geometry and physics), topology, abstract algebra, discrete mathematics, basic mathematical logic, and selected topics of numerical mathematics and statistics. Another goal is a deeper and broader study of the basic theoretical results of modern mathematics, as the initial phase of introducing the young mathematicians to scientific research in the field of mathematics. However, the intention of this programme is to develop the highest level of abstract, analytical and synthetic mental abilities, independence and initiative in solving mathematical problems, as well as a critical attitude towards the studied topics.

**General and course-specific competencies of students**Depending on the module selected, a Master in Mathematics will be able to do research, and advance in further self-development in the field of mathematics and other sciences. These include both scientific research activities at universities and research institutes, and participation in the implementation of development projects and other businesses, since it is expected that the flexibility, adaptability to new situations, and ability to apply theoretical knowledge, should be the main features of the Master in Mathematics. These professionals should be able to use computers in their work.

**Learning outcomes**

Upon completing the programme, successful students should master the basic concepts and theoretical principles of mathematical sciences, and should be well trained in all the skills necessary for mathematical research.

Compulsory courses carry a total of 79 ECTS (65,83%). Elective course 1 carries 5 ECTS, while the rest ECTS are accumulated through “free” elective courses (30%). Overall, elective courses carry 34,17% of the points. “Free” elective courses are by thematic and methodological closeness classified into two groups: A – mathematical courses (analysis, topology and geometry, algebra, logic and discrete mathematics, numerical mathematics) and B – methodical and psychological-pedagogical courses.

For example, a student can make a following selection of courses:

– First semester (32 ECTS): Partial Differential Equations, Numerical Analysis 2, Topology, Rings, Fields and Galois Theory, Mathematical Logic (all compulsory courses);

– Second semester (28 ECTS): Mathematical Logic, Functional Analysis (compulsory courses), Graph Theory (Elective course 1), Decision Theory, Mathematical Physics Equations (elective);

– The third semester (30 ECTS): The Theory of Curves and Surfaces, Measure and Integral, Algebraic Topology (compulsory courses), Fixed Point Theory, Formal Language Theory (elective);

– The fourth semester (30 ECTS): Master Thesis (compulsory), Universal Algebra, Set Theory (elective)

**A Distribution of the Courses into Semesters and Academic Years**

**The curriculum scheme**

**Module: Computer Science**

No. | Course Code | Course Title | Semester | Course Type | Course Status | Active teaching hours | Other lessons | ECTS | ||

Lectures | Exercises | OFT | Other lessons | ECTS | ||||||

FIRST YEAR | ||||||||||

1 | MA-01 | Partial Differential Equations | I | SP | C | 4 | 2 | 7 | ||

2 | MB-01 | Numerical Analysis 2 | I | SP | C | 4 | 2 | 7 | ||

3 | MA-02 | Topology | I | SP | C | 3 | 3 | 7 | ||

4 | MA-03 | Rings, Fields and Galois Theory | I | SP | C | 3 | 1 | 5 | ||

5 | MA-04 | Mathematical Logic | I | SP | C | 3 | 2 | 6 | ||

6 | MA-05 | Functional Analysis | II | SP | C | 3 | 3 | 7 | ||

7 | Elective course 1 | II | EB | 2 | 2 | 5 | ||||

Active teaching hours per year– total | 37 | |||||||||

SECOND YEAR | ||||||||||

1 | MA-06 | Algebraic Topology | III | SP | C | 2 | 2 | 5 | ||

2 | MA-07 | Measure and Integral | III | SP | C | 2 | 2 | 5 | ||

3 | MA-08 | Theory of Curves and Surfaces | III | SP | C | 3 | 1 | 5 | ||

Active teaching hours per year– total | 12 | |||||||||

Total: 29 | Total: 20 | Total: | ||||||||

Active teaching hours – total | 49 | |||||||||

MA-09 | Master Thesis | IV | 20 | |||||||

ECTS – total | 79 |

- Course type: AO-academic and general education, ТМ-theoretical-methodological, SP-scientific-professional, PА-professional applicative
- Course status: O-obligatory, E-elective block
- Teaching hours: L-lecture, E-exercise, АE-auditory exercises, LE-laboratory exercises, OTF-other teaching forms (seminar work, etc.), SRW-study research work

**Elective courses in the Study Program**

No. | Course Code | Course Title | Semester | Course Type | Course Status | Active teaching hours | ECTS | |||

Lectures | Exercises | OFT | SRW | ЕСПБ | ||||||

Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 | Elective Course 1 |

1 | MA-10 | Number Theory | II | SP | EB | 2 | 2 | 5 | ||

2 | MA-11 | Graph Theory | II | SP | EB | 2 | 2 | 5 | ||

Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A | Elective block A |

3 | MB-12 | Selected Topics in Applied Algebra | I or III | SP | EB | 3 | 1 | 5 | ||

4 | MA-12 | Theory of Algorithms | I or III | SP | EB | 3 | 1 | 5 | ||

5 | MA-13 | Theory of Formal Languages | I or III | SP | EB | 2 | 2 | 5 | ||

6 | MA-14 | Fixed Point Theory | I or III | SP | EB | 2 | 2 | 5 | ||

7 | MA-15 | Operator Theory | I or III | SP | EB | 2 | 2 | 5 | ||

8 | MB-02 | Stochastic Analysis | I or III | SP | EB | 4 | 2 | 7 | ||

9 | MB-14 | Operations Research | I or III | SP | EB | 4 | 2 | 7 | ||

10 | MP-08 | Numerical Solving of Equations | I or III | SP | EB | 3 | 1 | 5 | ||

11 | MB-17 | Information Systems | I or III | AG | EB | 2 | 3 | 6 | ||

12 | MB-18 | Seminar in informatics | I or III | АG | EB | 1 | 3 | 4 | ||

13 | MB-10 | Equations of Mathematical Physics | II or IV | SP | EB | 3 | 1 | 5 | ||

14 | MA-16 | Semigroups | II or IV | SP | EB | 3 | 1 | 5 | ||

15 | MA-17 | Set Theory | II or IV | SP | EB | 3 | 1 | 5 | ||

16 | MA-18 | Universal Algebra | IV | SP | EB | 3 | 1 | 5 | ||

17 | MA-19 | Differential Geometry | IV | SP | EB | 3 | 1 | 5 | ||

18 | MB-21 | Decision Theory | II or IV | PA | EB | 2 | 2 | 5 | ||

19 | MA-20 | Numerical Solving of Partial Differential Equations | II or IV | SP | EB | 3 | 1 | 5 | ||

Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block | Student has to choose at least 4 courses from this block |

Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B | Elective block B |

20 | MP-01 | Teaching of Mathematics 1 | I or III | TM | EB | 2 | 2 | 5 | ||

21 | MP-02 | Evolutionary and Educational psychology | I or III | TM | EB | 3 | 1 | 5 | ||

22 | MP-03 | Pedagogy | II or IV | TM | EB | 4 | 0 | 5 | ||

23 | MP-05 | Teaching of Mathematics 2 | II or IV | TM | EB | 2 | 2 | 5 | ||

24 | MP-06 | School Practice | II or IV | TM | EB | 0 | 6 | 6 |

- Course type: AO-academic and general education, ТМ-theoretical-methodological, SP-scientific-professional, PА-professional applicative
- Course status: O-obligatory, E-elective block
- Teaching hours: L-lecture, E-exercise, АE-auditory exercises, LE-laboratory exercises, OTF-other teaching forms (seminar work, etc.), SRW-study research work