greg mc

Dec 2020

\(\sum\frac{1}{1 + e^{\ell_\gamma}}=\frac{1}{2}\)

Is a formula but it is describing a geometric **truth**

Illustrator 88 circa 1990

This was hacked together

- in
`nvim`

- using Markdown
- and revealjs

and my first time on Zoom :(

2 dimensional space (manifold)

- disc (is a subset of)
- plane (is the image under the stereographic projection of)
- sphere
- torus (which is a branched cover of the sphere)

Study them by embedding

them in \(\mathbb{R}^3\).

- disc (polar)
- plane (cartesian or polar)
- sphere (spherical polar \(0 \leq \theta \leq \pi, 0\leq \phi\leq 2\pi\))
- torus (spherical polar \(0 \leq \theta \leq 2\pi, 0\leq \phi\leq 2\pi\))

- what are the isometries?
- what are the geodesics?
- what are the optimal maps to other spaces?

The disc, plane and sphere are simply connected.

Torus is not simply connected.

- Fundamental group is isomorphic to \(\mathbb{Z}^2\)
- Is a quotient of the plane by a group of translations \(\Gamma\) generated by...
- \(z \mapsto z + 1\) and \(z \mapsto z + \tau\) where \(\tau \in \mathbb{H} \subset \mathbb{C}\)

from a parallelogram with corners \(0, 1, 1 + \tau, \tau\)

by glueing opposite edges

Geodesic = length minimising curve

- straight line in the (euclidean) plane
- great circle on the sphere

Geodesic foliation: \(\mathbb{R}^2\) is partitioned by parallel families of geodesics.

Being geodesic is a local property so under the projection map straight lines map to geodesics

\((\theta, \phi) \mapsto (e^{i\theta},e^{i\phi})\), \(\mathbb{R}^2 \rightarrow \mathbb{R}^2 / \Gamma\) = torus

- rational slope => closed
- irrational slope => dense

- closed geodesic \(\gamma\), rational slope = p/q,
- where \(p,q\) are coprime integers
- length of the geodesic is \(\ell_\gamma = | p \tau + q |\)
- this is a closed formula for the length of a closed geodesic

to see how they fill outspace

Recursive enumeration coprime pairs \(p,q\)

There are holes but they will get filled in

metric for which the orientation preserving isometries are Moebius transformations : \(z \mapsto \frac{a z + b }{cz + d}\)

the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.

area = \(\pi\), all angles = 0

\(z \mapsto \frac{a z + b }{cz + d}\)

- rotations : single fixed point in \(\mathbb{H}\)
- parabolic : single fixed point on \(\mathbb{R}\)
- loxodromic : pair of fixed points on \(\mathbb{R}\)

Glue opposite sides ideal quadrilateral.

- closed geodesics
- dense geodesics
- all sorts of other geodesics

on the left simple, on the right non simple

most geodesics aren't simple they have self intersections.

81 shortest geodesics on torus

on punctured torus

The union of all complete **simple** geodesics is

- closed
- nowhere dense i.e. there are holes
- Hausdorff dimension 1, so measure zero

Pseudo sphere embedded in \(\mathbb{R}^3\). \((\mathrm{sech}(u)\cos(v),\mathrm{sech}(u)\sin(v),u-\tanh(u) )\)

Pseudo sphere as a quotient

The pseudo sphere is foliated by *vertical geodesics*

- every punctured torus contains a pseudo sphere of area 1
- no simple closed geodesic enters this pseudo sphere
- any simple geodesic that enters the pseudo sphere is vertical

The pseudo sphere is a bit like a black hole once past it's horizon there is no escape for the geodesic.

Intersection of Birman Series set with pseudo sphere.

\(K \times \mathbb{R}^+ = K_{ess} \sqcup K_{iso}\)

- \(K\) = closed nowhere dense subset of circle

- \(K_{iso}\) = isolated points
- "shallow" points in closure of interval \(\subset K_{ess}^c\)
- also "deep" points in \(K_{ess}\)

\(\frac{2}{1 + e^{\ell_\gamma}}\)

is the length of an interval in \(K_{ess}\)

what are the geodesics like that come from

- isolated points
- "shallow" points?
- "deep" points?

\(x \in K \rightarrow \gamma_x\) the geodesic "accumulates" somewhere.

Surface cut along accumulation set

- \(\gamma_x\) returns to pseudo sphere <=> x isolated
- \(\gamma_x\) spirals to a closed geodesic <=> x "shallow"
- otherwise x "deep"

The gaps in \(K_{ess}\) are 1-1 with closed geodesics

- Look what I proved I think you're going to like it.
- Either false or Bob Penner has done it already

(Cohn) the Markoff numbers are \(2/3 \cosh(\ell_\gamma/2)\) for \(\gamma\) a closed simple geodesic on the punctured torus.

- recursive enumeration, no closed formula

- x "deep" point not isolated or shallow
- Even-numbered convergents are smaller than the original number,
- while odd-numbered ones are larger.

partition of unity for integration

3 glueing parameters + 1 relation Shear coords

2 parameters - length and twist