Modern Aspects of Complex Geometry

A conference in honor of

Taft Professor David Minda


14-17 May 2015

Funded by the National Science Foundation, the Charles Phelps Taft Memorial Fund, the UC McMicken College of Arts & Sciences, and the UC Department of Mathematical Sciences




Room 800 Swift Hall

Start of Conference

14 May 2015

8:30-8:50
Dean Ken Petren
Welcome to UC

Schedule of Talks

14-15 May 2015ThursdayFriday
9:00-9:40Roger BarnardJang-Mei Wu
9:50-10:30Eric SchippersMario Bonk
10:30-11:00CoffeeCoffee
11:00-11:40Alex SolyninHuy Tran
11:40-1:50LunchLunch
1:50-2:30Marshall WilliamsDaniel Meyer
2:40-3:20Marie SnipesDavid Freeman
3:20-3:50CoffeeCoffee
3:50-4:30Pietro Poggi-Corradini Jani Onninen
4:40-5:20Hrant Hakobyan Mattti Vuorinen
16-17 May 2015SaturdaySunday
9:00-9:40Ken StephensonAimo Hinkkanen
9:50-10:30Donald E. MarshallJohn Parker
10:30-11:00CoffeeCoffee
11:00-11:40Steffen RohdeWilliam Cherry
11:40-1:50LunchLunch
1:50-2:30Dick CanaryOliver Roth
2:40-3:20Petra Bonfert-TaylorToshiyuki Sugawa
3:20-3:50CoffeeCoffee
3:50-4:30Zair IbragimovPatrick Ng

Titles and Abstracts of Talks

Professor Roger Barnard, Texas Tech University, TX

Title Bohr's phenomenon for power series
Abstract Given the power series $f(z)=\sum_{k=0}^\infty a_k z^k$ its majorant series is defined by $M_f(z)=\sum_{k=0}^\infty |a_k| |z|^k$. In 1914, Harold Bohr proved a weaker version of the following.
Theorem If $|\sum_{k=0}^\infty a_k z^k|\le 1$ in the unit disk $\mathbb{D}$ then $M_f\le 1$ in the disk $\mathbb{D}_{1/3}$ centered at $0$ with radius $1/3$. The radius $1/3$ is the best possible.
The notion of Bohr's radius, initially defined for mappings from $\mathbb{D}$ to itself, can be generalized by rewriting Bohr's inequality in the equivalent form as $\sum_{k=1}^\infty |a_k| |z|^k\le 1-|a_0|$. Then, the right-hand side $1-|a_0|$ can be interpreted as the distance $\text{dist}(f(0), \partial\mathbb{D})$. In this form, the notion of Bohr's radius can be generalized for a class of functions $f$ analytic in $\mathbb{D}$ which take values in a given domain $G$ as follows:
Problem For a given domain $G\subset\mathbb{C}$, find the largest radius $r_G > 0$ such that \[ \text{dist}(M_f(z); |f(z)|)\sum_{k=1}^\infty |a_k||z|^k\le \text{dist}(f(0),\partial G) \] for all $|z|\le r_G$ and all functions $f$ analytic in $\mathbb{D}$ such that $f(\mathbb{D})\subset G$.
We discuss our solution to this problem in a number of classes and show an interesting connection of the solution to this problem to the Brannan Conjecture on the odd coefficients of the power series for the function \[ \left(\frac{1+e^{i\theta}z}{1-z}\right)^\alpha,\qquad \qquad \qquad 0<\alpha<1. \] This is joint work with R. Ali and A. Yu. Solynin.

Professor Petra Bonfert-Taylor, Dartmouth College, NH

Title A survey of quasiconformal homogeneity
Abstract In 1976, Gehring and Palka introduced the notion of (uniform) quasiconformal homogeneity for subsets of $\overline{\mathbb{R}}^n$. Their definition was later extended to hyperbolic manifolds: A (complete) hyperbolic manifold M is said to be $K$-quasiconformally homogeneous if for all $x,y\in M$, there exists a $K$-quasiconformal homeomorphism $f:M\to M$ such that $f(x) = y$. In this talk we will survey the theory of quasiconformally homogeneous subsets of $\overline{\mathbb{R}}^n$ and quasiconformally homogeneous hyperbolic manifolds. We will furthermore include a discussion of open problems in the theory.

