Lectures

### Recent lectures (most recent at top)

Harmonic measure, conformal maps and optimal meshing, Friday, February 28, 2020, the Simons Collaboration on Algorithms and Geometry, the Simons Collaboration on Algorithms and Geometry.

First talk:
Both this and the following talk are colloquium style overviews, aiming to show how disparate areas come together in certain computational problems. I will start with the definition of harmonic measure and its connection to conformal mappings. I plan to discuss a few methods for computing conformal maps including my own 'fast mapping algorithm' whose formulation uses ideas from computational geometry (the medial axis), and ideas from quasiconformal mapping and hyperbolic manifolds to measure and prove convergence. In particular, this gives a linear time method to approximate the parameters in the Schwarz-Christoffel formula (linear in the number of vertices of the image polygon, with multiplicative constant that depends on the desired accuracy). At the end I will mention an application to optimal quad-meshing of planar domains.

Second Talk:
Picking up where the last talk ended, I will discuss what is known about optimal quad-meshing and optimal triangulation of planar domains. Here optimal refers to both the number of elements needed and the quality of these elements in terms of keeping angles bounded away from 0 and 180 degrees (well shaped mesh elements are important in various applications). Also important is the difference between meshing a domain whose boundary is a simple polygon (the easy case) or a planar straight line graph (harder, since there can be holes, omitted points, slits,...). For quad-meshes the optimal lower and upper angle bounds are 60 and 120 degrees and the optimal size is O(n) or O(n^2) for simply polygons and PSLGs respectively. Sharp algorithms are known in both cases. For triangulations, the situation is less complete. It is easy to see that 90 degrees (non-obtuse triangulation) is the best possible upper angle bound and that no positive lower bound is possible, but only recently has a polynomial time algorithm achieving this bound been discovered (the NOT theorem). The method involves a flow associated to any triangulation, and perturbing this flow to form closed loops (a sort of discrete version of Pugh's closing lemma). Sharpness of the algorithm remains open, as do analogous questions for triangulated surfaces and for polyhedra in 3-space. The triangle flow introduced in the proof is apparently a new object and its properties also remain to be investigated.

True Trees and Transcendental Tracts , December 4, 2019, Bernoulli Center at EPFL, Lausanne, Switzerland. Two talks given as part of the workshop Low-dimensional and Complex Dynamics .
Abstract: These two lectures will motivate, state and apply a method for constructing holomorphic functions called "quasiconformal folding". I will start with a quick review of harmonic measure and consider the problem of constructing conformally balanced trees, i.e., planar trees for which every edge gets equal harmonic measure and the two restrictions of harmonic measure to each side of an edge are equal. Such "true trees" are a special case of Grothendieck's theory of "dessins d'enfants" and correspond in a natural way to polynomials with exactly two critical values. After discussing the possible combinatorics and geometry of finite true trees, I will consider infinite planar trees that correspond, in the same way, to transcendental entire functions with exactly two singular values (singular values include both critical and asymptotic values). Quasiconformal folding is a method of constructing such functions with good geometric control near infinity. I will state the QC-folding theorem, explain the proof in a simple case and briefly sketch the general case. I will also state some variations of basic construction that are useful in practice, including one that constructs functions with bounded singular sets and almost no restrictions on the geometry. The final part of the talk will discuss applications of folding including: constructing new examples of entire functions with wandering domains, building transcendental Julia sets have dimension close to 1 (the minimal possible), and prescribing the post-singular dynamics of a meromorphic function.
Video of first CIB lecture , Video of second CIB lecture

The talk by Lasse Rempe-Gillen immediately following my second talk is closely related to my lectures: it describes joint work that applies QC folding to prove that every non-compact Riemann surface can built out of equilateral triangles. Video Rempe-Gillen's lecture

Weil-Petersson curves and finite total curvature , University of Geneva, Dec 2, 2019.
Abstract: In 2006 Takhtajan and Teo defined a Weil-Petersson metric making universal Teichmuller space (essentially the space of planar quasicircles) into a disconnected Hilbert manifold. The Weil-Petersson class is the unique connected component containing all smooth curves, and consists of certain rectifiable quasicircles. There are several known function theoretic characterizations of WP curves, including connections to Loewner's equation and SLE. In this lecture, I will describe some new geometric characterizations that say a quasicircle is WP iff some measure of local curvature is square integrable over all locations and scales. Here local curvature can be measured using quantities such as: beta-numbers, Menger curvature, integral geometry, inscribed polygons, tangent directions, and associated convex hulls and minimal surfaces in hyperbolic 3-space.

