Harmonic measure, true trees
and quasiconformal folding,
International Congress of Mathematicians, Rio de Janerio,
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.

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 algorithims
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 PLSG 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, Edelsbruner 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 comapring harmonic
measures on two sides of a closed curve. I will then consider
harmnoic 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 hyperpolic 3-mainfolds.
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 likelyhood of disease clusters occuring 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 PLSG 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 intrisically interesting because is it known to be
the smallest angle bound that is consistent with uniform polynomial
size bounds for the mesh.
Optimal algroithms 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 combintorics
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 gthe 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 combintorics 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 gthe 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 trinagulation of a PLSG 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 intrisically interesting because is it known to be the smallest angle bound that is consistent with uniform polynomial size bounds for the mesh. Optimal algroithms 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 applciation that I will desribe 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 enite 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 algoithms 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 Sierpinsky 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 explaination, 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 transparences 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 transparences 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 transparences for my talk at Minnesota, April 14, 2005, ``Conformal Mapping in Linear Time''.

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

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

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

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

Here is some material on Kleinian groups I perpared 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.