Professor Mario Bonk, UCLA, CA

Title Dynamics and quasiconformal geometry
Abstract Questions in geometry group theory or complex dynamics lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston's characterization of rational functions with finite postcritical set. Fractal 2-spheres, Sierpi\'nski carpets, or continuum trees are typical spaces for which a deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will give a survey on some recent developments in this area.

Professor Richard Canary, University of Michigan, MI

Title The Möbius metric and convex cores of hyperbolic 3-manifolds
Abstract We will discuss the Möbius metric on a hyperbolic domain in the Riemann sphere(aka the Kulkarni-Pinkall or Thurston metric) . If the domain is uniformly perfect, then the nearest point retraction onto the convex hull of the complement of the domain is a 1-Lipschitz (9,8)-quasi-isometry with respect to the Möbius metric. One may use this result to obtain bounds on the quasiconformal dilatation of the extremal map between the conformal boundary and the boundary of the convex core of a hyperbolic 3-manifold. One may also bound the difference between the volume of the convex core and the renormalized volume of a convex cocompact hyperbolic 3-manifold. We will highlight related work of Herron-Liu-Minda and Herron-Ma-Minda. (Joint work with Martin Bridgeman)

Professor William Cherry, University of North Texas, TX

Title Fubuni-Study derivatives of holomorphic maps to hyperplane complements and speculation on the extremal maps
Abstract I will recall joint work with Eremenko on effective bounds of the Fubini-Study derivative at the origin of holomorphic maps from the disc to projective space omitting sufficiently many hyperplanes in general position. I will focus on the sharpness of our bounds as the hyperplanes degenerate, and I will discuss the result of a recent REU project done by M. Fincher and H. Olney on the geometry of projective simplicies to motivate some speculation on the nature of the extremal mappings.

Professor David Freeman, University of Cincinnati Blue Ash, OH

Title Inversion Invariant Homogeneous Metric Spaces
Abstract Our talk will investigate homogeneous metric spaces that are invariant (up to equivalence via bi-Lipschitz homeomorphisms) under metric inversion. By the term "metric inversion" we refer to a metric generalization of classical Mobius inversion in Euclidean space. By the term "homogeneous" we refer to spaces that admit transitive families of self-isometries or of uniformly bi-Lipschitz self-homeomorphisms. Our aim is to consider the extent to which Euclidean spaces and certain classes of Carnot groups are characterized by their homogeneity and inversion invariance.

Professor Hrant Hakobyan, Kansas State University, KS

Title Modulus of measures and dimension distortion
Abstract We show that under quite general conditions a quasiconformal map preserves Hausdorff dimension of a generic Ahlfors regular subset of a metric space. In particular, if E is a bounded Ahlfors regular subset of a Euclidean space then dimension of a random translate of E is preserved by quasiconformal mappings. The main technique used in this work is Fuglede's modulus of measures. This is joint work with Chris Bishop and Marshall Williams.

Professor Aimo Hinkkanen, University of Illinois at Urbana-Champaign, IL

Title Another look at the complex dilatation
Abstract We show how the measurable Riemann mapping theorem can also be proved by using the Cartan-Kähler theory together with standard results on approximation of quasiconformal mappings in the plane.

Professor Zair Ibragimov, California State University, Fullerton, CA

Title The Cassinian metric
Abstract The Cassinian metric of a proper domain of the Euclidean space was introduced by the author in 2009. Geometrically, it can be defined by means of Cassinian ovals and hence the name. In this talk I will give latest results on the geometry of the Cassinian metric including its relations to other well-known metrics as well as its distortion properties under Moebius transformations.

Professor Donald E. Marshall, University of Washington at Seattle, WA

Title Conformal Welding and Planar Graphs
Abstract We will discuss the recent application of conformal welding to the construction of Grothendieck's dessins d'enfants, and associated Belyi rational functions. Every planar graph can be perturbed by a homeomorphism of the plane so that it becomes the preimage of the interval $[0,1]$ for a rational function with only three critical values. We use the zipper conformal mapping technique to approximate the associated rational function. The solutions are close enough that Newton's method improves the accuracy to thousands of digits. This gives a fast ``hands-free'' method sufficient to construct a catalog of all planar graphs (up to planar homeomorphism) with a small number of edges. An introduction to these graphs and their application to Galois Theory, Algebraic Geometry, and Quantum Fields can be found in ``Graphs on Surfaces and Their Applications'', by S. Lando and A. Zvonkin. Our talk will focus on the computational aspects of this new application of conformal maps. This is joint work with S. Rohde.

