Ulrich Oertel


Fall, 2014
Office - Smith Hall 322

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 appointment

Office Phone - 973-353-3909;           Email - oertel@rutgers.edu

Department of Mathematics and Computer Science, Rutgers-Newark


Current Research 


Measures with values in ordered abelian semi-groups,
transverse measures for laminations

In an attempt to understand "finite height (or finite depth) measured laminations," I have used 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 have defined 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 give a wide variety of orderd 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 having values in the ordered algebraic structure.

The methods have further applications in measure theory and probability theory, though the simplicity 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: 

Measured lamination spaces for surface pairs.  This paper is preparatory for the following paper.  Math arXiv

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