Lectures

Recent lectures (most recent at top)

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Rutgers colloquium, December 2, 2022.

Weil-Petersson curves, knot energies, traveling salesman theorems, and minimal surfaces, SCGP Workshop on Computational Differential Geometry and it's Applications in Physics, Simons Center for Geometry and Physics, Thur. Nov 17, 2022. Video recording of lecture

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Vanderbilt University, Colloquium, Thur. Oct. 8,, 2022.

Abstract: Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other. The lecture will contain many pictures, several definitions, but not too many proofs or technical details.

Optimal Triangulation of Polygons , Texas A&M Colloquium, Thur. Oct 12, 2022.

Abstract: It is a long-standing problem of computational geometry is to triangulate a polygon using the best possible shapes, e.g., to minimize the maximum angle used (MinMax), or maximize the minimum angle (MaxMin). Besides the problem's intrinsic interest, well formed meshes give better results in various numerical algorithms, such as the finite element method. If we triangulate using only diagonals of the polygon, then there are only finitely many possible triangulations and the famous Delaunay triangulation solves the MaxMin problem. When extra vertices (Steiner points) are allowed, the set of possible triangulations becomes infinite dimensional, but I recently proved that the optimal angle bounds for either the MinMax or MaxMin problems can be easily computed, and are (usually) attained by some triangulation. I will prove some previously known necessary conditions on the angle bounds using Euler's formula for planar graphs, and briefly describe the new theorem that they are also sufficient; the proof of this uses conformal and quasiconformal mappings, but our discussion is independent of the previous lecture. Several surprising consequences follow, and many related problems remain open.

Conformal Mapping in Linear Time , Texas A&M Colloquium, Wed Oct 11, 2022.

Abstract: What do hyperbolic 3-manifolds have to do with the Riemann mapping theorem? In this talk, I will explain how a theorem of Dennis Sullivan (based on an observation of Bill Thurston) about convex sets in hyperbolic 3-space leads to a fast algorithm for computing conformal maps. The conformal map from the unit disk to the interior of a polygon is given by the Schwarz-Christoffel formula, but this formula is stated in terms of parameters that are hard to compute. I will explain a fast way to approximate these parameters: the speed comes from the medial axis, a type of Voronoi diagram from computational geometry, and the accuracy is proven using Sullivan's theorem. At the end of the lecture, I will mention various applications to discrete geometry and optimal meshing; one of these will be the subject of the second lecture.

Optimal Triangulation of Polygons , Stony Brook Computational Geometry Group (online) , Tuesday, Oct 4, 2022.

Abstract: It is a long-standing problem of computational geometry is to triangulate a polygon using the best possible shapes, e.g., to minimize the maximum angle used (MinMax), or maximize the minimum angle (MaxMin). Besides the problem's intrinsic interest, well formed meshes give better results in various numerical algorithms, such as the finite element method. If we triangulate using only diagonals of the polygon, then there are only finitely many possible triangulations and the famous Delaunay triangulation solves the MaxMin problem. When extra vertices (Steiner points) are allowed, the set of possible triangulations becomes infinite dimensional, but I recently proved that the optimal angle bounds for either the MinMax or MaxMin problems can be easily computed, and are (usually) attained by some triangulation. I will prove some previously known necessary conditions on the angle bounds using Euler's formula for planar graphs, and briefly describe the new theorem that they are also sufficient; the proof of this uses conformal and quasiconformal mappings, but our discussion is independent of the previous lecture. Several surprising consequences follow, and many related problems remain open.

Optimal Triangulation of Polygons , Stony Brook Analysis Seminar , Fri, Sept 30, 2022.

Abstract: It is a long-standing problem of computational geometry is to triangulate a polygon using the best possible shapes, e.g., to minimize the maximum angle used (MinMax), or maximize the minimum angle (MaxMin). Besides the problem's intrinsic interest, well formed meshes give better results in various numerical algorithms, such as the finite element method. If we triangulate using only diagonals of the polygon, then there are only finitely many possible triangulations and the famous Delaunay triangulation solves the MaxMin problem. When extra vertices (Steiner points) are allowed, the set of possible triangulations becomes infinite dimensional, but I recently proved that the optimal angle bounds for either the MinMax or MaxMin problems can be easily computed, and are (usually) attained by some triangulation. I will prove some previously known necessary conditions on the angle bounds using Euler's formula for planar graphs, and briefly describe the new theorem that they are also sufficient; the proof of this uses conformal and quasiconformal mappings, but our discussion is independent of the previous lecture. Several surprising consequences follow, and many related problems remain open.

