My main research interests are in complex analysis and related areas with an emphasis on its applications to probability theory, low dimensional topology and the analysis of very non-smooth structures (e.g. fractals). In particular, I have worked on Brownian motion, random growth models such as DLA (diffusion limited aggregation), potential theory, conformal mappings, and the analysis of Kleinian groups. Below I will discuss some of these areas in a little more detail.
Brownian motion: Brownian motion is the mathematical way of describing a random path in space, first described by Einstein and put on a rigorous basis by Wiener. A typical Brownian path in the plane is very rough, i.e., nowhere differentiable, and quite large in the sense that it has dimension 2. Based on scaling arguments from polymer physics, Benoit Mandelbrot had conjectured that the so called ``outer boundary'' of a Brownian path should have fractal dimension $4/3$ (the outer boundary is a model for a self-avoiding random path in the plane; in three dimensions this models a long random polymer). Jones, Pemantle, Peres and I have provided the first progress in proving the lower bound portion of Mandelbrot's conjecture by showing the outer boundary has dimension $>1$. Our key idea is to use an estimate for the minimal length of curve running through a given set of points, i.e, the traveling salesman problem. In this case we can estimate the probability that the boundary looks very flat, show this is unlikely and deduce that the boundary has infinite length. A refinement of the argument gives the dimension estimate. I am working to improve the estimate in the case of Brownian motion and to apply the technique to other problems in statistical physics. One such example is DLA (diffusion limited aggregation) in which a random set is built by attaching disks together at random. Some of my other work on Brownian motion involves the following question: given a set in space what is the probability that a random path will hit the set? If it does hit the set, how ``frequently'' does it visit? The question of whether the set is hit was settled by Kakutani in the 1940's, but only recently have Peres and I described how frequently the set is visited in terms of its fractal dimensions. One interesting aspect of the frequency problem is that the best possible result involves two kinds of fractal dimension; Hausdorff dimension in the time variable and packing dimension in the space variable.
Harmonic measure: If Brownian motion is started in bounded domain $\Omega$, then eventually it will hit the boundary, and describing where it is most likely to hit is one of the fundamental problems of potential theory. The hitting is described mathematically as a probability distribution on the boundary; the probability of a set $E$ is the probability that a random Brownian path first hits the boundary at a point of $E$. This probability distribution is called ``harmonic measure'', and describing this measure in terms of the geometry of the domain is one of the fundamental problems of analysis. Much of my work has been devoted to describing this measure when the domain is as general as possible, e.g., is bounded by a very non-smooth curve such as the von Koch snowflake.
In 2 dimensions a great deal is known about harmonic measure, because of the close relationship to complex analysis. For example, no matter what the domain $\Omega$ is, harmonic measure gives full measure to a subset of the boundary of dimension at most $1$ (which is surprising since a boundary like the snowflake may have dimension bigger than $1$). There are still are large number of related open problems. Most come down to describing as precisely as possible the set on the boundary where Brownian motion is most likely to land. I conjecture that the measure looks ``1-dimensional'' exactly on the set which accessible via cones in $\Omega$ and have proven this in most cases. I have also shown that harmonic measure is very unstable on ``fractal'' boundaries; small perturbations such a boundary can give a completely different measure.
In higher dimensions, very little is known in comparison to the planar case. There are many conjectures on how the 2-dimensional results should extend to $3$-dimensions and higher, but the tools to prove them still have to be developed. For example, in the plane I proved there are at most two ways that a Brownian path might approach a typical boundary point (e.g. if the boundary has a slit then the path could approach from either side). Technically this is expressed as saying that there are at most 2 Martin boundary points for almost every usual boundary point. The corresponding problem in three (or higher) dimensions is open, but I have the best results on the problem so far. This particular problem is also interesting because it leads to a class of unsolved problems involving geometry in a difference sense; estimating the base eigenvalue of a domain (i.e., the base frequency if the domain was a vibrating membrane) in terms of its shape. Given a sphere, how do we cut it into 3 (or more) pieces so that the sum of the base frequencies for each is minimized?
Kleinian groups: Since arriving at Stony Brook I have become involved in Kleinian groups, an area which is well represented here by Kra and Maskit. Kleinian groups are groups of isometries on hyperbolic space (the analogs of rigid motions in Euclidean space) and they have many connections to complex analysis, the topology of $2$ and $3$ dimensional manifolds and Teichm{\"u}ller theory. They give rise to a beautiful class of fractal sets called limit sets, and in the last few years, I have solved (in collaboration with P. Jones) several of the old problems about the geometry of these sets, particularly those concerned with their fractal dimension.
