### M Theory Lesson 2

Mulase and Penkava studied Ribbon Graphs and came up with a constructive proof of the correspondence between two kinds of moduli: a Riemann surface moduli space and a Ribbon Graph moduli space. The original theorem is due to Penner, Thurston and others, and relies on the study of Grothendieck's Children's Drawings.

One works with a category of Ribbon Graphs. An object is a collection of vertices and edges and an incidence map i. Arrows are pairs of arrows that form a commuting square with the two incidence maps. Vertices are always at least trivalent, but we then add bivalent vertices at the centre of each edge to create half edges. A cyclic ordering on half-edge vertices gives an orientation to the ribbon edges. By definition, a boundary of a graph is a sequence of directed edges which cycles back on itself. Then Euler's relation holds,

v - e + b = 2 - 2g

where g is the genus of the surface represented by the ribbon graph.

One works with a category of Ribbon Graphs. An object is a collection of vertices and edges and an incidence map i. Arrows are pairs of arrows that form a commuting square with the two incidence maps. Vertices are always at least trivalent, but we then add bivalent vertices at the centre of each edge to create half edges. A cyclic ordering on half-edge vertices gives an orientation to the ribbon edges. By definition, a boundary of a graph is a sequence of directed edges which cycles back on itself. Then Euler's relation holds,

v - e + b = 2 - 2g

where g is the genus of the surface represented by the ribbon graph.

## 3 Comments:

Another lesson, this one a little easier to understand.

I've made two important discoveries. First, the people who write the subtitles for Venezuelan television do not have as good a vocabulary for Spanish swear words as I learned growing up in New Mexico. Because of this, I have doubts about my ability to learn the Spanish language this way.

Secondly, I've discovered that I can cancel the twin obstructions of television and the internet. One puts on a movie, and then uses the internet to find the plot synopsis on Wikipedia. One can then spend ones time on physics until the start of the next movie.

Hi Carl

Brilliant! I usually don't turn on the TV (when I'm near one, which is rare, but usually in a hotel of some sort) because it costs money. I saw a nice Mongolian movie the other day, but I'm not getting out much. Can't wait for the Maths workshop next week.

Cool stuff about Fool's Gold, Carl. You know much more than me about symmetry. Don't tell Baez too much. Of course I agree that geometry is the true path. I wouldn't be talking to you otherwise, would I?

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