Geometric topology is often split into low dimensional (4 or less) and high dimensional. This split is based upon the techniques employed, the kinds of question that can be answered, and the state of knowledge. There were enormous advances in high dimensional topology during the 60’s including the solution of the high dimensional Poincare conjecture, and a good understanding of how differentiability enters into the picture, for example through the existence of exotic smooth structures on spheres.
Today a considerable effort is being made to better understand manifolds of dimensions 3 and 4. The techniques, conjectures and outlooks in these two areas are very different, although there have also been hints of various unifying themes. In the 80’s it was discovered by Donaldson, Freedman and Casson that Euclidean space has exotic smooth structures only in 4 dimensions.
The theory of 3 dimensional manifolds was revolutionized in the late 70’s by Thurston’s Geometrization Conjecture. There are eight geometries (homogeneous Riemannian metrics) which (appear to) play a similar role in 3 dimensions to the three constant curvature geometries in two dimensions. Some problems in 3-dimensions are best studied through combinatorial and topological techniques using surfaces and their generalizations. Many problems in knot theory are of this type.
There are many connections to number theory, Riemannian geometry, geometric group theory and dynamical systems to name only a few.
- Stephen Bigelow
- Daryl Cooper
- Michael Freedman
- Darren Long
- Fedya Manin
- Jon McCammond
- Ken Millett
- Zhenghan Wang
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.
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.
I mostly think about questions that connect topology and metric geometry, or either of these to the theory of computation or probability. By historical accident, most of my past work has explored such questions in the context of spheres and other high-dimensional manifolds and simply-connected spaces. But the same ethos can be applied to many research areas, and I will give examples from an area I haven't (successfully) worked in: knot theory.
Traditionally, knot theorists are concerned with finding algebraic invariants of knots and using them to classify knots up to isotopy type. But you can also study metric invariants. For example, the ropelength of a knot is the minimal number of inches of 1-inch-thick rope that you need to tie that knot. Rather than trying to learn how to tell apart all isotopy types, can we learn enough to say how roughly how many knots have ropelength ≤ L, as a function of L?
One can also ask a relative version of this problem: given two isotopic knots which are tied using a 1-inch-thick rope of length L, how much do you need to stretch the rope to isotope them?
This is related to the problem of algorithmically determining whether two knots are isotopic. The worst vaguely sensible algorithm is to try all possible isotopies until either you find one, or you've tried everything that could possibly work. It turns out that for this specific problem, you can do significantly better than that, but there are many undecidable questions in topology, for which there is no algorithm to determine the answer. This in turn implies that there is no “reasonable” place for an exhaustive search to terminate: in other words, that certain pairs of knotted 3-spheres in (for example) are isotopic, but only in unimaginably convoluted ways.
Finally, another way to play with knots is to try different methods of generating random ones; or, in other words, contemplate average-case rather than worst-case geometry. For example, you could take a random big pile of knotted rope (assuming you can make this into a mathematically rigorous construction). Is its ropelength usually going to be comparable to its length, or is it often possible to untangle most of it?
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 interests lie in low dimensional topology, quantum topology/algebra, and topological physics. I am currently thinking about connections between quantum topology and classical topology such as the volume conjecture. I am also working on mathematics of quantum field theories and their applications to quantum computing. One problem is how to reconstruct conformal field theories from their modular tensor categories. T. Gannon and I both conjectured that every unitary modular tensor category can be obtained as the representation category of a chiral conformal field theory such as a vertex operator algebra or local conformal net. The approaches that I am taking are lattice models such as anyonic chains, and vector-valued modular forms.