# Mathematics (second cycle)

**Study program cycle:**

Second cycle study program.

**Anticipated academic title:**

In Slovenian: magister matematike, magistrica matematike, abbreviated to mag. mat.

**Duration:**

2 full years (4 terms) based on 120 ECTS credits

**Basic goals:**

This program is designed for Bachelors in Mathematics (academic) wishing to take active part in planning, engineering and development in professional environment, or engage in scientific research in mathematics, theoretical computer science or theoretical mechanics.

**Subject specific competences developed by the student:**

- familiarity with classical and modern major results in pure and applied mathematics as well as closely related areas such as computer science and mechanics,
- ability of following and understanding hard mathematical proofs,
- ability of abstraction of practical problems,
- ability of adequate use of mathematical literature,
- ability of application of different mathematical methods in order to solve concrete problems,
- ability of computer programming using appropriate programming software,
- ability of critical evaluation and presentation of research results,
- ability of continuous self-education and keeping in touch with the expert literature.

**Employment possibilities:**

Graduates of this program can find employment in:

- research and development companies (or their branches) using mathematical modelling,
- software development companies,
- mathematical or mathematics related research institutions,
- universities.

## Admission requirements

**Admission to the study program is open to the following:**

- Graduates of the Academic study program in Mathematics or Mathematics Education.
- Graduates of the University undergraduate study programs in Applied mathematics, Pure Mathematics, Mathematics Education, Computer Science with Mathematics. In this case, applicants are exempted from taking some exams.
- Graduates of some accredited first cycle study program. (A technology or natural sciences program with a basic amount of mathematical analysis and linear algebra is recommended.) From the mathematical exams, a grade point average (GPA) of 8.0 or higher is recommended.

**Additional requirements and exemptions:**

Applicants under 2 are exempted from 60 to 90 ECTS credit requirements, and also from the final exam. The exact requirements and exemptions are determined by the department study committee for each applicant individually. Applicants have to file a formal written request and attach the relevant accredited transcripts. It is obligatory for graduates of the University undergraduate study program in Computer Science with Mathematics to complete the Point-set topology course.

For applicants under 3, the List of minimal mathematical competences required for admission to a second-cycle study program in mathematics (regardless of the program's specialization) is relevant. Beside these obligatory minimal requirements there are also additional requirements depending on the second-cycle program's specialization.

For Bachelors in Financial Mathematics, the course goals and competences of Algebra 2, Algebra 3, and Point-set topology are required. (In the List of minimal mathematical competences, the three courses belong to the Algebra package and the Theory of metric spaces and topology package).

**The minimal competences can be validated or acquired in one of the following ways:**

**(a)** The applicants who were previously enrolled in an accredited study program may submit the relevant
transcripts in order to request validation of the completed courses.

**(b)** For applicants who have acquired the necessary competences by self-study, the department organizes
one or more ad hoc examinations where the competences are tested. The examination(s) must be
completed prior to admission.

**(c)** Applicants can acquire the requisite competences by attending lectures and problem sessions of one
or more courses offered at the mathematics department of the FMF; the competences are tested by
exams at the end of the course(s).

The mathematics department offers the possibility of admission to the second-cycle study program prior to verification of the requisite competences. Applicants taking advantage of this offer acquire the missing competences by way of (c). They have to complete the corresponding examinations either prior to enrollment in the 2nd year or prior to completion of the study program, subject to decision of the department study committee.

Applicants under (b) above receive the list of minimal mathematical competences at the Student Office.

## Admission limitation measures

Applicants are selected according to their GPA of exams, problem sessions and seminars' grades obtained at their respective first-cycle study programs.

## Enrollment requirements

To enroll in the second year, students must earn at least 50 ECTS credits from courses and exams in the first year.

## Re-enrollment requirements

To re-enroll in the current study year, a student needs to earn at least half of all possible credits of the current study year (30 ECTS credits). Re-enrollment is only possible once in the course of the study program; any change in a study program as result of disability of enrollment in the second year is automatically counted as re-enrollment.

