## Month: May, 2013

### A bit of computations

Today, I did a pretty nifty computation to explore the behavior of $f(z) = \frac{z}{z+1}$. This transformation is a linear fractional transformation on the upper half plane with fixed point 0. The blue circle is the horocycle on which the points $\{f^n (i) | n \in \mathbb{Z} \}$ lies. The yellow semi-circle is the unit circle. The region bounded by the vertical red line, vertical blue line and the exterior of the yellow semi-circle, is the modular figure, which is the fundamental domain of $\Gamma = PSL(2, \mathbb{Z})$.

Some questions I have about the group $\Gamma$ is regarding the parabolic transformations: how many degrees of freedoms do we have in the “lattice”? In other word, since the parabolic transformations are the hyperbolic analog of translations, there should be a set of directions to travel in. Of course, none of the directions should cross path. Besides parabolic transformations, the hyperbolic transformations are also not torsion. Hence, the term “direction” should be a bit more complicated here.

Actually, does it even make sense to speak of a hyperbolic lattice? In the Euclidean plane, the translation subgroup of a wallpaper group can generate a lattice by translating the origin. Similarly, we would like to study the lattice generated by translating $i$. I believe this question will illuminate the structures of the congruence subgroups of $\Gamma$.

### More Hyperbolic Geometry

Today, I went to the first lecture with my REU advisor. I think he really likes number theory and told us a great deal about geometry, number theory, group theory and everything else. I have to say that there are insane things that occurs inside strange spaces that I cannot fathom even the tip of the iceberg because how ridiculous insane it is. Insane in a good way, of course.

First really surprising thing are the principal (principle?) congruence subgroups of $PSL(2, \mathbb{Z})$. These just have the strangest group structure. For one, $PSL(2, \mathbb{Z})$ has torsion element but some of these principal congruence subgroup (all of them?) are torsion free. The cooler thing is that whether the subgroup is torsion free or not relates to something about the fundamental domain of the subgroup. $PSL(2, \mathbb{Z})$ contains 2 elliptic point, namely, $i$ and $(1/2, \sqrt{3}/2)$. These two points somehow corresponds to $PSL(2, \mathbb{Z})$‘s torsion element of order 2 and 3.

Equally surprising are the fundamental domains of these principal congruence subgroups, and their quotient spaces. Much like in the Euclidean plane, in the hyperbolic plane, we can form these quotient spaces. Whereas $\mathbb{Z}^2 \backslash \mathbb{R}^2$ is a torus, $\Gamma(N) \backslash \mathbb{H}^2$ are these punctured spheres and toruses of genus 2, 3 or beyond.

Another thing that I can not yet grasp is the different isometric classes of toruses. A torus made from a square lattice has less symmetry than a hexagonal lattice torus. Similarly, there are quotient spaces in $\mathbb{H}^2$ that have a lot more symmetry than the others. The meaning of this is something I will explore in the next month or so.

Finally, what is the curvature on a torus? In fact, what is curvature? I have no studied this property yet, but I am seeing it everywhere. Gauss discovered that curvature is intrinsic to the object. Yet, is torus flat in $\mathbb{R}^4$ but has no imbedding into $\mathbb{R}^3$ that is flat? The meaning of this I will explore in the next month or so.

### Hyperbolic Geometry

One of the oddest thing that happens in this world is coincidences. Can it be a mere coincidence that our solar system, the milky, the universe are held together in equilibrium (relative to our lifetime of course)? Physicists are interested in this question and developed theories to explain all this. More alarming to me is how the mathematical universe are held together. The existence of the hyperbolic geometry is remarkable. Even more remarkable is how it can be modeled inside our intuitive Euclidean geometry.

Sitting inside $GL(2, \mathbb{R})$ is the group of isometries on the hyperbolic plane, $SL(2, \mathbb{R})$.  When we first conceived the idea of matrices, it is highly doubtful that anyone had the slightest clue that a subset of matrices are actually distance preserving maps for a very bizarre geometry! After all, $SL(2, \mathbb{R})$ are linear maps on the plane that are possibly squeezes or stretches, it is astonishing that we can curve our space a slight bit and get isometries. If this isn’t coincidental, then I don’t know what is. It is almost a miracle. I imagine that when the mathematicians (Poincare?) discovered this, the feeling he felt was comparable to climbing to the top of Mount Everest, or being the first crew that flies to the edge of the galaxy. Mathematics is remarkable.