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<p>

My primary research interest is in low-dimensional
topology, particularly the topology of 
3-dimensional manifolds.<p>

There is a long tradition of using codimension-one 
objects to help
understand the topology and geometry of 
3-manifolds. The first and perhaps
most successful such object was, and is, the 
embedded incompressible surface. In fact, nearly 
every open question in the theory of 3-manifolds 
has been settled for  
Haken manifolds, which are those that contain an 
incompressible surface. For example, Waldhausen 
proved in the 1960's that a Haken manifold M is 
determined up to homeomorphism by its 
fundamental group (`algebra determines topology').
More recently Thurston proved the
geometrization theorem, which states that 
every Haken manifold can be canonically 
split open along tori so that the pieces 
are geometric: each admits a metric modeled 
on one of 8 model geometries (`topology 
determines geometry'). But since these results require an incompressible 
surface, such success has its limits; it is now generally 
accepted that most 3-manifolds,
in some sense, do not contain incompressible 
surfaces.<p>

Recently, more general codimension-one 
objects have been introduced to study 3-manifolds. 
Among them are $\pi_1$-injective immersed surfaces, 
taut foliations, and essential laminations. 
It is these last two that much of my research 
has focussed on.<p>


Essential laminations are hybrid objects 
which lie between incompressible surfaces and 
taut foliations, and which generalize both. 
The idea is that they
will provide tools for 3-manifold topology which are
as powerful as incompressible surfaces have 
proven to be. But they have the further advantage  
of being far more common than incompressible 
surfaces. As an example, Delman and Roberts 
have shown that every
non-trivial Dehn surgery along a non-torus 
alternating
knot yields a manifold containing an 
essential lamination (or {\it laminar} 
manifold, for short). By contrast, Hatcher and 
Thurston showed that only finitely many 
Dehn surgeries, on each of a certain class of 
alternating knots known as 2-bridge knots, 
yield Haken manifolds.<p>

The paragraphs that follow briefly 
describe four of the projects that I 
am currently working on.<p>


<H3>Knot theory</H3>


Knot theory is one of the subjects in 
3-manifold topology that has benefited the
most from recent research into essential 
laminations. For example, since Gabai
and Oertel proved that a laminar 
manifold has universal cover $\bf R^3$, Delman
and Roberts' result immediately proves 
both the Property P Conjecture (non-trivial Dehn surgery never yields
a simply-connected manifold) and the Cabling Conjecture (Dehn surgery on a non-cabled knot cannot yield a reducible manifold)
for alternating knots. Finding essential laminations in knot exteriors, which survive (i.e., remain essential after) non-trivial Dehn surgery, thus provides one approach to solving such conjectures, for many classes of knots. Such laminations
are called persistent. Recently I was able to show how one construction
of an essential lamination in one knot complement, due to Oertel, could be 
adapted to find infinitely-many knots whose complements contain a persistent lamination. Since then, Roberts and I have found a more general construction, which gives many more
examples of knots whose complements contain persistent laminations.<p>

<H3>Exceptional fillings of hyperbolic 3-manifolds</H3>

One problem that has received much recent
attention is the question of when the 
geometry (in the sense
of Thurston's geometrization theorem)
of a 3-manifold with boundary a 2-torus can 
degenerate (i.e., change) after Dehn
filling - the act of gluing a solid torus to the 
boundary of the knot exterior. Thurston has shown that 
for a hyperbolic 3-manifold all but finitely many Dehn fillings remain 
hyperbolic. Much recent research has focused on determining when and 
how often other geometric structures arise when Dehn filling a 
hyperbolic manifold. Many
general results are now known, with the single 
exception of when the Dehn-filled
manifold might have 
$\rm{\bf H^2}\times{\bf R}$ geometry and 
no essential tori; that is, when the 
manifold might be an exceptional
Seifert-fibered space. Essential laminations provide a tool
for answering this question, since there is a very good
structure theory for essential laminations in Seifert-fibered 
spaces, worked out in my thesis. In particular, one can often tell, by looking at an essential lamination $\cal L$, that the manifold containing $\cal L$ is \u{not} Seifert-fibered.
Together with constructions of essential laminations, due to 
Gabai and Mosher, this allowed me to show, for example, that at most 
20 Dehn fillings on a hyperbolic manifold are Seifert-fibered. The 
previous best bound was 24, due to Bleiler and Hodgson; their 
result figured strongly in mine. Ying-Qing Wu and I have also used this structure theory to show 
that an exceptional 
Seifert-fibered space cannot 
occur after Dehn-filling on a
2-bridge knot exterior, unless the knot is a torus or twist knot. This leads to a complete classifiaction of Dehn surgeries on 2-bridge knots.
This approach is quite general, so 
similar results should be obtainable 
wherever essential laminations can be constructed.<p>

<H3>Algebra determines topology</H3>

One of the most important open questions in 
the theory of essential laminations
is whether or not
homotopy equivalent laminar manifolds are homeomorphic. 
In my previous work I showed
that if a homotopy equivalence f:M\rightarrow
N, with N laminar, is sufficiently nice - specifically, if 
the pullback $\rm f^{-1}({\cal L})$, of 
an essential lamination $\cal L$ in N,
is essential in M - then f is homotopic to 
a homeomorphism.<p>

I am working to show that any homotopy 
equivalence is homotopic to one with an 
essential pullback; this has proved to be
surprisingly difficult. It
might be considered to be the most important unsolved
problem in the theory of essential 
laminations. My most recent work on 
this problem has focussed instead on 
avoiding this step altogether, by exploring 
to what extent the requirement of 
essentiality is needed
in the constructions of my previous work. 
Evidence suggests that it is not nearly 
as much as one might originally have thought.<p>


<H3>A hyperbolic, non-laminar 3-manifold?</H3>

Laminar manifolds are far more plentiful than Haken 
manifolds. But not all 3-manifolds, even those with universal 
cover $\bf\rm R^3$, are laminar. The first 
examples were found in my thesis, where
I showed that most exceptional Seifert-fibered 
spaces are not laminar. But to date no other examples are 
known; in particular, no hyperbolic example is known. 
The problem is 
that there is as yet no effective way to show that 
a 3-manifold doesn't contain an essential 
lamination (in contrast with the situation 
for incompressible surfaces). Using recent 
refinements in the structure theory 
for laminations in Seifert-fibered spaces, 
Naimi, Roberts, and I have established the intriguing result 
that the manifold obtained by 37/2 surgery 
on the (-2,3,7)-pretzel knot admits, essentially, exactly 
\u{one} essential lamination.
This manifold is a graph manifold, however, and so 
is not hyperbolic. However, current sentiment 
favors the existence of a non-laminar, 
hyperbolic 3-manifold; some of the best 
prospects, in fact, seem to be among some 
of the other surgeries on the (-2,3,7) pretzel knot.
One possible approach to the non-existence 
question is to use normal forms. In
previous work I have shown that for any 
triangulation $\tau$
of a laminar 
manifold M, there is an essential lamination 
which is in Haken normal form with respect 
to $\tau$. 
This implies that, if there is an essential 
lamination in M, then there is one which is 
carried by one of a finite, constructible collection of normal 
branched surfaces. The last major hurdle in applying 
this result to find non-laminar 3-manifolds 
is the current lack of an effective procedure for 
determining when a branched surface carries
a lamination; this is probably the other most 
important open problem in the theory of essential 
laminations. There are several sufficient 
conditions known, due to myself, Christy, and 
Naimi, as well as several well-known necessary 
conditions; the challenge is to bridge the 
gap between them.