Umeå University, Faculty of Science and Technology

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The Department of Mathematics and Mathematical Statistics conducts research in computational mathematics, discrete mathematics, mathematical modelling and analysis, and mathematical statistics. Our teaching is conducted at all levels and includes mathematics, mathematical statistics and computational science. Among our partners are international research groups, academic institutions, public organizations and companies.

The Department of Mathematics and Mathematical Statistics is opening a PhD position in mathematics, focusing on geometric deep learning. The position involves four years of doctoral studies, including participation in research and postgraduate courses. The last day to apply is August 15, 2022.

Project description and tasks
Deep learning models, particularly deep convolutional neural networks (CNNs), have enjoyed tremendous success on an impressive number of complex problems. However, a fundamental understanding of the mathematical descriptions of the models and their extensions to non-flat data are still lacking, presenting an exciting research problem spanning several areas such as differential geometry, numerical analysis, and dynamical systems.

A promising approach, referred to as geometric deep learning, is to account for the geometry of the input data by making the networks equivariant with respect to the symmetries of the data. This means that a transformation of the input produces a corresponding transformation of the output. Making such geometric structures manifest amounts to incorporating prior knowledge of the system to facilitate learning.

Another development explores the connection to differential equations and dynamical systems in the limit of infinitely deep networks. Formulated in terms of the dynamics propagating information through the network, the learning problem becomes amenable to powerful numerical techniques for differential equations and a rich theory of dynamical systems.

The project aims to unify the two approaches and develop the mathematical foundations of the emerging field of neural differential equations by exploring the connection to geometric deep learning. The objective is to develop a manifestly geometric formulation of equivariant neural differential equations, unifying symmetries of the data manifold and symmetries of the differential equations, and to investigate the properties and applications of such a formulation.

The project is a part of the AI-Math track within Wallenberg AI, Autonomous Systems and Software Program (WASP). The PhD student will participate in the WASP graduate school, see

The doctoral student will be admitted for studies at third-cycle level in Mathematics. To fulfil the general entry requirements, the applicant must have qualifications equivalent to a completed degree at second-cycle level or completed course requirements of at least 240 ECTS credits, including at least 60 ECTS credits at second-cycle level.

To fulfil the specific entry requirements to be admitted for studies at third-cycle level in mathematics, the applicant must have completed at least 60 ECTS credits within the field of mathematics, of which at least 15 ECTS credits shall have been acquired at second-cycle level. Applicants in some other system, either within Sweden or abroad, who have acquired largely equivalent skills are also eligible.

Good programming skills (preferably Python) and a good knowledge of the English language, both written and spoken, are required. Documented knowledge and experience in machine learning, differential equations, differential geometry or related areas are merits.

You are expected to take on an active role in developing this doctoral project and in departmental work. Therefore, you are expected to have excellent communication and collaboration skills. You have a scientific mindset, can work independently, and are structured, flexible and solution-oriented. Above all, you are analytical, creative, and committed to continuously developing your skills and contributing to the mathematical foundations of machine learning.

Assessments of the applicants are based on their qualifications and ability to benefit from the doctoral study they will receive.

About the position
The position is intended to result in a doctoral degree. The main task of the doctoral student is to pursue their doctoral studies, including active participation in research and doctoral courses, and participate in the WASP graduate school. The duties may include teaching or other departmental work (up to 20%). The employment is limited to four years of full-time (48 months) or up to five years when teaching part-time. Salary is set by the established salary ladder for PhD positions at Umeå University. The employment starts in January 2023 or according to an agreement.

Application is made through our recruitment system by August 15, 2022. Log in and apply via the button at the bottom of the page. The application must consist of the following documents written in English or Swedish:

  • Personal letter with a brief description of your qualifications and research interests. Justify why you are applying for the position and describe how your qualifications and merits are relevant to the employment.
  • Curriculum vitae.
  • Authenticated copies of degree certificates, diplomas or equivalent, including documentation of completed academic courses, received grades, and possibly other certificates.
  • Copies of relevant work such as master’s thesis or articles you have authored or co-authored. If the master’s thesis has not been completed before the application deadline, a summary of the master’s thesis project and current progress shall be included. The summary can be at most five pages, including figures and references.
  • GMAT (or GRE) and TOEFL/IELTS test scores, if available.
  • Contact information to at least two reference persons.

The Department of Mathematics and Mathematical Statistics values the qualities that a gender balance brings to the department, and therefore we particularly encourage female applicants.

The procedure for recruitment for the position is in accordance with the Higher Education Ordinance (chapter 12, 2§), and the decision regarding the position cannot be appealed.

More information
Further information can be provided by Associate Professor Fredrik Ohlsson ( or Professor Jun Yu ( You can also contact the Head of Department Professor Åke Brännström (, for additional information.

More information about the Department of Mathematics and Mathematical Statistics:

We look forward to receiving your application!

Type of employment Temporary position
Employment expires 2027-12-31
Contract type Full time
Number of positions 1
Full-time equivalent 100%
City Umeå
County Västerbottens län
Country Sweden
Reference number AN 2.2.1-1163-22
  • Fredrik Ohlsson, universitetslektor, 090-7865389,
  • Jun Yu, professor, 090-7865127,
Union representative
  • SACO, 090-7865365
  • SEKO, 090-7865296
  • ST, 090-7865431
Published 22.Jun.2022
Last application date 15.Aug.2022 11:59 PM CEST

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