Doctoral Programme Mathematics and Physics - subprogramme Mathematics

The main goal and the most important activity of doctoral study is the doctoral dissertation, which must include original scientific results and result in at least one publication in a scientific journal. The student prepares this under the guidance of their mentor whom they choose before enrolment into the first year of study.

Individual research work towards the doctoral dissertation counts for 174 credit points, and the submission and defence of the dissertation for 6 credit points. The remaining 60 credit points of the doctoral degree are obtained from the following study obligations:

• A first-year Predoctoral exam (6 credit points). This is a pass/fail exam, which the student must pass. The aim of the exam is to check the basic understanding of mathematical concepts. Students with a solid mathematical background should not need to prepare for this exam. Three sample past exams are: 2018, 2019, 2020.
• Successfully completing five courses in the first two years of study (each subject contributes 6 credit points, making a total of 30 credit points). These are the only study obligations in the doctoral programme for which the student receives a numerical grade. At least three of the courses must be organised taught courses on the taught doctoral programme in mathematics. In addition, at most two of the courses can be taken as consultation courses, chosen with the help of the supervisor, who may not themself be the teacher of the consultation course. The names of the subjects are sufficiently general (e.g., "Selected topics in analysis"), that they can encompass any mathematical topic
• Individual studies in the first two years (in each year worth 6 credit points, making a total of 12 credit points). Under the guidance of their mentor, the student is introduced to the broader area around their research field (for example, by studying graduate textbooks from the area, solving exercises, etc.). This does not however include the study of scientific papers directly related to the topic of PhD research.
• Active participation in one of the seminars of the department of mathematics, in the first two years of study (in each year worth 6 credit points, making a total of 12 credit points).

The main objective of the study programme in Mathematics and Physics is to produce highly qualified researchers and experts capable of conducting independent research and development in the fields of physics, astrophysics, nuclear physics, meteorology, mathematics, and physics and matematics education, at a level comparable to and competitive with current top-quality research pursued in the most advanced parts of the world. In this regard, independent research towards the doctoral thesis, which is the main component of the doctoral study programme, develops in-depth knowledge in the area of immediate specialization. This is supplemented by the organized part of the study programme, which develops the broader knowledge that is indispensable for any young researcher to be able to efficiently respond to rapid development and change in modern science, or for them to participate in larger interdisciplinary research teams.

• Capacity for abstraction and problem analysis
• Gathering, critical evaluation and synthesis of data, measurements and solutions
• Identification of data needed for obtaining new knowledge
• Forming new knowledge on the basis of existing theories and available data
• Use of knowledge in practice (especially in modern technologies)
• Capacity for interdisciplinary integration of scientific knowledge
• Capacity for both autonomous research and development, and for working in an (international) group
• Communicating scientific knowledge to the general public
• Ability to use modern research methods and procedures
• Ability to critically evaluate and present one’s own results
• Capacity for further independent education, research and following of the literature

• In-depth understanding of the physical laws of nature
• Combining the basic laws of nature and observable properties of the world
• Ability to creatively identify physical problems and analyse them
• Capacity for the mathematical formulation of physical problems
• Deduction of the physical basis of practical problems
• Ability to model problems
• Advanced physical experimental skills
• Critical evaluation of the results of measurements and their use in the construction of models
• Understanding the principles of operation of technical devices on the basis of fundamental laws
• Presentation of physical methods and results adapted to the target public
• Ability to convey knowledge of physics
• Thorough understanding of the results of research in the field of interest
• Independence in research
• Capacity for abstraction of practical problems
• Capacity for critical and independent work and for consultation in the field of physics education.

Candidates can enrol in the doctoral study programme if they have successfully completed:

• A second-cycle (i.e., masters) study programme.
• A university-level degree programme accredited before 11.6.2004.
• A study programme for specialisation following completion of a profession-oriented higher education programme. In this case, the relevant doctoral studies board (either mathematics or physics) will determine any additional study requirements that need to be fulfilled. These requirements may be tailored to the individual, or they may involve between 30 and 60 ECTS points of additional courses to study.
• A profession-oriented study programme that is either regulated by directives of the European Union or is a masters programme, and which carries at least 300 ECTS credit points.
• A Masters level study programme or a study programme for specialization, having previously completed another University-level degree programme. In accordance with the law, such candidates may request up to 60 ECTS credit points in recognition of their previous education.

Enrolment in the doctoral study programme in Mathematics and Physics is open also to graduates of foreign universities. In accordance with the Statutes of the University of Ljubljana, the equivalence of previous education acquired abroad is established as part of the process of recognition of foreign education for the continuation of education at University of Ljubljana.

In the case of a size-limited enrolment process, the selection of the candidates will be based on:

• The average mark in previous studies (15%)
• The mark for the diploma or masters work (5%)
• The score achieved in a written selection exam (80%) in physics or mathematics. The score from this exam can be partially (up to 40%) be replaced by the candidate’s research and professional achievements in the area of the study programme.

Note: By the end of the application deadline, the candidate must have an agreement with a higher education teacher or researcher to act as mentor / co-mentor for their doctoral studies. The written consent of the mentor / co-mentor must then be submitted at the time of enrolment, together with a short high-level outline of the planned research topic of the doctorate.

A student can be awarded credit for knowledge acquired in different forms of previous education that corresponds in content to courses in the study programme. Decisions concerning the recognition of such knowledge are made by the Department of Physics and Department of Mathematics Study Boards at FMF, on the basis of the student's written application with attached testimonials and other documents confirming successful acquisition of such knowledge and its content.

