General Study Syllabus for Mathematics

Mathematics

Swedish title: Matematik

TNMATE00

Responsible department: Department of Mathematics

Subject-specific parts (*) in Swedish adopted by the Faculty Board (the Doctoral Education Board) on 2014-05-14.

Faculty-common parts approved by the Faculty Board of Science and Technology on 2022-04-26. Revision on 2023-02-07.

The faculty-common parts go into force on 1 July 2022. The Faculty-common parts also apply to those who started their doctoral education prior to 1 July 2022, except if this would have a negative impact on the conditions for doctoral students.

Subject description *

Mathematics is a science of abstract nature developed for problem solving and development of various methods and models which aims toward studying and building structures of various kinds. It studies not only solutions of specific problems but deals also with questions about which problems admit and do not admit solutions, respectively. Mathematical structures are often motivated by concrete problems in other sciences but are most commonly formulated and studied from a more general viewpoint without immediate connections to applications.

The mathematical sciences include the following main areas: algebra, analysis, combinatorics, probability theory and geometry and topology. These are further subdivided into a vast number of subareas which, in spite their rich variation, have very strong interconnections. The choice of research area must be made individually by every doctoral student.

Information about current research at the Department of Mathematics is available at its homepage http://www.math.uu.se/

Aim and objectives for the education

Overall aim and objectives of the education

Doctoral education shall develop the knowledge and skills needed to be able to conduct research independently (Higher Education Act, 1992:1434). The Higher Education Ordinance’s System of Qualifications (Appendix 2, 1993:100) sets out the requirements to be met for a doctoral and licentiate degree, respectively (see individual study plan template).

Subject-specific objectives *

The doctoral level education should further expand the insights into the most central parts of the subject with advanced (Master’s) level education as starting point as well as lead to deep knowledge in at least one of its special areas. Through guidance and thesis work, the student will be well prepared to work as an independent researcher with the ability to critically analyze and evaluate the research of others or in other professions that require deep insights into the subject and research experience. The thesis should demonstrate independent thinking and include new results.

By conducting a degree project guided by a supervisor, the doctoral student shall also be able to present her/his goals and results orally and in writing to different target groups in English and, in the case of Swedish-speaking doctoral students, in Swedish.

Entry requirement

General entry requirements

General entry requirements for doctoral education are regulated in the Higher Education Ordinance as follows:

An applicant shall be considered as meeting the general entry requirements if they have

completed a degree at the advanced level (Master’s level), or
completed course requirements of at least 240 credits, of which at least 60 credits are at the advanced level (Master’s level), or
acquired substantially equivalent knowledge in some other way in Sweden or abroad.

The University may permit an exemption from the general entry requirements for an individual applicant, if there are special grounds (Chapter 7, § 39 of the Higher Education Ordinance).

In order for a person who has completed course requirements of at least 240 credits (under the second item above) to be considered eligible at Uppsala University, the 60 credits at advanced level must include a degree project of at least 15 credits (AFUU § 2, UFV 2022/729).

Special entry requirements *

A person who has passed at least 90 higher education credits worth of courses in mathematics or in subjects relevant for mathematics, or who has corresponding education from abroad, has special eligibility. Normally, a thesis of at least 15 higher education credits is also required.

Examples of educational programs which give special eligibility are the following:

Bachelor's program in mathematics (provided that basic eligibility is fulfilled)
Master's program in mathematics
Civil engineering with specialisation in technical physics

For other programs additional mathematics courses may be needed for special eligibility. For questions in individual cases contact the professor responsible for doctoral education.

Advertisement, selection and admission

Information and advertisement

Admission shall be made on a competitive basis following open advertisement of a doctoral place, with the exception of that which is specified in Chapter 7, § 37 of the Higher Education Ordinance. The advertisement shall be made available locally and on the University’s website (www.uu.se) at least three weeks before the application deadline and should be given appropriate national and international dissemination.

Selection

Selection among the applicants shall be carried out with consideration given to their ability to successfully conduct their studies. The assessment criteria for selection are:

knowledge and skills relevant to the thesis work and the subject/specialisation
assessed ability to work independently, for example
- ability to formulate and address scientific problems
- written and oral communication skills
- ability to carry out independent critical analysis
other experience relevant to doctoral studies, e.g. professional experience.

The assessment criteria may be demonstrated, for example, by supporting documents, an interview or a skills test.

In addition, an assessment is made of the applicant’s general competence and personal qualities, as well as their ability to collaborate. If a number of applicants are judged as being equal, preference shall be given to applicants from the underrepresented legal gender among the doctoral students in the subject/specialisation.

The mere fact that an applicant is deemed to be able to be awarded credits for prior studies or professional experience for the purpose of doctoral studies shall not give an applicant preference over other applicants during selection (Chapter 7, § 41 of the Higher Education Ordinance).

