Courses-
26:645:641:01 Graduate Topology I; Monday, Wednesday 10:00 am - 11:20; Smith 204
21:640:327:01 Probability and
Statistics; Monday 2:30 pm - 3:50, Wednesday 1:00 pm - 2:20; Smith
242
Course web pages on Blackboard
Office Hours: Monday 4:00 pm - 5:00, Wednesday 2:45 pm - 3:45, and by appointmentOffice Phone - 973-353-3909; Email - oertel@rutgers.edu
Department
of
Mathematics
and Computer Science, Rutgers-Newark
In an attempt to understand "finite height
(or
finite depth) measured laminations," I use certain ordered
abelian semi-groups and ordered semi-rings. The finite
height
laminations are characterized by having transverse measures with
values
in an appropriate ordered semi-rings. Using the ordered
semi-ring
as a parameter space, I define a space of finite height measured
laminations in a given surface, generalizing William Thurston's
projective measured lamination space for a surface. The
new space
is probably related to hyperbolic geometry on the surface, just
as in
the classical case. In the most recent paper on this
subject, I describe a wide variety of ordered algebraic
structures which can be used
to
give transverse structures to laminations in more general
settings. Associated to the transverse structures, we have
actions on certain trees. The trees can either be thought
of as
metric spaces with metrics having values in the ordered
algebraic
structure, or they can be thought of as order trees with
measures on
segments, the measures having values in the ordered algebraic
structure.
The methods have further applications in measure theory and probability theory, though the simplicity of some of the applications makes me suspect that they are known in some form.
My motivations for this research were originally more related to
3-manifold topology. Namely, I hope to obtain a better
understanding of the measured lamination space of a 3-manifold by
understanding a larger space of finite height measured
laminations. Also, in the study of automorphisms of
3-manifolds, it appears that there exist invariant lamination-like
objects of mixed dimension for automorphisms of 3-manifolds, and
these
should have finite height measured structures.
There are three related papers on this subject:
Finite height lamination spaces
for
surfaces. Math
arXiv
Applications of ordered abelian semi-groups and ordered semi-rings. Math arXiv
Automorphisms of 3-manifolds
The most interesting automorphisms (self-homeomorphisms) of 3-manifolds occur in boundary-reducible and reducible manifolds. The paper Automorphisms of 3-dimensional handlebodies develops a theory of automorphisms of handlebodies and compression bodies analogous to the Nielsen-Thurston theory of autmorphisms of surfaces. In the theory, the analogue of the pseudo-Anosov automorphism is the generic automorphism of a handlebody or compression body, and generic automorphisms turn out, as one would expect, to be the most interesting and mysterious. Automorphisms of irreducible 3-manifolds with non-empty boundary can then be understood by combining the new methods with older methods in 3-manifold topology, involving the Jaco-Shalen-Johannson characteristic submanifold and Bonahon's characteristic compression body. Much remains to be understood about generic automorphisms, which are currently also being studied by former Rutgers graduate student Leonardo Navarro de Carvalho.
In work with Carvalho, the ideas used for the classification of automorphisms of handlebodies and compression bodies are extended to classify, in a certain sense, the automorphisms of any compact 3-manifold (satisfying the Thurston Geometrization Conjecture). The current form of the paper is not satisfactory, and we plan to revise it extensively before publishing.
Automorphisms of 3-dimensional handlebodies and compression bodies, Topology 41 (2002) 363-410: Postscript file PDF file
A classification of automorphisms of
compact
3-manifolds, with Leonardo Navarro de
Carvalho. Math
arXiv
Mapping class groups of compression bodies and 3-manifolds
In work related to the classification of automorphisms of 3-manifolds, I obtained results concerning the mapping class associated to a compression bodies. This is applied to obtain results concerning the mapping class group of a reducible 3-manifold M. The posted manuscript has been changed extensively, and the goals for the final version have become more and more ambitious. This paper is helpful for making a classification of automorphisms (as described above) more natural and appealing.
Mapping class groups of compression bodies and
3-manifolds.
Math
arXiv
Normal surfaces
Charalampos Charitos and I have a new normal surface
theory
for incompressible (but not necessarily boundary incompressible)
surfaces
in irrecucible (but not necessarily boundary irreducible) compact
3-manifolds
with boundary. It may be possible to apply this theory to
obtain
better invariant lamination-like objects for generic automorphisms
of
handlebodies
and compression bodies. One of the motivations for the study
of
finite height measured laminations (above) is that these invariant
lamination-like objects should be finite height measured in the
appropriate sense.
Essential disks and semi-essential surfaces in 3-manifolds, with Charalampos Charitos. Essential disks and semi-essential surfaces in 3-manifolds, Topology Appl. 159 (2012), no. 8, 2174-2186. Math arXiv
Contact structures and contaminations in 3-manifolds
In a project with Jacek Swiatkowski (Wroclaw University) I am studying contact structures using branched surface techniques. A natural generalization of the contact structure and the confoliation is the contamination, an object which can be carried by a branched surface. Contact structures can be represented by contaminations. The goals of this research are 1) to understand contact structures combinatorially; 2) to understand the relationships between foliations, confoliations, contact structures, laminations, and contaminations.
A contamination carrying criterion for branched surfaces, with J. Swiatkowski, Ann. Global Anal. Geom. 34 (2008), no. 2, 135-152. Math arXiv
Correction: "A contamination carrying criterion for branched surfaces," with J. Swiatkowski, Ann. Global Anal. Geom. 44 (2013), no. 2, 137-150.
Contact structures, sigma-confoliations, and contaminations in
3-manifolds, with J. Swiatkowski, Commun. Contemp. Math. 11
(2009), no. 2, 201-264.. Math
arXiv
Some Other Papers