A random walk runs through it: a portfolio of probabilistic pictures . Illustrating Dynamics and Probability ICERM, Providence RI, Nov 11 - 15, 2019.
I like to put lots of pictures in my papers and lectures and I have chosen several of my favorites (either images or theorems) to present, all having something to do with random walks, Brownian motion or harmonic measure. After showing the images themselves, I will explain a little of the 'how' and 'why' they were drawn. Since I have prepared much more material than I can discuss in 45 minutes, the audience will have to help select which pictures get explained.
Video of my ICERM lecture

Weil-Petersson curves and finite total curvature , Modern aspects of complex analysis and its applications, a conference , in honor of John Garnett and Don Marshall
Abstract: In 2006 Takhtajan and Teo defined a Weil-Petersson metric making universal Teichmuller space (essentially the space of planar quasicircles) into a disconnected Hilbert manifold. The Weil-Petersson class is the unique connected component containing all smooth curves, and consists of certain rectifiable quasicircles. There are several known function theoretic characterizations of WP curves, including connections to Loewner's equation and SLE. In this lecture, I will describe some new geometric characterizations that say a quasicircle is WP iff some measure of local curvature is square integrable over all locations and scales. Here local curvature can be measured using quantities such as: beta-numbers, Menger curvature, integral geometry, inscribed polygons, tangent directions, and associated convex hulls and minimal surfaces in hyperbolic 3-space.

New Constructions in Transcendental Dynamics , Analytic Low-Dimensional Dynamics: a celebration of Misha Lyubich's 60th birthday
Abstract: I will describe the structure of Eremenko-Lyubich functions: a class of transcendental entire functions that has been intensely investigated over the last 30 years. In particular I will discuss some differences between this class and the more restrictive Speiser class of functions, and how to construct a wide variety of examples in either class using the method of quasiconformal folding. If time permits we will mention a number of examples, such as constructing Eremenko-Lyubich functions with wandering domains, constructing entire functions with Julia set of small dimension, and constructing meromorphic functions with prescribed post-singular dynamics.
Video of my above talk for the Lyubich-60 conference at the Fields Institute.

Random thoughts on random sets , IPAM Workshop on Geometry and Analysis of Random Sets UCLA, Jan 7-11, 2019.
Abstract: I will discuss some open problems that have I have thought about over the last 30 years. Some are well known (e.g., growth rate of DLA), but a new may be novel, such as the flow associated to a planar triangulation. There will be many pictures, results of some computer experiments, but very few theorems or proofs.

Video of my above talk at IPAM.

Schedule and links to all videos from IPAM workshop.

MATLAB Scripts used to generate examples in the IPAM lecture above.

Harmonic measure, true trees and quasiconformal folding, International Congress of Mathematicians, Rio de Janeiro, Brazil, August 6, 2018
Abstract: I will start by giving several equivalent definitions of harmonic measure on planar Jordan domains and then briefly review a few well known theorems about harmonic measure on Jordan domains. Then we will consider harmonic measure on finite planar trees and define a conformally balanced tree, which is the same as the true form of a tree. We will see that all finite trees have true forms, and that these true forms can approximate any continuum we want. Finally, we turn to the analogous questions for infinite planar trees. Although the corresponding theorems are not as definitive as for finite trees, they still suffice to answer a number of open questions in geometric function theory and conformal dynamics.

Video of my ICM talk.

My ICM lecture covers (roughly) Section 4 of my contribution to the ICM proceedings, which is a survey that also discusses other topics including algorithms for computing harmonic measure, applications to computational geometry, connections to hyperbolic geometry and hyperbolic manifolds and conformal weldings. See Harmonic measure: algorithms and applications .

The first few slides of my ICM lecture show a Brownian motion hitting the side of planar domain. The file Brazil.m is the MATLAB file that was used to draw these pictures. Inside MATLAB type "Brazil" and you will get a picture of one Brownian motion running until it hits the boundary. The program then stops and asks for an input: 1 or 2. If you enter 1, then the program will draw another path in a separated figure. It will keep do this each time you enter a 1. If you enter a 2, it will stop drawing the paths and compute random paths up to some preset number (currently set to 100 in the code) and then draw all the points where these paths hit the boundary. The limit=100 and the starting point can be changed by editing the file.