Admission

A doctoral student is admitted to a doctoral programme in a doctoral subject/specialisation. Admission of a doctoral student with full-time employment in their doctoral studentship at Uppsala University is decided by the head of the relevant department, except in cases specified in the Faculty’s guidelines for doctoral education (TEKNAT 2021/301). Admissions with other forms of funding are decided by the Faculty Board’s Working Committee after preparation in the Doctoral Education Board.

Structure and content of doctoral education

Doctoral education consists of courses and research work, and can take various forms, as specified in the individual study plan.

The research project must be well-defined and the level of ambition must be set taking into account both the degree objectives of the programme and the net study time (maximum 48 months for a doctoral degree or maximum 24 months for a licentiate degree).

Requirements for doctoral degree *

The requirements for the doctoral degree consist of passed examinations in the courses included in the approved individual study plan of each doctoral student, as well as a passed public defense of the degree project. The studies awarded a doctoral degree comprise 240 higher education credits (four years of full-time studies), of which the doctoral thesis comprises 120 higher education credits and the course part 120 higher education credits.

Requirements for licentiate degree *

A doctoral student who has acquired at least 120 higher education credits (two years of full-time studies) is eligible for a licentiate degree. The requirements consist of passing the examinations included in the program stage and receiving a passing grade on an academic paper of at least 60 higher education credits. The part of the course amounts to a minimum of 120 higher education credits.

Supervision

The head of department is responsible for ensuring that sufficient time is allotted for the department’s doctoral students to receive the necessary supervision. A doctoral student has the right to request a change of supervisor (Chapter 6, § 28 of the Higher Education Ordinance).

Individual study plan

The principal supervisor, in consultation with the professor responsible for doctoral studies (FUAP), is responsible for drawing up an initial individual study plan prior to admission. The head of department approves the study plan in connection with admission. The individual study plan shall contain a timetable for the doctoral studies, specification of how supervision is organised, and a description of the undertakings of the doctoral student and the department during the period of studies. The individual study plan also specifies the courses included in the doctoral student’s education.

The individual study plan must be revised at least annually in collaboration between the doctoral student and their supervisor. The revision involves following up on the doctoral student’s progress in relation to the degree objectives and previous plan, as well as planning for further studies. The FUAP approves the revised study plan.

Courses

Faculty-common courses

A course in research ethics, with a minimum of 2 credits and content equivalent to the courses provided by the Faculty, shall be included for the licentiate and doctoral degrees. An introductory course to doctoral studies and a course in scientific writing are also recommended.

Doctoral students who teach should undergo teacher training for higher education. This is specified in the individual study plan, and can either be a credit-bearing course element or take place within the framework of the doctoral student’s departmental service.

Teacher training for higher education, lasting 5 weeks, is equivalent to 7.5 credits at the Faculty, and may be included in the doctoral programme.

Basic Swedish language training for doctoral students who do not have Swedish as their first language may be a credit-bearing course element or take place within the framework of the doctoral student’s departmental service.

Subject-specific courses *

The doctoral studies may include different kinds of courses, such as lectures, literature studies, practical training, field studies, etc. The courses are intended to provide necessary background for the thesis work as well as wider insights into the subject and thereby serve as a complement to the more specialized thesis work. Courses should range over a variety of different subareas, in particular each student should pass at least one course in algebra, analysis, geometry and probability theory. The courses may either be doctoral level or advanced level courses in mathematics or other courses in mathematics and other subjects.

The courses in the individual study plan are to be decided by the student and the main supervisor.

The range of courses offered is revised continuously. Examples of advanced courses are the following: Homological algebra, Lie groups, Measure theory and integration, Functional analysis, Partial differential equations, Analysis on manifolds. Examples of pure research courses are the following: Doctoral course in Algebra; Doctoral course in Analysis; Doctoral course in Topology and Geometry; Doctoral course in Probability Theory and Statistics; Algebraic geometry, Category Theory, Symplectic Geometry.

Course examiner

For doctoral courses, the teacher responsible for the course normally serves as the examiner. The head of department appoints an examiner for a doctoral course. The task of the examiner is only to determine grades on examinations. The principal supervisor, in consultation with the FUAP and other supervisors, decides which courses and to what extent (number of credits) the doctoral student is allowed to be credited for in their doctoral studies, and this is documented in the individual study plan.

Credit transfer

It is up to the principal supervisor, in consultation with the FUAP and other supervisors, to decide which examinations (courses or other elements carried out during the study period of the doctoral education) the doctoral student will be allowed to count towards their doctoral studies and to what extent (number of credits). This shall be documented in the individual study plan. The task of the examiner is only to determine grades on examinations.

An assessment must be made as to whether credits can be awarded for prior studies or professional or vocational experience (that have been completed prior to admission to doctoral education) (Chapter 6, § 8 of the Higher Education Ordinance). At the Faculty, the FUAP is responsible for the assessment.