Quasiconformal folding: trees, triangles and tracts , Conference on “Complex Analysis, Geometry, and Dynamics”, Portorož (Slovenia), June 20th to June 24th, 2022.

Abstract: I will discuss quasiconformal folding, a type of quasiconformal surgery that carefully introduces critical points in order to construct holomorphic functions of one variable with good control of both the geometry and the singular values. As applications I will discuss true trees (i.e., a special case of Grothendieck's dessins d'enfants on the sphere), equilateral triangulations of Riemann surfaces, a strengthening of Runge's theorem, and the construction of entire functions with specified geometry and bounded singular sets.

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Univ. of Illinois, Urbana-Champaign , Colloquium, Wed. April 27, 2022.

Abstract: Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, probability theory, knot theory, applied mathematics, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than thirty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other.

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Texas A&M, Colloquium, Thur. Feb 24, 2022.

Abstract: Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, probability theory, knot theory, applied mathematics, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than thirty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic space that has finite total curvature. I will discuss several such characterizations and sketch why they are all equivalent to each other.

Dessins and Dynamics , Texas A&M, Groups and Dyanamics Seminar, Wed. Feb 23, 2022.

Abstract: After defining harmonic measure on a planar domain, I will discuss "true trees", i.e., trees drawn in the plane so that every edge has equal harmonic measure and so that these measures are symmetric on each edge. True trees on the 2-sphere are a special case in Grothendieck's theory of dessins d'enfant, where a graph on a topological surface induces a conformal structure on that surface. I will recall the connection between dessins, equilateral triangulations and branched coverings (Belyi's theorem). I will also describe some recent applications of these ideas to holomorphic dynamics: approximating continuua by polynomial Julia sets, finding meromorphic functions with prescribed postcritical orbits, constructing new dynamical systems on hyperbolic Riemann surfaces, building wandering domains for entire functions, and estimating the fractal dimensions of transcendental Julia sets. There will be many pictures, but few proofs.

SODA 2022, Jan 9-12, Alexandria VA

Lecture slides for: Optimal angle bounds for Steiner triangulations of polygons.

Conference website: Symposium on Discrete Algorithms (SODA22), Jan 10 -12, 2022,

MP4 recording of talk (22 minutes). (video/audio recording stored on www.stonybrook.edu)

Zoom recording of talk (22 minutes) (copy stored on Zoom cloud; include links to ful video, audio only, and transcript file)

Papers this presentation is based on: Full version with all details (52 pages) and summary of main results and proof sketch (19 pages)

Abstract: or any simple polygon $P$ we compute the optimal upper and lower angle bounds for triangulating $P$ with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an $N$-gon are computable in time $O(N)$, even though the number of triangles needed to attain these bounds has no bound in terms of $N$ alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle $\theta$ has a triangulation with all angles in the interval $I=[ \theta , 90^\circ - \min(36^\circ, \theta)/2]$, and for $\theta \leq 36^\circ$ both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver.

Fast conformal mapping via computational and hyperbolic geometry. Computational methods and Function Theory, Jan 10-14, 2022 .

Abstract: The conformal map from the unit disk to the interior of a polygon P is given by the Schwarz-Christoffel formula, but this is stated in terms of parameters that are hard to compute from P. After some background and motivation, I explain how the medial axis of a domain, a concept from computational geometry, can be used to give a fast approximation to these parameters, with bounds on the accuracy that are independent of P. The precise statement involves quasiconformal mappings, and proving these bounds uses a result about hyperbolic convex sets originating in Thurston's work on 3-manifolds. If time permits, I will mention some applications to optimal meshing and triangulation of planar polygons.

Conformal removability is hard MAT 626 guest lecture, Monday Nov 29, 2021, 9:45 am. Online.

