This years logo was designed by Professor Daniel Cohen of the LSU mathematics department.
Each of these numbers is the topological complexity of the corresponding surface. This number gives a measure of the "navigational complexity" of the surface. Roughly speaking, it is the smallest number of rules required to robustly construct paths between pairs of points on the surface. The values for the sphere, the torus, and the real projective plane have been known for some time, but finding the topological complexity of the Klein bottle proved to be difficult. It was only determined recently, in a 2017 paper written by Lucile Vandembroucq and Dan Cohen.
Many thanks go to graduate student Samuel Wilson who modified the original design and made it suitable for the website and t-shirts.