Thesis, doctoral defence and licentiate seminar

Thesis

The research work shall result in a scientific thesis that must be defended at a public doctoral defence. The research task may be carried out individually or in collaboration with others within or outside the department. However, the doctoral student must be trained to conduct independent research.

The doctoral thesis can be designed either as a monograph, i.e. a unified, coherent scientific work, or as a compilation thesis, i.e. a compilation of scientific papers with a summary of these. The thesis work must be equivalent to at least 120 credits (Chapter 6, §§ 4–5 of the Higher Education Ordinance, and Appendix 2, System of Qualifications). Theses within the Faculty shall include a popular science summary that is in Swedish and at least two pages long.

The doctoral thesis must either meet the requirements for publication in an international scientific journal with independent quality review, or be a summary of scientific papers with equivalent quality requirements. If the doctoral student has co-authored a paper with another person, this may be taken into account only to the extent that the individual effort can be distinguished. This should be done through a description of the doctoral student’s contribution in the papers on which a compilation thesis is based. If parts of the work have previously been published by the doctoral student in a licentiate thesis, this shall be made clear.

The licentiate thesis is smaller in scope, but is subject to the same quality requirements for constituent papers as apply for the doctoral thesis.

In consultation with the FUAP and other supervisors, the principal supervisor shall assess when the thesis work has progressed to the point that a date for doctoral defence or licentiate seminar can be set.

Doctoral defence

The forms of defence and the grading of doctoral theses are regulated in the Higher Education Ordinance, the Faculty’s Guidelines for doctoral education (TEKNAT 2021/301) and Admission and grading regulations for doctoral studies at Uppsala University (UFV 2022/729).

The doctoral thesis shall be defended orally in a public doctoral defence. An examining committee appointed by the Faculty decides on the grade for the doctoral thesis.

Licentiate seminar

The Faculty’s guidelines for doctoral education (TEKNAT 2021/301) summarise the rules for the licentiate seminar.

The grade for a licentiate thesis shall be determined by the FUAP, or another professor delegated this duty, in consultation with the principal supervisor and the external reviewer. The principal or assistant supervisor for the doctoral student may not serve as examiner.

Degree*

The following degree titles have been established for Mathematics:

Filosofie licentiat- och doktorsexamen

The English translation is Degree of Doctor/Licentiate of Philosophy.

Both the subject and the specialisation are listed on the degree certificate. The degree title (Teknologie/Filosofie licentiat- och doktorsexamen [Degree of Doctor/Licentiate of Philosophy]) shall be determined by the content of the doctoral education, and not by the doctoral student’s degree from a qualifying programme. If a doctoral student wishes to change their degree title to one that differs from that established for the doctoral subject dispensation is required from the Faculty Board (Working Committee). The request for a change of degree title must be made no later than the time of opponent and examining committee appointment or submission of the thesis for printing, whichever occurs first.

Doctoral and licentiate degree certificates are issued upon application in Ladok.

Regulatory framework and responsibilities for doctoral education

Doctoral education is regulated in the Higher Education Act (1992:1434) and the Higher Education Ordinance (1993:100). These are supplemented by the following local regulations: Guidelines for doctoral studies at Uppsala University (UFV 2022/728), Admission and grading regulations for doctoral studies and study programmes at Uppsala University (UFV 2022/729) and Guidelines for doctoral (third cycle) education at the Faculty of Science and Technology (TEKNAT 2021/301).

Responsibility for doctoral education ultimately rests with the University Board and the Vice-Chancellor (Chapter 2, §§ 2–3 of the Higher Education Ordinance). Through delegation, the Disciplinary Domain Board or Faculty Board has overarching responsibility and supervisory responsibility, but the day-to-day responsibility is exercised by the department in which the doctoral student is registered. Key functions in doctoral education are the head of department, professor responsible for doctoral studies (FUAP), director of doctoral studies, and the supervisor. See the Faculty’s Rules of Procedure (TEKNAT 2019/177) for a description of roles and responsibilities.

Other*

Research in mathematics is often conducted in cooperation with researchers from other countries and other subjects. It is therefore desirable that the student has the ability to read scientific works from other disciplines and in other languages. The student should know how to write mathematical texts in English and Swedish.

The thesis work is individual but cooperation with other doctoral students is very valuable. This is encouraged e.g. through doctoral student seminars.

During the education doctoral students should take part and present their research results at international research conferences.

The doctoral students should also take notice of the mathematical activities at the Mittag-Leffler Institute in Djursholm, Stockholm University and the Royal Institute of Technology in Stockholm. In order to widen the perspective, to learn about applications, and to increase the ability to use mathematics outside pure mathematics, contacts with other subjects at the university are valuable. This applies to mathematical statistics and applied mathematics, as well as to subjects outside the department, e.g. biology, computer sciences, economy, geo sciences, chemistry, medicine, technical sciences, theoretical physics, and theoretical philosophy.