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The central topic of the dissertation, large scale geometry, has a link to Copernican principle known in physics and also in philosophy. This principle is named after Nicolaus Copernicus who realised at the beginning of 16th century that Earth is not the centre of the universe that the other celestial bodies are orbiting. When astronomy has continued to progress, it has become ever more clear that Earth is not a special place in the universe. Nowadays, Copernican principle is understood to mean that the space is the same everywhere, i.e., the space is homogeneous.

Taken precisely, the space is not the same everywhere of course. Our own Sun, as well as all the stars and smaller celestial bodies, make the space curved around them. Generally speaking, the space is the more curved the closer one gets to a star. However, cosmology has shown that Copernican principle works extremely well on the large scale, that is, ignoring "small local phenomena" like star systems or galaxies. At the large scale the space appears as a wide field of grass that can contain small irregularities when examined close by but that appears the same everywhere looking from further away. In other words, the large scale geometry of our space is homogeneous.

Different spaces

In common usage, the space refers to the emptiness that surrounds us and to the place where the movement of bodies happens. The structure of the space is described by the general theory of relativity by Albert Einstein. In mathematics, however, there are multitude of spaces. A space may refer to any idealised structure with geometric phenomena. The dimension of a space may be any number whatsoever, a space may have arbitrary curvature properties, and it may even posses movement restrictions or other peculiar characters.

We may illustrate a space with movement restrictions by examining a car driving on a parking lot. By the state of the car we refer to the information about the location of the car at the parking lot and to its orientation. The state of the car is therefore determined by three parameters: two position coordinates for the centre of the car and one angular coordinate to describe where does the front of the car point to.
The space of states is therefore 3-dimensional, but the space has movement restrictions, since the car cannot move sideways or rotate on spot. By driving the car by a suitable manner, it is still possible to reach any state of the car at the parking lot as it is easy to show in practice.

Treating such spaces with movement restrictions requires more advanced mathematical tools than those used traditionally in geometry, and some results of the thesis indeed generalise results known in "usual" curved spaces (s.c. Riemannian manifolds) to this kind of more general setting.

Listing homogeneous spaces

The thesis investigates in particular what kind of homogeneous spaces (at the large scale) can exist, mathematically speaking. This mathematical existence refers to finding out which of such spaces are logically possible, without paying attention whether they appear in our physical reality or in models of it. An important class of homogeneous spaces is formed by Lie groups that are spaces introduced by the Norwegian mathematician Sophus Lie at the end of 19th century.

One of the results of the thesis shows that Lie groups may model homogeneous spaces under certain assumptions. The other results of the thesis concentrate on the question when two different Lie groups may model the same homogeneous space. When one restricts to consider for example only 4-dimensional spaces, these model spaces can be explicitly listed. The thesis presents such lists in dimension 4 and 5 under certain assumptions on Lie groups in question. Corresponding lists are previously known only in dimensions 1, 2 and 3.

The research is published JYU Dissertations series, number 376, Jyväskylä, 2021.
ISBN 978-951-39-8629-2 (PDF), URN:ISBN:978-951-39-8629-2, ISSN 2489-9003
Link to publication:

M.Sc. Ville Kivioja defends his doctoral dissertation in Mathematics "On metric relations between Lie groups" on Friday 14 May 2021 starting at 2 pm. at the Ä¢¹½Ö±²¥. Opponent Professor is Xiangdong Xie (Bowling Green State University, USA) and Custos Senior Lecturer Katrin Fässler (Ä¢¹½Ö±²¥). The doctoral dissertation is held in English.

The audience can follow the dissertation online.
Link to the Zoom Webinar (Zoom application or Google Chrome web browser recommended): 

Phone number to which the audience can present possible additional questions at the end of the event (to the custos): +358 50 4673382