The presentation on the Lefschetz fixed point theorem reminded me about Riemann Hurwitz, something I discussed with Thomas Tucker in 2012. It deals with an important topic in geometry: what happens if one quotients out a group acting on a geometry. This is very rich because it covers things like orbifolds, the case where a finite group acts on a manifold or a branch cover, or a fibre bundle in which the base can be seen as quotioning out the fibre. In a finite setting, one can look at a finite group A acting on a simplicial complex G. Riemann Hurwitz tells that the Euler characteristic of G/A times the order of A is equal to the Euler characteristic of G plus a correction R, which is the ramification part.
The prototype is to see the q-sphere G as a 2:1 ramified cover over the q-ball where the ramification part is the rim of the ball, the place where the 2:1 cover degenerates to a 1:1 cover. That obviously is the fixed point set of the nontrivial part of the 
I got recently reminded about Tom Tucker in the context of the Moebius-Kantor graph, as Tucker is known to have proven that the automorphism group of that graph is the only genus 2 group. This group is now known as the Tucker group. I find it always remarkable if in some category, a single object can be singled out by a single property. An other example is that the 3-sphere is the only compact manifold that can be made into a nonabelian group such that the group actions are smooth. The father of Thomas Tucker was Albert Tucker, who was the PhD dad of eminent game theorists like David Gale, John Nash, Lloyd Shapley and Marvin Minsky. That naturally came up in our “god number project” from last spring even so the focus on that work was mostly on solitaire and 2 player games.
I’m at the moment trying to finish 12 topics in geometry. The Riemann Hurwitz theme is planned to be included in a second batch of 12 topics if fate will allow this. But too ambitious project almost always tank, so that I focus now on the first 12.
