A gallery of some nice pictures that showed up in my math research.
This picture shows the set of Anosov representations from a triangle reflection group into the Lie group SL(3,R). The character variety (or more precisely one of its components) is homeomorphic to the twice punctured Riemann sphere. The two punctures 0 and infinity are at the top and bottom of the bright vertical lines in the middle. The central black region is an approximation of the set of Anosov representations, obtained by checking all words up to length 23 for equal eigenvalue moduli. The black vertical line between the two punctures is the (real) Hitchin component. This is joint work with David Dumas.
This picture is like the above, but shows a different component of the (5,5,5) triangle group. Here the central horizontal line corresponds to real representations, and we know exactly which representations on this line are Anosov: this is the result of my paper with Gye-Seon Lee and Jaejeong Lee. The situation becomes much more complicated in the complex numbers, and the set of Anosov representations seems to have a fractal boundary. The picture is again joint work with David Dumas.
This picture shows a phenomenon I am investigating together with Jeff Danciger. It shows the 2d character variety of a triangle rotation group into the group SL(3,R). For each representation, the color codes which element of the group maximizes the “Jordan slope”, that is the quotient of the logarithms of the highest and lowest eigenvalue. It seems like only a special selection of group elements appears, and we think we understand which ones.
This is the same as the previous picture, but “zoomed in”, by looking at more elements of the group (and only looking at the upper left quadrant). Our guess still holds up.