Abstract: Suppose E is a compact set in the complex plane and U is its complement. The set E is called removable for a property P, if any holomorphic function on U with this property extends to be holomorphic on the whole plane. This is an important concept with applications in complex analysis, dynamics and probability. Tolsa famously characterized removable sets for bounded holomorphic functions, but such a characterization remains unknown for conformal maps on U that extend homeomorphically to the boundary. We offer an explanation for why the latter problem is actually harder: the collection of removable sets for bounded holomorphic maps is a G-delta set in the space of compact planar sets with the Hausdorff metric, but the collection of conformally removable sets is not even a Borel subset of this space. These results follow from known facts, but they suggest a number of new questions about fractals, removable curves and conformal welding.

Random walks, true trees and equilateral triangulations. Seminar on graphs on surfaces and curves over number fields, Seminar website and Video Recording Wed. Nov 17, 18:30pm (10:30am New York).

Abstract: I will start by reviewing the definition and basic properties of harmonic measure on planar domains, i.e., the first hitting distribution of a Brownian motion on the boundary of a domain. For example, how does this distribution depend of the starting point? For a tree embedded in the plane, can both sides of every edge have equal harmonic measure? If so, we call this the ``true form of the tree'' or a ``true tree'' for short. These are related to Grothendieck's dessins d'enfants and I will explain why every planar tree has a true form, and what these trees can look like. The proofs involve quasiconformal maps and will only be sketched. I will also discuss the application of these ideas to Belyi functions and building Riemann surfaces by gluing together equilateral triangles. If time (and the audience) permits, I will briefly describe a generalization from finite trees and polynomials to infinite trees and entire functions.

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Brazilian Approximation Theory and Harmonic Analysis Webinar, Nov 11, 2021, 2pm (NY), 4pm (Brazil). video of talk

We describe several new characterizations of Weil-Petersson curves. These curves are the closure of the smooth planar closed curves for the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Their work was motivated by problems in string theory, but the same class arises naturally in geometric function theory, Mumford's work on computer vision, and the theory of Schramm-Loewner evolutions (SLE). The new characteriztions include quantities such as Sobolev smoothness, Mobius energy, fixed curves of biLipschitz involutions, Peter Jones's beta-numbers, the thickness of hyperbolic convex hulls, the total curvature of minimal surfaces in hyperbolic space, and the renormalized area of these surfaces. Moreover, these characterizations extend to higher dimensions and remain equivalent there.

Dimensions of transcendental Julia sets. At the conference “On geometric complexity of Julia sets - III”, September 26 to October 1, 2021. Stefan Banach International Mathematical Center, IMPAN Mathematical Research and Conference Center in Będlewo, Poland. Conference Website .

Abstract: This is a survey of some known results about Hausdorff and packing dimension of Julia sets of transcendental entire functions. After reviewing the basic definitions and some analogous results for polynomials, I will discuss Baker's theorem that the Hausdorff dimension is always at least 1, and describe examples showing that all values in the interval [1,2] can be attained. Whereas in polynomial dynamics it is hard to construct Julia sets with dimension 2 or positive area, in transcendental dynamics the difficult problem is to build "small" examples, e.g., dimension close to 1 or having finite (spherical) length. I will end by stating some open problems.

Transcendental functions with small singular sets at the confrence "Advancing Bridges in Complex Dynamics" in Luminy, September 20-24, 2021. Conference Website .

Video of lecture

Abstract: Abstract: I will introduce the Speiser and Eremenko-Lyubich classes of transcendental entire functions and give a brief review of quasiconformal maps and the measurable Riemann mapping theorem. I will then discuss tracts and models for the Eremenko-Lyubich class and state the theorem that all topological tracts can occur in this class. A more limited result for the Speiser class will also be given. I will then discuss some applications of these ideas, focusing on recent work with Kirill Lazebnik (prescribing postsingular orbits of meromorphic functions) and Lasse Rempe (equilateral triangulations of Riemann surfaces).

Fast conformal mapping via computational and hyperbolic geometry. Here is the Zoom recording of the lecture. This talk was in the CAvid seminar, 9:00am (New York time) September 14 , 2021.