From fractals to phones: hyperbolic ideas in Euclidean geometry , Dartmouth, Wed. Feb. 28, 2018.
Abstract: Back in the 1990's I was mostly working on problems involving hyperbolic geometry and fractal sets known as Kleinian limit sets. More recently (2016) I have published a couple of papers proving polynomial complexity for certain algorithms solving problems of Euclidean geometry related to meshing, triangulations and Voronoi diagrams. In this talk I will explain the path I followed from one problem to the other, and try to convince you that each step was perfectly natural. In particular, several ideas from hyperbolic geometry proved crucial to obtaining results stated purely in terms of Euclidean geometry. There will be plenty of pictures but few formulas or precise definitions. For the first part of the talk it would be helpful, but not essential, to know the definition of hyperbolic distance in the unit disk and in the 3-dimensional upper half-space, and what a conformal map is.

Keeping your soul in the devils' abode: a simple tale of geometric complexity, Topics in Geometric Function Theory, Les Diablerets, Switzerland, February 11-16, 2018 }
Abstract: This talk is about connections between conformal geometry (harmonic measure, conformal maps, hyperbolic geometry, ...) and computational geometry (Delaunay triangulations, Voronoi diagrams, optimal meshing, ...) and how ideas from each area have proven useful in the other. In particular, I will describe how questions about limit sets of Kleinian groups led me to think about fast numerical computation of conformal mappings, which led to new optimal results about meshing planar domain and finally to polynomial complexity bounds for an old problem about Voronoi diagrams. There will be lots of pictures, a several algorithms, few precise definitions, and almost no formulas.

NOT theory (not knot theory), University of Washington, Feb 23, 2017
Abstract: Here "NOT" means a Non-Obtuse Triangulation, i.e., a triangulation where no angles are bigger than 90 degrees. A PSLG is a planar straight line graph; any finite collection of points and disjoint edges. A triangulation of a PSLG is a triangulation of the point set so that the edges of the triangulation cover all the given edges of the PSLG; we sometimes say the triangulation "conforms" to the PSLG. For various reasons, given a PSLG we would like to construct a conforming triangulation that uses a "small" number of triangles that have "nice" shapes. Here "small" means there is a polynomial bound for the number of triangles in terms of the number of elements in the given PSLG, and "nice" means the triangles are not long and narrow, i.e., there an upper angle bound strictly less than 180 degrees. I will explain why 90 degrees (giving non-obtuse triangles) is the best bound we can hope for and still get polynomial complexity. Linear sized NOTs for polygons were constructed in the early 1990's but the existence of polynomial sized NOTs for PSLGs has remained open until recently (July 2016 DCG). I will show how to construct NOTs in a special case (refining a triangulation of a polygon by diagonals to a non-obtuse triangulation in quadratic time), and then I will discuss some of the difficulties that must be overcome in the general case and how to deal with them. Our results improve various optimal meshing results of Bern, Eppstein, S. Mitchell, Edelsbrunner and Tan, including improved complexity bounds for conforming Delaunay triangulations.

Some Random Geometry Problems , Math Day at the Simons Center for Geometry and Physics, November 12, 2016
I will discuss some open problems about the sizes of some random sets such as the simple random walk, loop erased walks and DLA. There are many pictures and few proofs.

True Trees , Geometry Seminar, Courant Institute , March 29, 2016
I will start with an intuitive introduction to harmonic measure and then discuss finite trees in the plane such that every edge has the same harmonic measure from infinity and each edge is equally likely to from either side; these are the true trees of the title and are special cases of Grothendieck's "dessins d'enfants". If time permits, I will discuss the possible shapes of such trees, the analogous problem for infinite trees, and some examples that arise in the theory of entire functions and transcendental dynamics.

Planar maps with at most six neighbors on average , AMS Sectional Meeting, Stony Brook March 19-20, 2016.
Given a decomposition of the plane into infinitely many cells or countries, how many neighbors can a country have, on average? Suppose that the diameters of the countries are bounded above, that the areas are bounded away from zero, and that we compute averages over the sub-maps defined by containment in an expanding region. We show that the limsup of the averages is less than or equal to 6. The area and diameter conditions are both sharp in the sense that dropping either one allows counterexamples. A weaker conclusion still holds if we don't bound the cell sizes, but control their shapes instead (e.g., convex with bounded aspect ratio). In this case, there is some sequence of expanding sub-maps along which the average number of sides tends to a limit less than or equal to 6.