Professor Daniel Meyer, University of Jyväskylä, Finland

Title From fractal spheres to rational maps to groups
Abstact We consider certain self-similar surfaces. These may be realized as rational maps. Conversely, for a rational map that is postcritically finite one may construct a so called visual metric on the sphere with many interesting properties. The iterated monodromy group is a group that is assigned to such a rational map. It is a so called self-similar group. The map may in turn be described via such a group. This is partly joint work with Mario Bonk, and partly joint work with Mikhail Hlushchanka.

Professor Patrick Ng, University of Hong Kong

Title Polynomials Versus Finite Blaschke Products
Abstact In this talk, we shall compare polynomials of one complex variable and finite Blaschke products and demonstrate that they share many similar properties. In fact, we will see that one can establish a dictionary between polynomials and finite Blaschke products. This also motivates one to consider some problems in finite Blaschke products, for example, Smale's mean value conjecture for finite Blaschke products.

Professor Jani Onninen, Syracuse University, NY

Title Existence of 2D traction free minimal deformations
Abstract In Geometric Function Theory we seek, as a generalization of the Riemann Mapping Problem, homeomorphisms that minimize certain energy integrals. No boundary values of such homeomorphisms are prescribed. This is interpreted as saying that the deformations are allowed to slip along the boundary, known as traction free problems. This leads us to determine the infimum of a given energy functional among homeomorphisms from $X$ onto $Y$. Even in the basic case of the Dirichlet energy, it is certainly unrealistic to require that the infimum energy be attained within the class of homeomorphisms. Of course, enlarging the set of admissible mappings can change the nature of the energy-minimal solutions. To avoid the Lavrentiev phenomenon one is forced to show that the minimizing sequence actually converges strongly, which is the subject of my talk.

Professor John Parker, Durham University, England

Title Non-arithmetic lattices
Abstract If G is a semi-simple Lie group, it is known that all lattices are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1). Non-arithmetic lattices have been constructed in SO(n,1) for all n and there are infinitely many non-arithmetic lattices in SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability classes of non-arithmetic lattices in SU(2,1) and a single example in SU(3,1). The problem is open for n at least 4. I will survey the history of this problem, and then describe recent joint work with Martin Deraux and Julien Paupert, where we construct 10 new commensurability classes of non-arithmetic lattices in SU(2,1). These are the first examples to be constructed since the work of Deligne and Mostow in 1986.

Professor Pietro Poggi-Corradini, Kansas State University, KS

Title Modulus of walks as a tool for studying epidemics and vice versa
Abstract Given an epidemic simulation on a network, suppose that one is allowed to vaccinate 10% of the nodes. What is the best way to pick these nodes so as to best mitigate the spread of the disease? Although several different schemes have been proposed in network science, we propose new ones based on modulus computations. In this talk, I will describe this and other applications of modulus to network science. Intuitively, p-modulus is a measure of the richness of a given family of walks where lots of short walks is preferred to fewer longer ones. On the theoretic side, I will show that p-modulus is an ordinary convex optimization problem and I will use this fact to show that p-modulus generalizes and interpolates three well-known graph-theoretic quantities, while being a much more versatile tool. Looking more deeply into the convex duality we will see how the well-known Beurling Criterion fits into this picture and how Beurling families suggest an algorithm for scientific computation. This work is part of an interdisciplinary effort at Kansas State comprising Nathan Albin (Math), Caterina Scoglio (Electrical and Computer Engineering), Faryad D. Sahneh (Postdoc-ECE), Heman Shakeri (PhD graduate student - ECE), and Max Goering (Master graduate student - Math), as well as a team of undergraduate math REU students.

Professor Steffen Rohde, University of Washington at Seattle, WA

Title Random Carpets
Abstract The Conformal Loop Ensemble CLE is a "random Sierpinski-like carpet", more precisely a (random) countable collection of disjoint Jordan curves in a simply connected domain, defined in terms of conformal maps. It is a fundamental object of importance in statistical physics and probability, and a close relative of the Schramm-Loewner Evolution SLE. I will give an introduction to CLE, and discuss an analog of Mario Bonk's carpet uniformization, based on joint work with Brent Werness.

Professor Oliver Roth, University of Wuerzberg, Germany

Title The Schramm-Loewner equation for multiple slits
Abstract We show that any disjoint union of finitely many simple curves in the upper half-plane can be generated in a unique way by the chordal multiple-slit Loewner equation with constant weights. This is joint work with Sebastian Schleissinger.

