Abstract:We describe a program to classify those 3-manifolds for which
the converse of Waldhausen's conjecture holds. That is, we describe
progress toward a classification of toroidal 3-manifolds that admit an
infinite number of non-isotopic Heegaard splittings of some genus. Many
of the proofs rely heavily on use of the Farey graph of the torus. This
work combines several results, including joint work with Saul Schleimer
and Eric Sedgwick, and joint work with with Ryan Derby-Talbot.
Tao Li (minicourse)
Title:Heegaard surfaces and measured laminations
Abstract:We will first review some basics of strongly irreducible Heegaard
splittings, branched surfaces and measured laminations. The main part of
the talks is the proof of two theorems on Heegaard splittings. The first
is the so-called generalized Waldhausen conjecture, which says that an
orientable irreducible atoroidal 3-manifold has only finitely many
Heegaard splittings in each genus, up to isotopy. We will also show that
a closed non-Haken 3-manifold has only finitely many irreducible Heegaard
splittings, up to isotopy.
Title:The Coarse Geometry of the Complex of Curves
Abstract:In 1996, Masur and Minsky showed that the curve complex of
a surface is hyperbolic in the sense of Gromov. Later, Bowditch gave
a somewhat more combinatorial proof. I will sketch the proof of this
important theorem, and discuss some related results. This is an
Title:Heegaard genus bounds distance (joint work with
Abstract:Kevin Hartshorn proved that the genus of an
surface in a closed 3-manifold M provides a limit on the distance of
any Heegaard splitting of M. We show that the same is true for
alternate Heegaard splittings. That is, the genus of one Heegaard
splitting puts a bound on the distance of any non-isotopic Heegaard
splitting. This is the most striking consequence of a somewhat more
general statement about bicompressible, weakly incompressible,
separating surfaces in compact 3-manifolds.
Saul Schleimer (minicourse)
Title 1: The mapping class group and the curve complex.
Abstract: This lecture will be a leisurely introduction to the mapping class group
of an orientable surface S and also to its curve complex, C(S). We'll
define pseudo-Anosov maps and "partial" pseudo-Anosov maps and discuss how
these act on the curve complex.
Title 2: Coarse geometry and the zoo.
Abstract: We'll discuss a few pieces of Gromov's framework of coarse geometry: the
aspects of a metric space that are always visible, even after you have
lost your glasses. This will lead us to the statement of the beautiful
result of Masur and Minsky: the curve complex is Gromov-hyperbolic.
We'll also explore the "zoo" of complexes related to the curve complex
(the arc complex, the separating curve complex, the disk complex, the
Hatcher-Thurston complex, etc). These are all related in fairly subtle
ways -- for example the complex of non-separating curves is
quasi-isometric to C(S) but the complex of separating curves is not.
Title 3: Subsurface projections and estimating distance.
Abstract: I'll give a careful proof of the sentence above using subsurface
projections, introduced by Ivanov. In fact, we can give a good
description of all ways that the complex of separating curves fails to be
quasi-isometrically embedded in the whole curve complex. Time permitting,
we'll also discuss "hierarchies" as defined by Masur and Minsky.
Title 4: The disk set
Abstract: The disk set, D(V), is defined as follows: let V be a handlebody with
boundary S. Then D(V) is the subcomplex of C(S) spanned by the curves in
S which bound disks in V. This complex is important in the study of
Heegaard splittings. Masur and Minsky have shown that D(V) is a
quasi-convex subset of C(S). Using the theory we've built up so far, it
will be easy to show that D(V) is not quasi-isometrically embedded.
We'll also discuss the problem of estimating distance in D(V) and, time
permitting, discuss some work-in-progress with Howard Masur: an
"algorithm" to compute the Hempel distance up to a bounded error.
notes on the Complex of Curves
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