# Mathematics (first cycle)

**Study program cycle and type:**

First cycle academic study program.

**Anticipated academic title:**

In Slovenian: diplomirani matematik (UN), diplomirana matematičarka (UN), abbreviated to dipl. mat. (UN).

**Duration:**

3 full years (6 terms) based on 180 ECTS credits.

**Basic goals:**

The principal goal of the academic study program in Mathematics is to qualify its graduates for solving hard mathematical problems arising in industry, in the public sector, and in sciences. At the same time, graduates of the program are equipped with the core knowledge necessary for studies in the second cycle.

**Generic competences developed by the student:**

- ability of abstract thinking and problem analysis,
- ability of sorting out effective solutions and of their critical evaluation,
- ability of application of knowledge in practice,
- ability of using and following the expert literature,
- ability to set forth both written and oral presentations of specialized topics,
- ability to work both individually and as part of an (international) team,
- ability of lifelong self-education.

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

Mathematics is the link between natural sciences, engineering, social sciences and computer sciences. Therefore, a graduate of the academic study program in Mathematics should be able to:

- model a practical problem mathematically,
- qualitatively analyze the obtained mathematical problems,
- conceive algorithms to solve them,
- implement those algorithms using appropriate programming tools,
- analyze and present the results

**Employment possibilities:**

Graduates of the academic study program in Mathematics can find employment in:

- the technology and logistic sector of the economy,
- banks and insurance companies,
- research and planning institutions, technology parks, and the public sector.

## Admission requirements and admission limitation measures

Admission to the study program is open to the following:

**a)** Holders of the matura certificate.

**b)** Holders of the vocational matura certificate obtained in any of the four-year high school programs.
In this case, an additional examination in one of the general matura subjects different from those of the
vocational matura is required. Either one of the vocational matura subjects or the additional one must
be mathematics.

**c)** Holders of the final examination certificate obtained in any of the four-year high school programs prior
to 1 June 1995.

In case the number of applicants exceeds the maximum availability, the applicants are selected according to their final matura (or vocational matura) grade, their mathematics matura (or vocational matura) grade, their grade point average (GPA) in the last two years of high school, and their final mathematics grades in the last two years of high school. These are weighted in the following way.

Applicants under **a)**

- Matura certificate grade - 30 % of points
- Matura mathematics exam grade - 30 % of points
- GPA in the 3rd and 4th years of high school - 20 % of points
- Final grade in mathematics in the 3rd and 4th years of high school - 20 % of points;

Applicants under **b)**

- Vocational matura grade - 20 % of points
- Matura or vocational matura mathematics exam grade - 40 % of points
- GPA in the 3rd and 4th years of high school - 10 % of points
- Final grade in mathematics in the 3rd and 4th years of high school - 30 % of points;

Applicants under **c)**

- Final examination grade - 30 % of points
- Mathematics final examination grade or mathematics grade in the 4th year of high school in case of exemption from the final exam - 30 % of points
- GPA in the 3rd and 4th years of high school - 20 % of points
- Final mathematics grade in the 3rd and 4th years of high school - 20 % of points

## Enrollment requirements

Enrollment in the first study year is granted upon admission.

For enrollment in the next study year it is necessary to earn 50 ECTS credits from courses and exams in the current study year.

In addition to the credit quota, the completions of the following exams are obligatory:

- for enrollment in the 2nd year: Analysis 1, Algebra 1 and Computer lab,
- for enrollment in the 3rd year: all exams of the 1st year, Analysis 2a, Analysis 2b, Algebra 2, Programming 1, Point-set topology and Seminar 1.

## Re-enrollment requirements

For re-enrollment in the same study year, a student needs to earn:

**a)** at least half of all possible credits of the current study year (30 ECTS credits), and

**b)** all credits from the previous study years.

Re-enrollment is only possible once in the course of studies. A change of the study program as result of disability of enrollment in the next study year is automatically counted as re-enrollment.

## Finishing requirements

To finish the program, students need to complete all exams.

## Validation of competences, knowledge, and skills acquired prior to admission to the study program

Students may apply for validation of their competences acquired previously by means of various forms of education if their competences match those of one or more courses offered within this study program. In a formal written request submitted to the mathematics department at the FMF, the applicant must specify the course(s) whose competences he or she had already mastered, and attach accredited transcripts proving it. When considering the possible validation of competences corresponding to a particular course, the department study committee bases its decision on a comparison of

- the duration of the educational process where the student acquired the competences with the duration of the respective course(s), and
- the scope of the previously acquired competences with the goals of the respective course(s).

If the study committee decides to validate the previously acquired competences, the student is awarded all ECTS credits that correspond to the respective course(s). In the validation process, the study committee follows The rules and guidelines for validation of informally acquired knowledge and skills, accepted by the Senate of The University of Ljubljana on 29 May 2007 (http://www.uni-lj.si/o_univerzi_v_ljubljani/statut_in_pravilniki.aspx).

