Mapping randomness in a flat world  

Erik Duse received his Ph.D. in Mathematics from KTH Royal Institute of Technology in 2015. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Kari Astala and Professor Eero Saksman at the Department of Mathematics and Statistics, University of Helsinki, Finland. 

Interesting mathematical patterns can be encountered in all natural sciences. Even in seemingly random phenomena, some regularity can be found. Many random physical processes can therefore be described by applying probabilistic methods. A relatively simple model examines a flat surface covered with regularly shaped tiles. Such probabilistic tiling models have proven an enormously fruitful source of mathematical methods for statistical mechanics. One of the advantages of such models is the great number of possible patterns they can form. For example, there are almost 13 million possible ways (12,988,816) to place domino tiles on a square board consisting of 64 fields. 

If a plane surface’s geometrical form is not regular, specific geometric patterns can be found at the boundaries between random and ordered areas. For example, if the surface is sufficiently large, the boundary can form a perfect circle. Studying the structures of large tiled surfaces is one of the goals of the project. 

Random tilings share some properties with physical models that can be described by variational theory. Researching variational problems is another goal of the project. Such methods, which attempt to establish the largest or the smallest value for a certain quantity, for example the lowest energy level, have a long history. Another typical example of such a problem in physics is explaining why a soap bubble is spherical – because a sphere has the minimum surface area.  

In order to study such problems, a surface is constructed to represent the tiling. As in the case of flat tilings, the surface is also random. Such surface construction gives rise to a natural variational problem. Researching this problem, its geometric interpretations and the relationship with the original random patterns is the goal of the project.