My main interest is the interaction between topology and geometry. Suppose that f is a diffeomorphism of the 3-sphere to itself and C is a knot in the 3-sphere such that that every point of C is mapped to itself by f. Also assume that there is p>1 such that f^p is the identity. Then C is (topologically) unknotted. This was the Smith Conjecture. It is now a consequence of Thuston's orbifold theorem. I have spent a number of years working (with Hodgson and Kerckhoff) on the proof of this theorem. Many of the most common 3-manifolds have some sort of symmetry. Under certain conditions on the manifold (compact, orientable, irreducible, atoridal) and on the symmetry (it has finite order and leaves fixed a non-empty 1-submanifold) then the orbifold theorem says the 3-manifold has a (typically unique) geometric structure (homogeneous Riemannian metric) for which the symmetry is an isometry.
With Darren Long I have studied the existence of various kinds of surface in 3-manifolds. Here is a sample of the work of some of my students:
- Give an algorithm to determine whether or not a 3-manifold whose fundamental group has solvable word problem is hyperbolic. (Manning)
- Every connected smooth 4-manifold is the quotient of Euclidean 4-space by a group of homeomorphisms. (Lawrence)
- Relationships between combinatorial and group-theoretic aspects of 3-manifolds (for example the length of a presentation of the fundamental group) on the one hand and geometric aspects (for example injectivity radius of a hyperbolic metric). (White)
- The geometry of certain kinds of metrics on Cantor sets. (Vuong, Cockerill)
- Are there exotic actions of a cyclic group on the 3-sphere (Maher)
My mathematical interests have all grown out of problems which arose in low dimensional topology, mostly connected with geometry and algebra and how they interact.
Algebra has appeared in many forms in my work over the years: braid groups, mapping class groups, p-adic Lie groups, and in work with Cooper and/or Reid, a good deal of work about representations of three manifold groups and the existence of surface subgroups. Currently I'm interested in subgroup separability questions and certain connexions between number theory and hyperbolic geometry.
As an undergraduate at MIT, I was drawn first to engineering, then physics and, ultimately, to mathematics, specifically geometry and topology, because these have served as the language of expression and the means to explore the mysteries of the natural sciences. This interest continued through my graduate study at the University of Wisconsin and still provides much of the stimulus for my interest in specific questions in low dimensional geometric topology. Although I have worked in several areas such as the topology of fiber bundles and foliations, I am currently exploring some novel aspects of knot theory.
A while ago, I was involved in the creation of knot invariants growing out of the celebrated discovery of the "Jones" polynomial. In joint work with Ray Lickorish and Bob Brandt, I discovered two classes of knot invariants and participated in the development of topological quantum field theory and spatial invariants of graphs. I am now working on applying the knot invariants to questions growing out of molecular biology, for example, the structure of DNA. This includes the development of methods appropriate for the models used in the study of macromolecules, at one extreme, or solar storms, at another end of the scale. I have been lead to new questions concerning polygonal models of knots and connections of these to aspects of classical knot theory. For example, what can one say about the local and global structure of the space of regular n-gons in 3-space? Which knot types occur with what probability, as a function of the number of edges in the polygon? What are the differences between the topological and the polygonal theory of knotting? What are the spatial characteristics of such knots that optimize an energy function or the thickness of an imbedded tubular neighborhood?
A former student, Jorge Alberto Calvo, and I have been interested in the question as to whether the knot, 8.19, can be constructed as an equilateral octagon. This seems to be a very delicate question and appears to require new methods. Eric Rawdon and I have been working in physical knot theory and the numerical analysis required to study knot energies, ropelength and other spatial characteristics aspects. We describe the relationship of the local structure of knot space to the thickness of the knot. I am also working, with colleagues in Switzerland and France, on measures of the complexity of DNA models and manifestations in experiments.
What all of these have in common is a curiosity about the nature of polygonal knot space, about the spatial properties of polygonal knots, especially those that appear to express characteristics that are tied to physical manifestations of these knots, whether at the scale of DNA or solar storms.
My mathematical interest is mostly in the highly visual field of "geometric topology". Although I started out thinking about 4-dimensional manifolds, I eventually found myself drawn to the elegant 3-dimensional problems that naturally arose when thinking about cross-sections. In particular I now get excited by the beautiful combinatorial patterns and problems that emerge when thinking about 3-manifolds and the surfaces they contain.
Here's an example: When can a graph imbedded in 3-space be moved in 3-space so that it lies in a plane? When I got interested in the problem there was already an ambitious conjecture of what the answer should be, but it had been verified for only a small number of graphs. I worked on it with a former Ph. D. student Abby Thompson (now a professor at UC Davis) and together we were able to prove the conjecture in complete generality, thereby answering the central question about planar graph placement in 3-space. Were the algorithm we verified ever made practical, it could have important consequences for real-world graphs, e. g. chemical molecules.
Most recently I've been exploring an idea called "thin position" and applying it to special sorts of graphs in 3-space called Heegaard spines. Thin position is an idea that's a bit like geometry and a bit like topology and the information I'm seeking is relevant to one of the most visual of mathematical fields: classical knot theory.
I am currently working on braid groups, and I am also interested in knot theory and 3-manifolds. The main result of my PhD thesis was the existence of a one-to-one map from the braid group B_n into a group of matrices. Since then I have been looking at topological ways of studying braid group representations, knot invariants and Hecke algebras.
- What does the subset XY=UV of Euclidean space of dimension 4 look like?
- How many disjoint copies of a figure eight "8" can be embedded in the plane, countably or uncountably many? What about if "8" is replaced by "Y" ?
- A planar surface is an open subset of the plane. Are there uncountably many connected planar surfaces no two of which are homeomorphic? Are there two connected planar surfaces without boundary having the same fundamental group but which are not homeomorphic?.
- Is it possible to have two topological spaces X and Y which are not homeomorphic but so that each space is (homeomorphic to) a finite sheeted covering space of the other?
- A table has 3 legs of equal length. Is it always possible to place the table on a convex hill so that the surface is level?
- There are equal numbers of black and white points in the plane. No 3 points lie in a straight line. Is it always possible to draw straight lines so that each straight line starts on a white point and ends on a black point, so that the lines do not meet, and so that every point is on exactly one line?.
- A finite graph is a finite set of points in space called vertices together with a finite number of edges connecting pairs of vertices. Each edge meets exactly 2 vertices, one at each of its endpoints. If one edge meets another edge they meet only at one endpoint. Prove that if a graph has more than 1 vertex, then there are 2 vertices with the same degree (=the number of edges meeting the vertex)
- If you take a square and look at it from some point in space it looks like a quadrilateral. What are the possible shapes of this quadrilateral?
- Is there a subset of Euclidean 3-space with an element of finite order (not the identity) in its fundamental group?