| Week |
Date of Thursday |
Thursday |
Friday |
Lecture notes |
Comments/Mistakes |
| 2 |
Jan. 21 |
Surfaces and curves. Intersection number. Bigon Criterion. C(S). |
Farey graph. Edge metric. Hempel's lemma. Continued fractions. |
Thursday
Friday |
Can Hempel's lemma be improved? Friday's lecture: I drew
-2/3 instead of 2/3! Oops! |
| 3 |
Jan. 28 |
Train tracks. Transverse measures. Foliated tie
neighborhoods. Carrying. |
Splittings. The basic observation. Nesting
criterion. Dimension. |
Thursday
Friday |
Fixed characterization of the Farey graph. |
| 4 |
Feb. 4 |
Keane property. Dense leaves. |
No lecture due to technical difficulties. |
Thursday
| In class we gave a non-constructive proof that pairs (track,
measure) exist having the Keane property. Give a constructive
proof. Thursday's lecture - Keane should also imply that w is
positive on every branch of τ. (Otherwise μ could contain
non-singular boundary leaves.) |
| 5 |
Feb. 11 |
Finish dense leaves. C(S) has infinite diameter. |
Coarse geometry. |
Thursday
Friday |
Measures decompose as a finite sum of atomic measures on
curves and minimal measures. Can you use "volume growth" to prove
that the Farey graph is not quasi-isometric to
H2 or T3. |
| 6 |
Feb. 18 |
Triangles. Hyperbolicity. Combings. |
Combing implies chords. Combinatorial area. Isoperimetric
inequalities. |
Thursday
Friday |
If X is a tree then δ = 0. The converse is essentially
true (R-trees). If (X, d) is Gromov hyperbolic then so is the
quasi-isometric space (X, d/n); note that δ scales as well.
Gromov-Hausdorff convergence. Question: are the paths in a slim
combing quasi-geodesics? |
| 7 |
Feb. 25 |
Chords imply subquadratic. Subquadratic implies linear.
Linear implies stable. |
Stable implies slim. Singular flat structures. |
Thursday
Friday |
A singular flat surface. Warning: some of
the vertices have cone angle 3π/2. Thursday's lecture - The
maximum of (x + a)^2 + ((1-x) + a)^2 occurs when x is as large (or
as small) as possible.
|
| 8 |
Mar. 4 |
Examples of half-translation surfaces. Geodesics. Flat
annuli. |
Length. Systole map. Balance time. |
Thursday
Friday |
Squared surfaces. Singular flat metrics. (Half-)translation
surfaces. Abelian/quadratic differentials. Rational billard
tables. Suspensions of interval exchange transformations... |
| 9 |
Mar. 11 |
Wide annuli. Separation lemma. Minimal spines and systole
length. |
Finish separation. Annulus inequality. Wide curves. |
Thursday
Friday |
The isoperimetic function for the plane is
x2/4π. Puzzle: what is the shortest arc dividing
the equilateral triangle into two pieces of equal area?
|
| 10 |
Mar. 18 |
Review. Set of systoles has bounded diameter. Weighted
multicurves and their squared surfaces. W ⊂ L part
II. |
The combing. Flat length is convex. Averaging trick. Combing
triangles are slim. |
Thursday Friday |
Have a good break. |
