Geometry at Infinity – Mathematicians of the University of Potsdam are researching actually unimaginable things
Prof. Dr. Christian Bär holds the professorship for geometry at the University of Potsdam and investigates differential geometry and its neighboring and application areas. As co-initiator and deputy speaker of the Priority Program (SPP) 2026 of the German Research Foundation (DFG), he describes the fascination of exploring infinitely extended geometric objects.
Research at infinity. Is that possible? How can people approach something that has no end? Investigate it, describe it, analyze it? Mathematics can do that, say mathematicians. “The strength of mathematics is that it can describe things completely beyond our imagination,” says Bär. However, he says, it is permissible to work with auxiliary ideas. “For example, if you take a particle in the plane that moves randomly, this particle always returns to its starting point.” This would not happen in an infinite, three-dimensional space, where the particle would get lost in infinity, so to speak. Not only space but also time can go into infinity. “We are investigating what this would mean for the solutions of important equations,” Bär describes. To do this, mathematicians work with proofs. “This is, of course, the tool of the trade of every mathematician, our daily bread,” he says. A proof is a mathematical derivation to decide whether a statement is true or false. “Let’s assume that we want to find out if a certain surface can be deformed so that it is positively curved everywhere. If I can’t get it right, that doesn't mean it can’t be done.” Bär radiates enthusiasm. “And now it gets really exciting: the solutions to the Dirac equation from physics can help us answer this purely geometric question.”
Whether there are solutions to such geometric questions and, if so, what they look like, is being analyzed in the SPP 2026. The six-year priority program comprises 80 individual research projects at more than 20 German and Swiss universities and brings together experts from differential geometry, geometric topology, and global analysis to address current issues across disciplines.
Waves on curved spacetime
In his own project during the first funding period 2017-2020 with the title “Index Theory on Lorentzian Manifolds,” Bär studied solutions of hyperbolic equations on curved spacetimes. “In general relativity theory, for example, you have a four-dimensional curved spacetime. We then look at how waves behave on it. These solutions describe what a physical system will look like in the future.”
Currently, Professor Bär and his team are working on the project “Boundary value problems and index theory on Riemannian and Lorentzian manifolds”. Here, a new aspect is added. “We mathematically study the question: what happens at the edge of space? Do the solutions of the equations behave nicely there, or do they get wild as we approach the edge?” the mathematician asks.
Two other projects are funded at the University of Potsdam within the current funding period 2020-2023. Prof. Dr. Jan Metzger and Prof. Dr. Carla Cederbaum from the University of Tübingen work on the topic “Geometrically defined asymptotic coordinates in general relativity”, Prof. Dr. Matthias Keller, Prof. Dr. Daniel Lenz, and Dr. Marcel Schmidt from Friedrich Schiller University Jena are investigating “Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces”
Key aspect: promoting junior researchers
As deputy speaker of the SPP 2026, Bär and speaker Prof. Dr. Bernhard Hanke (University of Augsburg) and Prof. Dr. Anna Wienhard (Heidelberg University) form the current program committee. In the first funding period, Prof. Dr. Burkhard Wilking (University of Münster) was also a member. “The four of us submitted the proposal to the DFG at that time,” he recalls. “The individual projects have their budgets, and there is an extra pot for funding workshops and conferences, which is at the disposal of the program committee.” These funds can be used very flexibly.
Our conversation takes place at the end of October 2021, shortly before the SPP 2026 kick-off meeting in Nuremberg where all project leaders will present their approved projects for the second funding period. Professor Bär is looking forward to it. “Contacts often happen by chance, during breaks, over coffee – you can’t simulate that online. Such networking events are very valuable, especially for young people. Quite a few of the project leaders are still at the postdoc level.” The priority program supports both individual research projects and overarching research activities such as seminars, conferences as well as invitations of cooperation partners from abroad. Professor Bär, together with his whole team, also organizes a block seminar once a year together with Bernhard Hanke’s research group at the University of Augsburg. “Bachelor and master students can also take part in this seminar and establish their first contacts.” Endless possibilities, in other words, for junior researchers in mathematics, who are particularly promoted within the SPP 2026.
“Boundary value problems and index theory on Riemannian and Lorentzian manifolds” within the DFG Priority Program „Geometry at Infinity”
Participants: Prof. Dr. Christian Bär (project leader), Dr. Lashi Bandara (until July 2021), Dr. Mehran Seyedhosseini, Penelope Gehring, Rubens Longhi, Sebastian Hannes
Funding: German Research Foundation (DFG)
Duration: Sep 2020 – Sep 2023
Prof. Dr. Christian Bär studied mathematics at the universities of Kaiserslautern und Bonn. Since 2003, he has been Professor of Geometry at the University of Potsdam, and since 2017, he has been Deputy Speaker of SPP 2026 “Geometry at Infinity”. In 2011/12, he was President of the German Mathematical Society.
This text was published in the university magazine Portal Wissen - Eins 2022 „Zusammen“.