The Department of Mathematics and Mathematical Statistics is offering a postdoctoral scholarship in mathematics focusing on geometric deep learning. The scholarship is full-time for two years commencing 1 September 2024 or by agreement. The last day to apply is 31 May 2024.

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.

Umeå offers excellent working and living conditions. The city is young and located right next to a large river. It is surrounded by forests and lakes and lies close by the sea. In the vicinity there are plenty of opportunities for both indoor and outdoor activities.

For further information and instructions on how to apply, see:
https://www.umu.se/en/work-with-us/postdoctoral-scholarships/6-867-24/