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21:640:135:07 Calculus I: Tuesday 10:00-12:00, Thursday 10:00-11:35 pm, Room B22, Smith Hall
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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
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
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