Snowflakes and Trees , Everything is Complex - A complex analysis conference in honor of N. Makarov, Saas-Fee, Switzerland, March 6-12, 2016
I will start with a very brief description of Makarov's LIL for harmonic functions, his results of the dimension of harmonic measure and some consequences for comparing harmonic measures on two sides of a closed curve. I will then consider harmonic measure on the two sides of each edge of a finite planar tree and make the connection to Grothendieck's dessins d'enfants. If time permits, I will conclude with a discussion of the analogous problems for infinite planar trees and some applications to holomorphic dynamics.

True Trees , SCGP Weekly Talk, Simons Center for Geometry and Physics, 1pm-2pm, Tuesday, December 8, 2015

A finite planar tree has many topologically equivalent drawings in the plane; is there a most natural way to draw it? One possible choice is called the "true form" of the tree. It arises from algebraic geometry and is closely related to Grothendieck's dessins d'enfants. I will describe the true form of a tree (a true tree) in different terms, using harmonic measure, Brownian motion, and conformal maps, and then prove that every planar tree has a true form by using the measurable Riemann mapping theorem. I will then discuss the possible shapes of true trees, e.g., can any compact connected set can be approximated by true trees? If time permits, I will mention the analogous problem for infinite planar trees and some applications to holomorphic dynamics.

Video of `True Trees' talk at SCGP.

Conformal maps and optimal meshes , Rainwater Seminar, University of Washington, Tuesday, Nov. 3 2015.

The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk, but how difficult is it to compute this map? It turns out that the conformal map from the disk to an n-gon can be computed in time O(n), with a constant that depends only on the desired accuracy.

As one might expect, the proof of this is somewhat involved and uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by 2-dimensional computational geometry and 3-dimensional hyperbolic geometry.

In the first part of the talk, I will make a few general remarks about conformal mapping and the Schwarz-Christoffel formula, and then show how some ideas from computational geometry can be used to define a `fast-to-compute' and `roughly correct' version of the Riemann map. In the second part of the talk, I will first explain why this map is `roughly correct' with estimates independent of the domain. The key fact comes from a theorem of Dennis Sullivan's about boundaries of hyperbolic 3-manifolds. I will then explain how fast, approximate conformal mapping leads to new results about optimal meshing, e.g., every simple polygon with n vertices has a quad-mesh that can be computed in linear time where every quadrilateral has angles between 60 and 120 degrees (except for smaller angles of the polygon itself).

Counting on Coincidences , CTY Program, Oct 10, 2016, Stony Brook. We discuss the Birthday problem, the likelihood of disease clusters occurring at random, estimating the size of a set via random samples and the number partitioning problem.

Conformal Mapping and Optimal Meshes , University of Cincinnati, April 10, 2015
Abstract: see same title below.

The NOT theorem , University of Michigan, April 9, 2015
Abstract: Here NOT = non-obtuse triangulation. A PSLG is a planar straight line graph; any finite collection of points and disjoint edges. A triangulation of a PSLG is a triangulation of the point set that covers all the given edges and non-obtuse means all angles are less than equal to 90 degrees. The non-obtuse condition is useful for a variety of practical reasons and is intrinsically interesting because is it known to be the smallest angle bound that is consistent with uniform polynomial size bounds for the mesh. Optimal algorithms for non-obtuse meshing of polygons were proven in the early 1990's but the case of PSLGs has remained open until recently. I will review what is known about such meshes and sketch a proof that a polynomial sized non-obtuse triangulation of a PSLG always exists (our bound is \$n^{2.5}\$ where \$n\$ is the number of vertices in the PSLG). This also gives an improved bound for the conforming Delaunay triangulations.

Conformal Mapping and Optimal Meshes , Georgia Tech, Feb. 18, 2015
Abstract: The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. This result is over a 100 years old, but the study and computation of such maps is still an active area. In this talk I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry.
If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all new angles between 60 and 120 degrees. A closely related result states that any planar triangulation of n points can be refined by adding vertices and edges into a non-obtuse triangulation (no angles bigger than 90 degrees) in time O(n^{2.5}). No polynomial bound was previously known.

