||Date of Thursday
||Surfaces and curves. Intersection number. Bigon Criterion. C(S).
||Farey graph. Edge metric. Hempel's lemma. Continued fractions.
||Can Hempel's lemma be improved?
Friday's lecture: I drew
-2/3 instead of 2/3! Oops!
||Train tracks. Transverse measures. Foliated tie
||Splittings. The basic observation. Nesting
||Fixed characterization of the Farey graph.
||Keane property. Dense leaves.
||No lecture due to technical difficulties.
||In class we gave a non-constructive proof that pairs (track,
measure) exist having the Keane property. Give a constructive
Thursday's lecture - Keane should also imply that w is
positive on every branch of τ. (Otherwise μ could contain
non-singular boundary leaves.)
||Finish dense leaves. C(S) has infinite diameter.
||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.
||Triangles. Hyperbolicity. Combings.
||Combing implies chords. Combinatorial area. Isoperimetric
||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
||Chords imply subquadratic. Subquadratic implies linear.
Linear implies stable.
||Stable implies slim. Singular flat structures.
||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.
||Examples of half-translation surfaces. Geodesics. Flat
||Length. Systole map. Balance time.
||Squared surfaces. Singular flat metrics. (Half-)translation
surfaces. Abelian/quadratic differentials. Rational billard
tables. Suspensions of interval exchange transformations...
||Wide annuli. Separation lemma. Minimal spines and systole
||Finish separation. Annulus inequality. Wide curves.
||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?
||Review. Set of systoles has bounded diameter. Weighted
multicurves and their squared surfaces. W ⊂ L part
||The combing. Flat length is convex. Averaging trick. Combing
triangles are slim.
||Have a good break.