Aristotelis Panagiotopoulos Explores Dynamics of Infinity

Aristotelis Panagiotopoulos' research is in mathematical logic and topological dynamics. He is interested in the foundations of mathematics and the complexities which arise in any attempt to formalize infinity.

"Ask a young child what the largest number is and then respond to their number by adding one. Their impish grin reveals that you share a universal human concept - that of infinity in its potential," said Panagiotopoulos, a postdoctoral associate in Carnegie Mellon University's Department of Mathematical Sciences. "Even though we never encounter anything infinite in the physical world, we still share a universal innate notion of infinity. I always found this fascinating. Perhaps this is what lead me into mathematics. Mathematics allow us to formalize infinity and do various things with it."

Panagiotopoulos came to Carnegie Mellon because of its long tradition in logic, set theory and combinatorics.

"At CMU there is a very strong group in logic," he said. "It's one of the best in the world. I wanted to be part of this, spend time here and work with the people here. The level of the undergraduate students at CMU is also very high and you can engage in research with them."

At Carnegie Mellon, Panagiotopoulos taught courses such as algebraic topology, calculus II and dynamics of Polish groups. Panagiotopoulos will be joining the University of Vienna as an assistant professor beginning in January 2024, but he will stay connected to Carnegie Mellon through a National Science Foundation grant. The NSF award will support his research related to dynamics beyond turbulence and obstructions to classification until 2025.

"One thing I find fascinating about logic is that it provides a framework for identifying incompleteness phenomena, and establishing which mathematical problems are too complex to be solvable within the context in which they were first conceived." he said. For example, a person given a straightedge and a compass will find it impossible to trisect an arbitrary angle with just those tools.

"The study of this sort of incompleteness phenomena is the 'obstructions' part in the title of the NSF award," he said. "The 'dynamics' part refers to the tools and methods that I use."

Panagiotopoulos worked with students during the summer of 2023 developing various examples where dynamical obstructions are applicable. One student worked to understand certain dynamical phenomena which appear in geometry and the other worked on similar phenomena in topology.

"I'm hoping there will be a couple of students next summer interested in working on the NSF research program either through Zoom, myself coming to Pittsburgh or them flying to Vienna," he said. "I often work on research projects where I can learn from the person I'm collaborating with. This is especially important with younger students. Very quickly they get a sense that they are not talking to an authority but a collaborator. That helps them stand on their feet and gain confidence."

Panagiotopoulos' interests in research extend beyond the field of mathematics and into other fields such as physics, where he recently published a paper in Physical Review Letters with collaborators from the University of Pittsburgh and the University of Vienna's Institute for Quantum Optics and Quantum Information. This paper is about incompleteness theorems for observables in general relativity.

"It has been a longstanding open problem general relativity how to identify which quantities are 'physical.' That is, which quantities do not depend on the coordinate system," he said. For example, in Newtonian mechanics, you can study the motion of a certain object by fixing a coordinate system and using it to write down equations for this motion. However, the physics of this motion should not depend on the coordinate system that you fixed.

"If you rotate the system and your axis rotates, the equations that you write for it to do the exact same thing will look a little bit different, but they will be equivalent," he said. "When you get to theories such as general relativity, what happens is that the change of coordinates involves much more complex dynamics. In Newtonian mechanics, you take space and time for granted. In general relativity you are trying to describe space/time itself, you cannot take it as a background, it's much more complex."

Panagiotopoulos and his collaborators will continue to explore the connection between theoretical physics and set theory.

Prior to arriving at Carnegie Mellon, Panagiotopoulos worked as a postdoctoral researcher at the University of Münster in Germany and as a research instructor at the California Institute of Technology. He earned his doctorate in mathematics from the University of Illinois at Urbana-Champaign.