## Grading system

The methods for testing the competences, knowledge, and skills are described in the courses syllabi. The basic knowledge testing rules are explained in the Exam guidelines of the FMF. Course examinations are either written or oral or both. They can have the form of midterm exams, oral defense of midterm exams, written exams, oral exams, seminar or project work and oral defense of seminar and project work. Grading is based on the grading scale determined in the Statute of The University of Ljubljana. All forms of examinations are graded by grades 1-10, out of which 6-10 are passing grades, and 1-5 are failing grades. The following grading scale is most commonly used for grading the score of a written course examination:

Score (in %) | grade |
---|---|

50 - 59 % | 6 |

60 - 69 % | 7 |

70 - 79 % | 8 |

80 - 89 % | 9 |

90 - 100 % | 10 |

## 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),

**Lab** = lab classes per week (in hours),

**S** = seminars 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 | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Analysis 1 | 4 | 4 | 0 | 0 | 9 | 270 | 4 | 4 | 0 | 0 | 9 | 270 | 18 | 540 |

Algebra 1 | 3 | 3 | 0 | 0 | 7 | 210 | 3 | 3 | 0 | 0 | 7 | 210 | 14 | 420 |

Logic and sets | 2 | 2 | 0 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Computer lab | 1 | 0 | 3 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Introduction to programming | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 3 | 0 | 6 | 180 | 6 | 180 |

Physics 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 6 | 180 | 6 | 180 |

Elective | 1 | 2 | 0 | 0 | 2 | 60 | 1 | 2 | 0 | 0 | 2 | 60 | 4 | 120 |

Weekly total |
11 | 11 | 3 | 0 | 30 | 900 | 13 | 12 | 3 | 0 | 30 | 900 | 60 | 1800 |

Term total |
165 | 165 | 45 | 0 | 195 | 180 | 45 | 0 |

**Electives**

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

Course | L | P | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Proseminar A | 2 | 2 | 0 | 0 | 2 | 60 | 0 | 2 | 0 | 0 | 2 | 60 | 4 | 120 |

Proseminar B | 0 | 2 | 0 | 0 | 2 | 60 | 2 | 2 | 0 | 0 | 2 | 60 | 4 | 120 |

*Remark: Proseminars A and B are electives. Proseminar A is intended as a review of selected high school topics in mathematics, while Proseminar B should be chosen by students with strong mathematical background. Problem sessions for both Proseminars are a supplement to the other courses in the first year's curriculum.*

#### 2nd year

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

Course | L | P | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Analysis 2a | 4 | 3 | 0 | 0 | 8 | 240 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 240 |

Analysis 2b | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 | 8 | 240 | 8 | 240 |

Algebra 2 | 2 | 2 | 0 | 0 | 6 | 180 | 2 | 2 | 0 | 0 | 4 | 120 | 10 |
300 |

Physics 2 | 3 | 2 | 0 | 1 | 6 |
180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Programming 1 | 2 | 0 | 2 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Point-set topology | 2 | 2 | 0 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Seminar | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 90 | 3 | 90 |

Discrete mathematics 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Elective 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Elective 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Weekly total | 14 | 10 | 2 | 0 | 30 | 900 | 13 | 11 | 0 | 3 | 30 | 900 | 60 | 1800 |

Term total | 210 | 150 | 30 | 0 | 195 | 165 | 0 | 30 |

**Electives**

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

Course | L | P | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Programming 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 5 | 150 | 5 | 150 |

Algebraic curves | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Introduction to geometric topology | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Affine and projective geometry | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

#### 3rd year

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

Course | L | P | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Analysis 3 | 3 | 3 | 0 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Analysis 4 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 6 | 180 | 6 | 180 |

Introduction to numerical methods | 3 | 3 | 0 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Probability | 2 | 2 | 0 | 0 | 5 | 150 | 0 |
0 | 0 | 0 | 0 |
0 | 5 | 150 |

Statistics | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Diploma seminar | 0 | 0 | 0 | 2 | 3 | 90 | 0 | 0 | 0 | 2 | 4 | 120 | 7 | 210 |

Mechanics 1 | 2 | 2 | 0 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Mathematics elective 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Mathematics elective 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Mathematics elective 3 | 2 | 2 | 0 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

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

Weekly total | 12 | 12 | 0 | 2 | 30 | 900 | 13 | 11 | 0 | 1 | 30 | 900 | 60 | 1800 |

Term total | 180 | 180 | 0 | 30 | 195 | 165 | 0 | 15 |

**Mathematics electives**

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

Course | Group | L | P | Lab | S | ECTS | TSW | L | P | Lab | S | ECTS | TSW | ECTS | TSW |

Numerical linear algebra | B1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Mechanics 2 | B1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Mathematical modelling | B1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Data structures and algorithms 1 | B2 | 2 | 1 | 1 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Data structures and algorithms 2 | B2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 5 | 150 | 5 | 150 |

Coding theory and cryptography | B2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Discrete mathematics 2 | B2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Optimization 1 | B2 | 2 | 2 | 0 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Algebra 3 | B | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Database basics | B | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 5 | 150 | 5 | 150 |

Financial mathematics 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Game theory | B | 3 | 3 | 0 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 180 |

Introduction to differential geometry | B | 2 | 2 | 0 | 0 | 5 | 150 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Stochastic processes 1 | B | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 5 | 150 | 5 | 150 |

Elementary number theory | B | 3 | 2 | 0 | 0 | 6 | 180 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 |

Elementary geometry | B | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 | 0 | 0 | 6 | 180 | 6 | 180 |

Selected topics in data analysis | B | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 | 5 | 150 | 5 | 150 |

*Remarks:
*

*A student earns 20 ECTS with elective courses. He or she must choose at least three Math electives (worth at least 15 ECTS). The remaining credits (corresponding to General elective) can be earned by completing courses in other study programmes at the University of Ljubljana. Ideally one Math elective should be chosen in the winter term. The student has to choose one Math elective from the group B1 and one from the group B2.**Electives can also be chosen from among the courses offered in the second cycle of this program.**Student's choice of electives must be approved by the department's study committee.*