Abstract: The conformal map from the unit disk to the interior of a polygon P is given by the Schwarz-Christoffel formula, but this is stated in terms of parameters that are hard to compute from P. After some background and motivation, I explain how the medial axis of a domain, a concept from computational geometry, can be used to give a fast approximation to these parameters, with bounds on the accuracy that are independent of P. The precise statement involves quasiconformal mappings, and proving these bounds uses a result about hyperbolic convex sets originating in Thurston's work on 3-manifolds. If time permits, I will mention some applications to optimal meshing and triangulation of planar polygons.

Conformal removability is hard Mathematical Congress of the Americas, 5:10 ET, Thursday July 15, 2021. Session: S13 - Harmonic Analysis, Fractal Geometry, and Applications

Abstract: Suppose E is a compact set in the complex plane and U is its complement. The set E is called removable for a property P, if any holomorphic function on U with this property extends to be holomorphic on the whole plane. This is an important concept with applications in complex analysis, dynamics and probability. Tolsa famously characterized removable sets for bounded holomorphic functions, but such a characterization remains unknown for conformal maps on U that extend homeomorphically to the boundary. We offer an explanation for why the latter problem is actually harder: the collection of removable sets for bounded holomorphic maps is a G-delta set in the space of compact planar sets with the Hausdorff metric, but the collection of conformally removable sets is not even a Borel subset of this space. These results follow from known facts, but they suggest a number of new questions about fractals, removable curves and conformal welding.

Mappings and Meshes I: connections between continuous and discrete geometry and the link to Video of talk 1
Mappings and Meshes II: connections between continuous and discrete geometry and the link to Video of talk 2
This is 2-part minicourse at the online conference FRG Workshop on Geometric Methods for Analyzing Discrete Shapes Harvard, Center of Mathematical Science and Applications, May 7-9, 2021. My lectures at 11am May 7 and 11am May 8 (Boston time). Links to videos of all talks may be found of this webpage.

Abstract: I will give two lectures about some interactions between conformal, hyperbolic and computational geometry. The first lecture shows how ideas from discrete and computational geometry can help compute conformal mappings, and the second lecture reverses the direction and shows how conformal maps can give meshes of polygonal domains with optimal geometry.

Lecture 1: The conformal map from the unit disk to the interior of a polygon P is given by the Schwarz-Christoffel formula, but this is stated in terms of parameters that are hard to compute from P. After some background and motivation, I explain how the medial axis of a domain, a concept from computation geometry, can be used to give a fast approximation to these parameters, with bounds on the accuracy that are independent of P. The precise statement involves quasiconformal mappings, and proving these bounds uses a result about hyperbolic convex sets originating in Thurston's work on 3-manifolds.

Lecture 2: I will show how conformal maps can be used to give meshes with sharp angle bounds on the mesh elements. For example: (1) every polygon P has a linear sized quad-mesh with all angles between 60 and 120 degrees (except at smaller angles of P), (2) every polygon has an acute triangulation and the optimal upper angle bound can be computed for each P, (3) every planar straight line graph (PSLG) has a conforming acute triangulation of polynomial complexity. In addition to conformal mappings, we also use some ideas from dynamics (a discrete closing lemma) and Riemann surfaces (the thick-thin decomposition).

Trees and Triangles
Abstract: This is an 80 minute guest lecture in Misha Lyubich's class on quaisconformal mappings. This follows the first Barcelona lecture described below, followed by a description of the folding theorem taken from the second Barcelona lecture (but much shorter). on planar domains and consider the problem of constructing conformally

Slides for Trees, Triangles and Tracts, Part I and Trees, Triangles and Tracts, Part II . Link to Video . (one YouTube video covers both talks).
This is 2-part minicourse at the online conference Transcendental Dynamics and Beyond: topics in complex dynamics 2021, (Centre de Recerca Matemàtica, Barcelona, April 19-23, 2021 (8:30am New York, 2:30pm Barcelona).

Links for all recorded conference talks

Abstract: These two lectures will motivate, state and apply a method for constructing holomorphic functions called quasiconformal folding.

In the first lecture, I will review the definition and some properties of harmonic measure on planar domains 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. We will address the questions of which trees can be drawn in this way, and what are the possible shapes of such trees. I will end by discussing joint work with Lasse Rempe showing that every non-compact Riemann surface is a branched cover of the 2-sphere with 3 branch points.