Dessins d'adolescents , Rice University, Monday, January 25, 2015, 4pm
I will start by describing the true form of finite planar tree and the connection to polynomials that have exactly two critical values and sketch the proofs that all possible combinatorics and all possible shapes actually occur. I will then describe a generalization to entire functions and the construction of various new examples. Perhaps the most interesting of these is an entire function with bounded singular set that has a wandering domain. It has been known since the 1980's that an entire function with a finite singular set can't have a wandering domain (Dennis Sullivan's proof for rationals extends to this case) and our example shows this is sharp in a precise sense.

Conformal Mapping and Optimal Meshes , UCSD, April 3 2014.
Abstract: The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. This result is over a 100 years old, but the study and computation of such maps is still an active area. In this talk I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry.
If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all new angles between 60 and 120 degrees. A closely related result states that any planar triangulation of n points can be refined by adding vertices and edges into a non-obtuse triangulation (no angles bigger than 90 degrees) in time O(n^{2.5}). No polynomial bound was previously known.

Conformal Mapping and Optimal Meshes , Duke University, Feb 24 2014.

An introduction to Besicovitch-Kakeya sets , Rainwater Seminar, University of Washington, Tuesday, Nov 11 2013
Abstract: It has been known for almost a 100 years that a needle can be moved continuously inside a planar set of arbitrarily small area so that it eventually reverses direction. Moreover, there are compact sets K of zero area that contain unit line segments in every direction. I will start by showing how each is possible, giving the classical construction for the first problem and a recent construction for the second. These are quite elementary and will be accessible to students (both graduate and undergraduate). For the second part of the talk I will discuss some more advanced topics such as
(1) showing a delta-neighborhood of our example has area O(1/log delta),
(2) prove that this is optimal for any Besicovitch set,
(3) discuss an application of Besicovitch sets to Fourier analysis (Fefferman's disk multiplier example),
(4) give Kahane's construction of K via projections of Cantor sets,
(5) discuss Nikodym sets (a zero area union of open half-rays, whose endpoints have full measure).
Even the latter part of the talk will be colloquium style in presentation and should be accessible to students.

Dessins d'adolescents , Rainwater seminar, Dept of Math, Univ. Washington, Nov 14, 2013

Quasiconformal Folding , IPAM Workshop on Dynamics, April 8-12, 2013 . I will start by describing the true form of finite planar tree and the connection to polynomials that have exactly two critical values and sketch the proofs that all possible combinatorics and all possible shapes actually occur. I will then describe a generalization to entire functions and the construction of various new examples. Perhaps the most interesting of these is an entire function with bounded singular set that has a wandering domain. It has been known since the 1980's that an entire function with a finite singular set can't have a wandering domain (Dennis Sullivan's proof for rationals extends to this case) and our example shows this is sharp in a precise sense.

A link to a video of the above talk can be found at the bottom of this page ,

For alternate presentations of the QC folding idea, see lecture slides by Lasse Rempe-Gillen , Xavier Jarque , Sebastien Godillon , Simon Albrech .

Dessins d'adolescents , Dynamics Learning Seminar, Stony Brook, April 3, 2013

Non-obtuse triangulations of PSLGs , Courant Institute, 6pm Tuesday, March 12, 2013 A PSLG is a planar straight line graph; any finite collection of points and disjoint edges. A triangulation of a PSLG is a triangulation of the point set that covers all the given edges and non-obtuse means all angles are less than equal to 90 degrees. The non-obtuse condition is useful for a variety of practical reasons and is intrinsically interesting because is it known to be the smallest angle bound that is consistent with uniform polynomial size bounds for the mesh. Optimal algorithms for non-obtuse meshing of polygons were proven in the early 1990's but the case of PSLGs has remained open until recently. I will review what is known about such meshes and sketch a proof that a polynomial sized non-obtuse triangulation of a PSLG always exists (our bound is \$n^{2.5}\$ where \$n\$ is the number of vertices in the PSLG). This also gives an improved bound for the conforming Delaunay triangulations.

Mappings and Meshes (invited talk, SoCG 2012) , This is general talk about how the medial axis plays a role in the asymptotically fastest known method of computing Riemann maps and how conformal and hyperbolic geometry play roles in optimal meshing results for polygons and PSLGs.

