• Weil-Petersson curves, traveling salesman theorems, and minimal surfaces, Purdue Geometry and Geometric Analysis Seminar, Purdue University, MArch 3, 2023.

• 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 2006. 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 2006. 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 2006. 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.

• 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.