The second lecture considers the analogous connection between infinite planar trees and entire functions with two singular values. I will state the Folding Theorem, which takes an infinite planar tree with certain geometric assumptions, and returns an entire function with two singular values. I will sketch the idea of the proof, and then describe some applications of the folding construction to transcendental dynamics, e.g., the construction of wandering domains in the Eremenko-Lyubich class, the construction of transcendental Julia sets with dimension close to 1, and constructing meromorphic functions with prescribed post-singular dynamics.

Random thoughts on random sets Online -- 6:35pm, Wednesday , April 7, 2021 Joint meeting of Stony Brook Math Club and University Indiana, Bloomington Math Clubs

Abstract: Many research level problems in mathematics require lots of technical looking definitions from many advanced classes just to state, but the goal of this talk is to discuss some easy-to-state open problems about random sets in the plane. Most of these problems are probably pretty hard: several have been attacked by Fields medalists without success. There will be lots of pictures, some numerical evidence, but not much in the way of proofs. I will start with an informal introduction to Brownian motion as the limit of discrete random walks, and define harmonic measure as the first hitting distribution of Brownian motion in a domain on the boundary. We will then discuss some known facts about Brownian paths in the plane and state some open problems about them, e.g., does a Brownian path cover a rectifiable path? We will end with a discussion of DLA (diffusion limited aggregation), a much studied random growth process about which very little is rigorously known.

Random walks, true trees and equilateral triangulations. Seymour Sherman Memorial Lecture, 4pm, Friday , March 19, 2021, Indiana University at Bloomington.

Abstract: I will start by reviewing the definition and basic properties of harmonic measure on planar domains, i.e., the first hitting distribution of a Brownian motion on the boundary of a domain. For example, how does this distribution depend of the starting point? What if two starting points are on opposite sides of a closed Jordan curve? For a tree embedded in the plane, can both sides of every edge have equal harmonic measure? If so, we call this the ``true form of the tree'' or a ``true tree'' for short. These are related to Grothendieck's dessins d'enfants and I will explain why every planar tree has a true form, and what these trees can look like. I will also discuss the application of these ideas to building Riemann surfaces by gluing together equilateral triangles, and, if time permits, applications to holomorphic dynamics via the folding theorem.

Random thoughts on random sets 1pm, Thursday , March 11, 2021 SCGP Weekly Meeting In Math

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.

Random thoughts on random sets 10am NY time, 3pm London time. Tues, March 2, 2021 St. Andrews Analysis Seminar,
Link to St Andrews lecture recording.

Weil-Petersson curves, beta-numbers, and minimal surfaces, Thur, Feb 18, 2021 Stony Brook Colloquium (online). Link to Video.

Weil-Petersson curves, beta-numbers, and minimal surfaces, Tue, Feb 16, 2021, CUNY Graduate Center Hyperbolic Geometry Seminar (online).

Weil-Petersson curves, beta-numbers, and minimal surfaces, Tue, Feb 9, 2021 MIT PDE and analysis seminar (online).

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Tue, Jan 12, 2021 Quasiworld seninar (online).

This is similar to the 60 minute Princeton talk below, but a few topics have been dropped to fit in the 50 minute format. Link to lecture recording.

Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Wed, Nov 18, 2020, Differential Geometry and Geometric Analysis Seminar (online), Princeton University.

We describe several new characterizations of Weil-Petersson curves. These curves are the closure of the smooth planar closed curves for the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Their work was motivated by problems in string theory, but the same class arises naturally in geometric function theory, Mumford's work on computer vision, and the theory of Schramm-Loewner evolutions (SLE). The new characteriztions include quantities such as Sobolev smoothness, Mobius energy, fixed curves of biLipschitz involutions, Peter Jones's beta-numbers, the thickness of hyperbolic convex hulls, the total curvature of minimal surfaces in hyperbolic space, and the renormalized area of these surfaces. Moreover, these characterizations extend to higher dimensions and remain equivalent there.

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.

Nick's favorite things (audio) , (words)

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

Optimal Meshing ,

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.

Wesleyan, April 6, 2006 :

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.