## Finishing requirements

To finish the program, students must:

- Successfully complete all exams.
- Successfully complete the final exam.
- Prepare and defend the master's thesis.

## Transition from other study programs

Graduates of the second-cycle program in Financial Mathematics must meet the admission requirements. Granted that, they are exempted from retaking courses they have already completed within the Financial Mathematics program. The exact requirements for enrollment in the second study year and completion of the program are determined by the department study committee.

Graduates of the second-cycle program in Mathematics Education are exempted from retaking courses they have already completed within the Mathematics Education program. The exact requirements for enrollment in the second study year and completion of the program are determined by the department study committee.

## Study program description

The study program comprises two full academic years based on 120 ECTS credits. Of these, the final "master's" exam and master's thesis account for 25 ECTS credits. All courses are single-term courses with 30 to 45 hours of lectures and 15 to 30 hours of problem sessions altogether (with a weekly load of 2/2 or 3/1 hours of lectures/problem sessions). Each course is worth 5 ECTS credits. A student's choice of courses has to be approved by the department study committee.

The courses are divided into the following groups:

**M1** – Analysis and mechanics

**M2** – Algebra and discrete mathematics

**M3** – Geometry and topology

**M4** – Numerical mathematics

**M5** – Probability, statistics and financial mathematics

**R1** – Computer mathematics

Other – Courses that do not belong to one of the above and courses that are offered by other departments (physics, chemistry, economics, education, linguistics, computer science, electrical engineering, etc.)

The requisite 120 ECTS are earned primarily by completion of exams, completion of the final (master's) exam, preparation and defense of a master's thesis, but possibly also by acquisition of work experience or publication of a scientific research paper.

A minimum of 75 ECTS credits must be earned by completing exams from the following categories:

**Basic:**at least one course from each of the groups M2-M5 and R1, and at least one of the courses Introduction to functional analysis and Measure theory from the group M1. Together, these courses account for at least 30 ECTS credits.**Specific:**at least 6 electives from groups M1-M5 and R1 which account for at least 30 ECTS credits.**Supplementary:**electives from the group Other which account for at least 15 ECTS credits.

The final exam and the master's thesis together account for 25 ECTS credits.

Of the other requisite credits a maximum of 10 ECTS credits can be awarded for work experience or a scientific publication. To be awarded credits, work experience must comprise at least 150 working hours as well as an obligatory presentation preparation. Meeting this requirement, 1 ECTS credit is awarded for each 30 working hours.

## Curriculum

The spreadsheet data are given for both the winter and the summer term.

Each term comprises 15 weeks of classes.

Abbreviations:

**L** = lectures per week (in hours),

**P** = problem sessions per week (in hours),

**ECTS** = ECTS credits worth,

**TSW** = estimated total student workload (in hours).

#### 1st year

Winter term | Summer term | Total | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Course | L | P | ECTS | TSW | L | P | ECTS | TSW | ECTS | TSW |

Basic elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-5 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Specific elective from groups M1-5 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Specific elective from groups M1-5 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Specific elective from groups M1-5 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Specific elective from groups M1-5 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

General elective | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Weekly total | 12 | 12 | 12 | 12 | ||||||

Term total | 180 | 180 | 30 | 900 | 180 | 180 | 30 | 900 | 60 | 1800 |

#### 2nd year

Winter term | Summer term | Total | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Course | L | P | ECTS | TSW | L | P | ECTS | TSW | ECTS | TSW |

Specific elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Specific elective from groups M1-5 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Master's thesis and final exam | 0 | 0 | 0 | 0 | 0 | 0 | 25 | 750 | 25 | 750 |

