The modular surface $\mathbb{H}/\Gamma,\, \Gamma= \mathrm{SL}(2,\mathbb{Z})$ has many covers - for example the three punctured torus $\mathbb{H}/\Gamma(2)$ and the once punctured torus $\mathbb{H}/\Gamma'$. We will discuss how classical Diophantine approximation can be interpreted in terms of the behaviour of geodesics on the once punctured torus and a geometric reformulation of the Frobenius uniqueness conjecture.
We will then give an account of two theorems of Fermat in terms of   the automorphisms of $\mathbb{H}/\Gamma(2)$:
- if $p$ is a prime such that $4|(p-1)$ then  can be written as a   sum of squares $p = c^2 + d^2$

- if $p$ is a prime such that $3|(p-1)$ then  can be written as  $  p = c^2 +cd +  d^2$
Finally we will discuss possible extensions to surfaces of the for  m $\mathbb{H}/\Gamma_0(N)$.

 

The four color theorem states that each bridgeless planar graph has a proper $4$-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with $m$ colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

In this talk, I show how to categorify the $m$-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the ``Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded $m$-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the $E_1$ page of a spectral sequence whose $E_\infty$ page has a basis that can be identified with proper $m$-face colorings, that is, each successive page of the sequence provides better approximations of $m$-face colorings than the last. Since it can be shown that the $E_1$ page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)

If time, I will relate this work to the study of the moduli space of stable genus $g$ curves with $n$ marked points. Using Strebel quadratic differentials, one can identify this moduli space with a subspace of the space of metric ribbon graphs with labeled boundary components. Proper $m$-face coloring in this setup is, in a sense, studying points in the space of metric ribbon graphs where similarly-colored boundaries (marked points) don’t get ``too close’’ to each other. We will end with some speculations about what this might mean for Gromov-Witten theory of Calabi-Yau manifolds.
 

Note to students: This talk will be hands-on with ideas explained through the calculation of examples. Graduate students and researchers who are interested in graph theory, topology, or representation theory are encouraged to attend.