A one page abstract for the above talk and some other analysis talks given at SoCG 12 can be found at analysis workshop webpage .

Constructing entire functions by QC folding , Dynamics Learning Seminar, Stony Brook, March 28, 2012. I describe some of the results in the preprint of the same title. We construct entire functions with finite or bounded singular sets using quasiconformal maps of a half-plane into itself that I call ``foldings''. One application that I will describe carefully is the construction of a entire function with bounded singular set that has a wandering domain. This has been open since the 1980's when Sullivan's proof for rational functions was extended to entire functions with finite singular set.

How to Draw a Conformal Map , Math Club, Stony Brook, Fall 2011, Intro to computing conformal maps using Schwarz-Christoffel formula and iterative algorithms for finding the parameters.

Conformal Maps, Optimal Meshing and Sullivan's Convex Hull Theorem , Math Dept Colloquium, Stony Brook, Thursday, March 3, 2011

Conformal Maps, Hyperbolic Geometry and Optimal Meshing , FWCG 2010, Stony Brook, Saturday, October 29, 2010

Nonobtuse Triangulation of PSLGs , CG problem group, Stony Brook, Tuesday, Oct 12, 2010

Conformal Mapping in Linear Time , FWCT 2009, Tufts, Saturday, Nov 14, 2009,

Random walks in analysis , Simons Center for Geometry and Physics, Tuesday, Nov 10, 2009,

Conformal Mapping in Linear Time , CG problem session, Oct 27, 2009, Stony Brook

Counting on Coincidences , CTY Program, Oct 3, 2009, Stony Brook

An A_1 weight not comparable to any QC Jacobian , Memorial Conference for Juha Heinonen, Ann Arbor, May 12-16, 2008. 12 pages. We sketch the proof of the claim in the title. The idea is to construct a Sierpinski carpet with the property that that certain QC images must contain a rectifiable curve. As a corollary, we show that that there is a surface in R^3 that is quasisymmetrically equivalent to the plane, but not biLipschitz equivalent.

University of Maryland, May 14, 2007 : Conformal welding and Koebe's theorem, PDF file

The following three files are talks on the same subject but with a slightly different emphasis and organization in each one. Many pages are simply figures which I explain in the talk; if you need further explanation, you can refer to my preprint of the same name, or email me.

Workshop on Computational and Conformal Geometry, Stony Brook April 20, 2007 : Conformal mapping in linear time. video of my talk and videos for all workshop talks

UW Seattle, Wed Jan 17 2007 : An A_1 weight not comparable to any quasiconformal Jacobian

Microsoft Research, Seattle, Tue Jan 16 2007 : Conformal mapping in linear time. This file is really a superset of talk.

Delaware.pdf : this is a pdf version of the transparencies for my talk at the University of Delaware, Nov 28, 2005 ``Conformal Mapping in Linear Time''. (This is a big file, about 5M, so may take some time to download).

ABcoll.pdf : this is a pdf version of the transparencies for my talk at the Ahlfors-Bers colloquium, May 21, 2005 ``Conformal Mapping in Linear Time''.

Minnesota.pdf : this is a pdf version of the transparencies for my talk at Minnesota, April 14, 2005, ``Conformal Mapping in Linear Time''.

postscript : this is a postscript version of the transparencies for my talk, ``A fast approximation of the Riemann map'' given at Brown University, Feb 2004.

Barrett lectures : this is a postscript version of the transparencies for my June 1998 Barrett lectures talk, "Measures, martingales, manifolds and mappings". Click here for the dvi version (no figures).

postscript , pdf : this is a postscript version of the transparencies for my talk ``Conformal welding and Koebe's theorem''

postscript : this is a postscript version of the transparencies for my colloquium, ``Conformal maps, convex hulls and Kleinian groups''.

postscript : this is a postscript version of the transparencies for my talk, ``Hausdorff dimension of limit sets''

Video : of my talk "Harmonic Measure, Arclength and Schwarzian Derivative" at the CUNY Einstein Chair Mathematics Seminar , March 10, 1992.

Here is some material on Kleinian groups I prepared for my lectures in Segovia in June 1996: dvi file with definitions related to Kleinian groups. dvi file with references related to Kleinian groups. dvi file with an outline of my lectures.