My primary interest is, very broadly,
actions of (mainly
semisimple Lie) groups acting on manifolds. More specifically, suppose G
acts on M, preserving a certain geometric structure on M. How
are the following related:
I'm a mathematics grad student at the University of Chicago, with advisor Bob Zimmer, expected graduation date June 2000. Following this paragraph are a few words concerning the type of math I'm studying. If you discover they make no sense to you, click here to read something much easier which relates to it.
My primary interest is, very broadly,
actions of (mainly
semisimple Lie) groups acting on manifolds. More specifically, suppose G
acts on M, preserving a certain geometric structure on M. How
are the following related:
Typical examples of geometric structures include Riemannian and pseudo-Riemannian metrics, invariant measures, and connections. Instead of the whole group G, one can be study actions of a lattice L.
My thesis will provide results on fixed points and isotropy subgroups of connection preserving actions, and show how this is related to the linear-representation-theoretic concepts of epimorphic and observable subgroups.
For more information, take a look at my 4-page research summary, available in PDF and TeX DVI format. There's also my research-oriented CV, in PDF format only.
If the above made no sense at all, click here to jump below. If it all makes complete sense, talk to me and explain it to me. If you're somewhere in the middle, you can look at the proposals I wrote for my topic examinations. [Warning: the PDF files are about 150k each since they include the Computer Modern fonts used by AmSTeX.]
An underlying theme through the above progression is "generalizations of Mostow rigidity", from the point of view of quasi-isometries in the second topic, and from the point of view of superrigidity and cocycles in the "third".For some other mathematical interests of mine, including preprints and papers, click here.
This is a gentler, more vague, and partially incorrect and outdated description of some of the questions I write about above. But it should give some idea of the sort of math I am interested in.
Consider the surface of the sphere. Curves which minimize the distance between two (nonantipodal) points, called geodesics, are great circles. If we travel along two geodesics which intersect at a point, we find the geodesics diverge less than they would in the (flat) plane. Alternatively, the interior angles of a triangle with given side lengths are greater than they would be in the plane. This is positive curvature. Negative curvature means the opposite; examples are harder to visualize; its sort of like a surface which has a saddle at every point.
In fact, we really consider this same idea in higher dimensions. We often allow the case when lower-dimensional slices of our space might be flat; this is what we pedantically call nonpositive curvature, that is negative-but-might-be-zero.
More generally, we consider nonpositively curved manifolds, which are objects which look myopically like nonpositively curved space. Such manifolds can be considered by identifying points in space via isometries, which are transformations of the space which preserve the geometry. It is useful to switch back and forth between considering the geometry of the manifolds, versus considering the algebraic structure of the group of isometries describing them. In fact, the amazing thing is how much information about one can be recovered from the other. Generalizations of this idea fall under what is called geometric group theory.
This interplay places considerably many restrictions on what sorts of manifolds and isometry groups can exist. A chain of results originating with Mostow in the late 60s in fact states that the geometry of such manifolds is fully determined by their topology. This is called Mostow rigidity. What does it mean? Well, an old joke says topologists are mathematicians who cannot distinguish between a doughnut and a coffee cup since both have one hole, even though they are geometrically quite different. Mostow rigidity says that somehow nonpositively curved space is such that you cannot have two geometrically different objects which are topologically the same: in such spaces, doughnuts are actually coffee cups no matter what sort of mathematician you are. Crucial in the proof of these results is showing that certain transformations which are known to distort scale by a uniformly bounded amount (called quasi-isometries) are actually isometries. More generally, the description of what groups can be related by quasi-isometries leads to the concept of quasi-isometric rigidity. Super-rigidity and the other incomprehensible terms I mention at the top of this page can be viewed as further generalizations of Mostow rigidity. This was the point of view taken in my topic exams mentioned above.
The mumbo-jumbo about geometric structures at the top of the page is a related but different generalization of what I just wrote above. I talked about a space with curvature (sphere or "space of non-positive curvature") and its isometries. This can be viewed as studying transformations (the isometries) of a space (the sphere, etc.) preserving a geometric structure (distance, curvature). Now "group actions" is a fancy word for transformations, and "manifold" a fancy word for space, and you get (generalizing like crazy) what I say my research interest is.
I welcome comments and suggestions re this explanation.