In awarding credit for an individual course, the Study Board will use the following criteria:

• comparability of the scope of previous education with the scope of the course for which credit is awarded,
• correspondence of the content of previous education to the content of the course for which credit is awarded.

In cases in which the Study Board determines that the acquired knowledge can be recognised, it is accredited with the number of ECTS points attached to the corresponding courses.

The assessment methods follow the Statutes of the University of Ljubljana, and are listed in the curricula.

Mathematics:

For enrolment in higher years it is necessary to meet the following conditions:

• For enrolment in the second year, it is necessary to obtain at least 48 ECTS from the programme requirements. This must include the predoctoral exam and at least an additional 24 ECTS from organised forms of study (courses, seminars, individual study) during the first year.
• For enrolment in the third year of study it is necessary to obtain at least 90 ECTS from the first two years. This must include the predoctoral exam and at least an additional 48 ECTS from organised forms of study (courses, seminar or disposition, individual study). In addition, before enrolment in the 3rd year, the student must obtain ratification by the senate of the Faculty (UL FMF) of a positive evaluation of the suitability of the subject of the doctoral dissertation from the Committee for monitoring the doctoral student (KSDŠ).
• For enrolment in the fourth year of study, it is necessary to meet all study requirements from the first three years and to obtain approval of the subject of the doctoral dissertation from the Senate of the University (UL).

Transfer to the doctoral (third-cycle) programme in Mathematics and Physics is possible:

• From other doctoral (third-cycle) programmes of UL. The candidate must fulfil the requirements for enrolment in the doctoral (third-cycle) programme in Mathematics and Physics. On the basis of credit points awarded for exams taken in the previous study programme, the FMF Study Board may require a candidate to take bridging exams (corresponding to up to 60 ECTS points) and decide in which year of study he/she may enrol.
• Under the same conditions as in the previous paragraph, transfer is possible also from comparable programmes at other universities if the candidate meets the requirements for enrolment in the doctoral (third-cycle) programme in Mathematics and Physics.

The requirements for completion of study and acquisition of the title Doctor of Science are that the candidate successfully meets all study requirements laid down in the programme and successfully defends the doctoral dissertation, amounting to a total of 240 ECTS. That is the student must fulfil all requirements for organised study (amounting to 60 ECTS), as well as for individual research work including the preparation and defence of the doctoral dissertation (amounting to 180 ECTS). The candidate must also publish or have accepted for publication at least one scientific article from the field of the dissertation in one of the journals from group I - III, as defined in the FMF Interpretation of the Criteria for Election of University Teachers, Researchers and Research Associates, as approved at the UL Senate on 22.11.2016. If it is not possible for the candidate to be the first author of the publication, the mentor should provide written confirmation of the significance of the candidate’s contribution. The research article must be published or accepted for publication prior to the doctoral dissertation being submitted for evaluation. For the student to complete the doctorate, the dissertation and its defence must receive positive evaluations from the examining committee.

International cooperation:

FMF participates in the following professional organizations:

• AAPT (American Association of Physics Teachers)
• AGU (American Geophysical Union)
• AMPERE (Association for Microwave Power in Europe for Research and Education)
• AMS (American Meteorological Society)
• AMS (The American Mathematical Society)
• APS (American Physical Society)
• Biophysical Society
• EAS (European Astronomical Society)
• ECMI (The European Consortium for Mathematics in Industry)
• EMS (The European Mathematical Society)
• ENEN (European Nuclear Education Network)
• EPS (European physical society)
• FuseNet Association
• HOPE (Horizons in Physics Education)
• ILCS (The International Liquid Crystal Society)
• IOP (Institute of Physics)
• MAA (The Mathematical Association of America)
• SIAM (The Society for Industrial and Applied Mathematics)

FMF collaborates with numerous academic and research institutions. For example:

• CERN (Conseil Européen pour la Recherche Nucléaire)
• ETH (Eidgenössische Technische Hochschule Zürich)
• LANL (Los Alamos National Laboratory)
• Ludwig-Maximilians-Universität München
• MIT (Massachusetts Institute of Technology)
• University of Oxford
• Stanford University
• University of Cambridge
• University of California
• Weizmann institute, Izrael

FMF international cooperation: https://www.fmf.uni-lj.si/en/research/

The mobility of students and teachers is supported by the Erasmus programme:

• KLASIUS-SRV: Education leading to doctorate of science (third Bologna cycle)/Doctorate of science (third Bologna cycle)
• ISCED: Teacher training and education science, Physical sciences, Mathematics and statistics
• KLASIUS-P: Science (broad programmes)
• KLASIUS-P-16: Physics, Mathematics
• Frascati: Natural Sciences
• SOK level: 10
• EOK level: 8
• EOVK level: Third cycle

• Topics in analysis: Positive operators and Banach lattices (second semester, lecturer Marko Kandić)
• Topics in discrete mathematics: Regular maps (second semester, lecturer Primož Potočnik)
• Topics in mathematical foundations of computer science: Formalization of mathematics and proof assistants (first semester, lecturer Andrej Bauer)

Students can also choose from the following courses, which will be offered in the Master's programme in Mathematics:

• Topics in analysis: Fibre bundles (first semester, lecturer Riccardo Ugolini)
• Topics in algebra: Expander graphs (first semester, lecturer Urban Jezernik)
• Topics in topology: Topological robotics (second semester, lecturer Petar Pavešić)
• Stochastic processes 3 (first semester, lecturer Matija Vidmar)

Student's choice validates the coordinator of the programme.