A dissertation work on theoretical shortcomings in geometries continues across the pond. The shift of focus from rocky to gas planets lays groundwork for future computations.
“Everyone who works on inverse problems has come across someone from Finland often through direct collaboration. We Finns are appreciated here in the US”, says postdoc Antti Kykkänen about his new academic community in Houston, Texas.
The Finnish expertise on inverse problems, cultivated over decades in close collaboration with different institutions, is highly appreciated and sought after internationally. With his research work, Kykkänen is also contributing to this pool of knowledge.
Kykkänen defended his doctoral thesis work at the University of Jyväskylä in early August. Supervised by FAME Principal Investigator Joonas Ilmavirta, Kykkänen’s dissertation “Geodesic X-ray Transforms in Non-smooth Riemannian Geometries” focused on a geometric generalisations of the classical problem of recovering an unknown function from its known integral data.
Problems such as these typically arise from problems in seismologic research. A good example is measuring the travel times of seismic waves after an earthquake, which can be modelled as geometric information on the boundary of a planet. This is where the theoretical work comes in.
“Usually, a theoretical mathematician makes use of idealizations, so we handle geometries of smooth surfaces or manifolds”, says Kykkänen. “However, the internal structure of a planet such as the Earth is anything but smooth, with multiple layers that cause rapid changes in the geometry. We took the theory’s shortcomings as a starting point and were able to introduce some new and fresh approaches.”
Hybrid theory
The example mentioned above is done by utilizing manifolds, a special type of mathematical structures called topological spaces. Many geometric shapes are manifolds. Classical examples of such are the line and the plane, the sphere, and the torus, which is the surface of a donut. In his research Kykkänen focused on so-called Riemannian manifolds. Named after German mathematician Bernhard Riemann, these manifolds have a built-in measuring stick known as Riemannian metric, which makes it possible to measure geometric quantities such as distances, angles, curvature, and volume on these various types of geometric shapes.
While this theoretical framework is well-known and utilised, the results of Kykkänen’s dissertation work introduces completely new approaches in the research of geometric inversion problems.
“Traditionally, mathematical theories in seismology have been worked on with terrestrial planets, which is evident in the fact that all classical models are based on them. So, what we did is that we started to think about what kind of changes the typical working process would go through should we shift our focus from rocky terrestrial planets to gas planets instead”, Kykkänen describes.
Since gas planets don’t have well-defined hard surfaces, the density of matter near the surface is close to zero, so seismic waves would travel significantly slower before finally stopping entirely. Because of this, Kykkänen participated in a process to build new geometric model that would be more suitable to investigate gas planets. One reason to do so is to understand how wave propagation on gas planets differs from that of terrestrial planets.
“While it may still take many years, my hope is that someday this theoretical work can be integrated into practical applications and real-world research.”
“Rapakon takana”
Kykkänen is currently working as a postdoctoral researcher at the Rice University’s Department of Computational Applied Mathematics and Operations Research. His supervisor, Professor Maarten V. de Hoop, had participated in Kykkänen’s dissertation work, and was eager to continue with theoretical changes that occurred when focusing on gas planets.
“This research is highly theoretical, but thanks to initiatives such as NASA’s Juno mission on Jupiter, we are now receiving seismological data from gas planets as well. This means that now is a good time to really dig into the theory, since this will ensure that the computational work will lead to accurate results once we begin to pick the data apart”, Kykkänen explains.
While the change of scenery did not happen without culture shocks, Kykkänen has adjusted well to his new home. While the overall tempo can get quite more intense than in the Finnish academia, the persistent mental image of productiveness-and-business-before-all that often gets attached to the American society and Americans has not materialized. Kykkänen gives praise to Rice’s academic community and how it provides time to focus on the research and opportunities to meet and socialize with fellow faculty members across the fields.
“Ideas flow from theoretical to applied mathematics, and vice versa. Everybody here understands that there is a long road from theory to practical applications and each step along the way is equally important.”
Photo: Antti Kykkänen / University of Jyväskylä