Weekly total | 12 | 12 | 2 | 2 | ||||||

Term total | 180 | 180 | 30 | 900 | 30 | 30 | 30 | 900 | 60 | 1800 |

#### M1 Analysis and Mechanics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Measure theory | 2 | 2 | 5 | 150 |

Introduction to functional analysis | 2 | 2 | 5 | 150 |

Functional analysis | 2 | 2 | 5 | 150 |

Introduction to C* - algebras | 3 | 1 | 5 | 150 |

Operator theory | 3 | 1 | 5 | 150 |

Introduction to harmonic analysis | 3 | 1 | 5 | 150 |

Special functions | 2 | 2 | 5 | 150 |

Partial differential equations | 2 | 2 | 5 | 150 |

Complex analysis | 2 | 2 | 5 | 150 |

Analytical mechanics | 2 | 2 | 5 | 150 |

Continuum mechanics | 2 | 2 | 5 | 150 |

Fluid mechanics | 2 | 2 | 5 | 150 |

Mechanics of deformable bodies | 2 | 2 | 5 | 150 |

Dynamical systems | 2 | 2 | 5 | 150 |

Industrial mechanics | 2 (seminar) | 2 (practice) | 5 | 150 |

#### M2 Algebra and Discrete Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Commutative algebra | 3 | 1 | 5 | 150 |

Associative algebra | 3 | 1 | 5 | 150 |

Non-associative algebra | 3 | 1 | 5 | 150 |

Ordered algebraic structures | 3 | 1 | 5 | 150 |

Group and semi-group theory | 3 | 1 | 5 | 150 |

Number theory | 3 | 1 | 5 | 150 |

Combinatorics | 2 | 2 | 5 | 150 |

Graph theory | 2 | 2 | 5 | 150 |

Cardinal arithmetic | 3 | 1 | 5 | 150 |

Topics in discrete mathematics | 2 | 2 | 5 | 150 |

Applied discrete mathematics | 1 | 3 | 5 | 150 |

Logic | 2 | 2 | 5 | 150 |

#### M3 Geometry and Topology

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Analysis on manifolds | 3 | 1 | 5 | 150 |

Introduction to algebraic geometry | 3 | 1 | 5 | 150 |

Convexity | 3 | 1 | 5 | 150 |

Algebraic topology 1 | 2 | 2 | 5 | 150 |

Algebraic topology 2 | 2 | 2 | 5 | 150 |

Differential geometry | 3 | 1 | 5 | 150 |

Lie groups | 3 | 1 | 5 | 150 |

Riemann surfaces | 2 | 2 | 5 | 150 |

#### M4 Numerical Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Numerical integration and ordinary differential equations | 2 | 2 | 5 | 150 |

Numerical solutions of partial differential equations | 2 | 2 | 5 | 150 |

Iterative numerical methods in linear algebra | 2 | 2 | 5 | 150 |

Computer aided (geometric) design | 2 | 2 | 5 | 150 |

Numerical approximation and interpolation | 2 | 2 | 5 | 150 |

Numerical methods for linear control systems | 2 | 2 | 5 | 150 |

#### M5 Probability, Statistics and Financial Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Probability theory 2 | 3 | 1 | 5 | 150 |

Statistics 2 | 3 | 1 | 5 | 150 |

Financial mathematics 2 | 2 | 2 | 5 | 150 |

Introduction to random processes | 2 | 2 | 5 | 150 |

Econometrics | 3 | 1 | 5 | 150 |

Random processes 2 | 2 | 2 | 5 | 150 |

Actuarial mathematics | 2 | 2 | 5 | 150 |

Modelling with random processes | 2 | 2 | 5 | 150 |

Topics in game theory | 2 | 2 | 5 | 150 |

Topics in financial mathematics | 2 | 2 | 5 | 150 |

Optimization in finance | 2 | 2 | 5 | 150 |

Time series | 2 | 2 | 5 | 150 |

Riesz spaces in mathematical economics | 2 | 2 | 5 | 150 |

#### R1 Computer Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Doing mathematics with a computer | 1 | 3 | 5 | 150 |

Theory of computability | 2 | 2 | 5 | 150 |

Computational complexity | 2 | 2 | 5 | 150 |

Topics in computer mathematics | 2 | 2 | 5 | 150 |

Topics in optimization | 2 | 2 | 5 | 150 |

Optimization 2 | 2 | 2 | 5 | 150 |

Data structures and algorithms 3 | 2 | 2 | 5 | 150 |