Postdoctoral scholarship (2 years) within mathematics focusing on geometric deep learning

This scholarship is part of a more significant expansion of the research group at the department investigating mathematical foundations of artificial intelligence. The group covers a diverse range of topics in modern machine learning research, including geometric deep learning, non-convex optimization problems and federated learning. In addition, this project is deeply connected with the geometry group at the department, a recently formed and inspiring research environment that is also expanding.

Project description

Deep learning has enjoyed tremendous success on an impressive number of complex problems. However, the fundamental mathematical understanding of deep learning models is still incomplete, presenting exciting research problems spanning areas such as differential geometry, numerical analysis, and dynamical systems. Neural ordinary differential equations (NODEs) mark a recent advance in geometric deep learning, the pursuit to incorporate symmetries and non-Euclidean structures in machine learning using geometrical principles. NODEs describe the dynamics of information propagating through neural networks in the limit of infinite depth using ordinary differential equations (ODEs) on manifolds and offer several appealing properties.

The dynamical systems in NODE models are constrained, however, in that the intrinsic nature of the dimension of a manifold fixes the dimension of their state vector. This limitation precludes the use of certain architectural elements, like the encoder-decoder structure used in autoencoders and sequence-to-sequence prediction, and applications where the dimensionality of the state space changes dynamically, like quantum mechanical systems interacting with classical external fields where quantization effects cause freeze-out of degrees of freedom.

To remedy these limitations, the overarching goal of this project is to accommodate variable dimension dynamics in geometric deep learning by extending NODEs from manifolds to M-polyfolds, a generalization of manifolds where the number of local coordinates is allowed to vary smoothly. This requires the development of a comprehensive geometric framework for flows and integral curves on M-polyfolds and a theory of group actions compatible with the M-polyfold structure.

The appointed candidate is expected to carry out collaborative research according to a mutually agreed research plan. They are also expected to present their research at international conferences and actively participate in joint workshops and seminars.

The scholarship holder will be based at the Department of Mathematics and Mathematical Statistics at Umeå University. The postdoctoral scholarship is funded and administered by the Kempe Foundation (JCSMK24-0043). The stipend will be 737 000 SEK for two years.

Qualifications

To qualify as a postdoctoral scholarship holder, the postdoctoral fellow is required to have completed a doctoral degree, or a foreign degree deemed equivalent to a doctoral degree. This qualification requirement must be fulfilled no later than at the time of the decision about the scholarship recipient.

The doctoral degree should be in mathematics or be deemed to provide equivalent academic competence.

Priority should be given to candidates who completed their doctoral degree, according to what is stipulated in the paragraph above, no later than three years prior. If there are special reasons, candidates who completed their doctoral degree prior to that may also be eligible. Special reasons include absence due to illness, parental leave, appointments of trust in trade union organizations, military service, or similar circumstances, as well as clinical practice or other forms of appointment/assignment relevant to the subject area.

The successful candidate is expected to have a strong background in differential geometry. Good programming skills and a strong interest in deep learning are required. A strong publication record and previous experience with differential equations and machine learning are meritorious.

A strong command of the English language, both written and spoken, is required, as are excellent communication and collaboration skills. The successful candidate should be committed to continuously developing their skills and contribute to the mathematical foundations of geometric deep learning.

Application

A full application should include:

• Cover letter in which you describe your research background and future ambitions and how they fit the described project (max three pages),
• Curriculum vitae (CV) with publication list,
• Verified copy of doctoral degree certificate or documentation that clarifies when the degree of doctor is expected to be obtained,
• Verified copies of other diplomas, list of completed academic courses and grades,
• Copy of doctoral thesis and relevant articles (max three articles),
• Other documents that the applicant wishes to claim,
• Contact information to two persons willing to act as references.

Submit your application as a PDF marked with the reference number FS 2.1.6-867-24, both in the file name and in the subject field of the email, to medel@diarie.umu.se. The application can be written in English or Swedish. Application deadline is 31 May 2024. 

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

Additional information

Additional information regarding the scholarship and the project can be provided by Associate Professor Fredrik Ohlsson (fredrik.ohlsson@umu.se).

More information about the Department of Mathematics and Mathematical Statistics can be found at https://www.umu.se/en/department-of-mathematics-and-mathematical-statistics/.

We look forward to receiving your application.