The best known problem that remains open is the so called Ahlfors conjecture. This has been open for 30 years and states that the limit set of a finitely generated group should have either zero or full area. I have reformulated this question in terms Brownian motion on a certain $3$ dimensional manifold and have obtained the best results yet on the precise size of the limit sets. Currently I am working on extending these results from special cases to the general setting, and in exploring the consequences of these results for the ergodic theory of such groups (already we have disproved a conjecture of McMullen about the rigidity of these groups).
Another fundamental result about limit sets is Bowen's dichotomy. It states that a limit set is either an ordinary circle or is a nowhere differentiable set with fractal dimension strictly bigger than 1. Bowen proved this in a special case in the 70's and it was extended by Dennis Sullivan and others in the 80's. The general result for all finitely generated groups is due to Jones and myself. Currently I am working on extending this to a larger class of groups called ``divergence type''. It is known that Bowen's dichotomy always fails beyond this class, so this work will give a complete understanding of Bowen's dichotomy in all cases.
Quasiconformal mappings: A mapping between spaces is called conformal if it preserves angles. This is a very important class of functions which is central to modern complex analysis. Due to the special nature of these maps, it is often convenient to consider a more general class, called the quasiconformal maps. These are well behaved in certain senses, but can still be quite wild. For example, the fractal ``snowflakes'' pictured earlier are all images of circles under quasiconformal mappings. The study of quasiconformal maps has been intense recently and they have many applications in complex analysis, Kleinian groups, dynamical systems and other areas. My main interest has been in determining the natural domains of such maps, or equivalently in describing their removable sets ($E$ is removable if any map which is quasiconformal off of $E$ must extend to be quasiconformal on the whole space). My best results concern the construction of non-removable sets in three dimensions which are sharp with respect to the known theorems (in fact, my examples are the only non-trivial examples known to exist). Besides answering some fundamental questions about such sets (e.g., there is a totally disconnected example) I provide some striking examples of new phenomenon (e.g., any two diffeomorphic open sets in $\Bbb R^3$ can be made quasiconformally equivalent by removing a ``small'' Cantor set from each). Besides building these examples which show the limits of any possible theory, I am very interesting in identifying specific removable sets which occur in dynamics. For example, various conjectures about the classification of complex dynamical systems could be proven if we can show that certain Julia sets were removable for quasiconformal mappings.
I have also been interested in how the Hausdorff dimension of a set can change under quasiconformal mapping. There are well know bounds in terms of the map, but I am interested in results for specific sets. In particular, I answered a question of J. Heinonen by showing that for any set $E \subset \Bbb R^d$ of positive dimension, there is a quasiconformal map $f$ so that $f(E)$ has dimension as close to $d$ as we wish, i.e., the dimension can always be raised. On the other hand, there are sets whose dimensions can never be lowered by any quasiconformal map (e.g., a line segment). However, all known examples have integer dimension, so I am trying to construct such an example with fractional dimension.
Rigidity: It is a fundamental result of Mostow that certain groups are ``rigid'', i.e., any attempt to slightly perturb the group must fail because there are no other groups nearby. Rigidity is false in some cases, but in a paper with Tim Steger we prove a representation theoretic version of Mostow rigidity which works in these cases. A geometric corollary of the result proves a conjecture of Tukia about the smoothness of conjugacies between Fuchsian groups. Our proof uses a version of the classical Poicar{\'e} series, but adapted to a pair of groups instead of just one. This gives us a function on a product of Teichm{\"u}ller spaces which blows up along the diagonal. Thus one suspects that the function we have defined may be some type of Greens function on Teichm{\"u}ller space. Its exact properties remain to be investigated.
Function algebras: This work mainly concerns algebras on planar domains generated by holomorphic and harmonic functions and the main technique involves finding bounded solutions to certain $\overline{\partial}$ problems. It includes results which generalize the Wermer maximality theorem to more general domains, explicit constructions of certain exotic function algebras, the solution to Sarason's problem of characterizing Blaschke products in the little Bloch space and structure theorems for certain algebras generated by bounded harmonic functions on the unit disk.