Lectures

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.
The version posted now is preliminary; I will definitely make some changes before giving the talk, but this about 90 percent correct.

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

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 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.

Wesleyan, April 6, 2006 :

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.