The Department of Mathematics and Mathematical Statistics is opening a PhD position in Mathematics focusing on complex differential geometry and optimal transport. The position covers four years of third-cycle studies, including participation in research and third-cycle courses. The last day to apply is June 10, 2024.

This recruitment is part of larger expansion of the research group in Geometry. The group, which currently consists of five senior researchers, three doctoral students and one postdoc, constitute a young environment with an inclusive and friendly atmosphere and during the following year it will increase in size by three doctoral students and three postdocs.

Project description and tasks:
In the 80’s and 90’s a surprising phenomenon was observed by physicists: It seemed like certain complicated geometric objects (Calabi-Yau manifolds) appeared in pairs, one taking the form of a mirror image of the other. The phenomenon was dubbed mirror symmetry and similarly as for other duality principles in mathematics, for example the duality of time and frequency in Fourier analysis, researchers quickly realized it could be very useful. An important branch of research with the aim of understanding mirror symmetry is the SYZ-conjecture, which gives a detailed description of a conjectural structure in Calabi-Yau manifolds. To show that this description is true has turned out to be very difficult. In essence, the problem consists of controlling the limit of solutions to certain (Monge-Ampère) partial differential equations when the dimension of their domain drops, which is a very challenging problem in general. However, due to a recent breakthrough the problem has been reduced to showing existence of solutions to a class of these equations in very singular settings. The purpose of this project is to show existence of these solutions and study their properties to learn more about mirror symmetry. The main new idea is to use optimal transport to achieve this. Optimal transport is a classical tool with roots in 18th century France, where it was studied in relation to engineering and geometry. Here, it will provide a link between the partial differential equations above and a very robust variational theory which, as indicated by preliminary results, might be very useful.

The doctoral student will carry out research in complex algebraic geometry and optimal transport as part of the project above, which is funded by the Swedish Research Council. As part of this, they will write articles (both as single author and in collaborations) and publish in international journals. The project will also present many opportunities to travel and take part in conferences and workshops with possibilities to initiate collaborations with researchers at other universities, both in Sweden and internationally. Upon completion of their doctoral degree, they will be in a good position to apply for national and international postdoc grants, for example the Mathematics Program by the Knut and Alice Wallenberg Foundation or the Marie Skłodowska-Curie Postodctoral Scholarships by the European Research Council.

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://umu.varbi.com/en/what:job/jobID:721682/