Professor Eric Schippers, University of Manitoba, Canada

Title Monotonic conformal invariants from quadratic differentials
Abstract In this talk, we define a family of conformally invariant functionals of pairs of nested simply connected domains. A different invariant is obtained from each choice of quadratic differential admissible for the outer domain. The conformal invariant is bounded and monotonic as a function of the inner domain for fixed outer domain. In particular, for each quadratic differential we obtain a sharp inequality for bounded univalent functions. These results arise from (and generalize) the Dirichlet energy method of Nehari. We discuss how these results are an "intrinsic" formulation of the Dirichlet energy method in the spirit of David Minda's work.

Professor Marie Snipes, Kenyon College, OH

Title Harmonic measure for doubly connected domains
Abstract The boundary of a planar domain can be equipped with the so-called harmonic measure, which is closely connected with the behavior of harmonic functions inside the domain. Harmonic measure is invariant under conformal maps, so for simply connected domains, one can use the Riemann map to calculate the harmonic measure of subsets of the boundary. For non-simply connected domains, the problem becomes more complex. In this talk, we will describe (and compare) four different approaches to calculating harmonic measure for doubly-connected domains, focusing on the example of the doubly-slit plane.

Professor Alex Solynin, Texas Tech University, TX

Title Comparison theorems for the Poincaré metric: David Minda's legacy
Abstract The Poincaré metric, named after Henri Poincaré, is of a great importance in non-Euclidean Geometry and Complex Analysis. In the disk model, the Poincaré metric, which is also called the hyperbolic metric, is given by its density $\lambda(z)=\frac{1}{1-|z|^2}$.
This metric is conformally invariant and can be transplanted to more general domains and surfaces; the transplanted metric, $\lambda_D(z)$, can also be obtained as a solution to Liouville's equation \[\Delta\log \lambda_D(z)=-4\lambda_D^2(z).\] Thus, $\lambda_D(z)|dz|$ is a metric of constant negative curvature $-4$. A fundamental result of modern Complex Analysis, the Ahlfors-Schwarz Lemma, states that every ultrahyperbolic metric $\mu(z)$ in a domain $D$ is dominated by the hyperbolic metric; i.e., \[\mu(z)\le \lambda_D(z).\] The latter also implies that every holomorphic mapping is a contraction with respect to the hyperbolic metric. In the early 1990's, when I started my work on certain extremal problems for the hyperbolic metric and searched the literature, I was very surprised to find that the name of one author was attached to literally every aspect related to the hyperbolic metric! And the name of that author was neither Poincaré nor Ahlfors, but rather the name of the author was David Minda. David established new results of solutions of equations similar to Liouville's equation. He also proved new comparison theorems resembling the Ahlfors-Schwarz Lemma. David worked on and solved several extremal problems on the hyperbolic metric. He has also applied the hyperbolic metric to study several special classes of analytic functions. In this talk, I will recall some of David Minda's results on the hyperbolic metric. Then I will discuss related comparison theorems and extremal problems studied in several papers where I am the author or a coauthor. Finally, I will mention a few open extremal problems on the hyperbolic metric, some of them are new and some are old resisting all approaches (known to me).

Professor Ken Stephenson, University of Tennessee, TN

Title Branched circle packings: Free at last
Abstract (Joint work with Edward Crane and James Ashe) Attempts to build a discrete theory for rational maps on the sphere via circle packing have foundered on discretization effects in locating branch points. The authors remove this impediment by introducing generalized branch points. A generalized branch point need no longer be attached to an individual circle, but with the help of chaperones and other devices, can be positioned anywhere that the geometry requires while remaining faithful to the conformal structure. The effects will be illustrated in images and videos as we use generalized branching to fix flaws in discrete Ahlfors and Weierstrass functions.

Professor Toshiyuki Sugawa, Tohoku University, Sendai, Japan

Title Uniform perfectness and convexity in spherical geometry
Abstract A hyperbolic domain in the complex plane possesses uniformly perfect boundary if, and only if, the product of its hyperbolic density and the distance function to the boundary has a positive lower bound. Unfortunately, this is no longer true for hyperbolic domains in the Riemann sphere. In this talk, we propose a similar characterization in terms of spherical geometry. We also consider characterizations of spherically convex domains in terms of the hyperbolic metric.

Professor Huy Tran, UCLA, CA

Title The topology of Continuum Random Tree
Abstract I will talk about joint work with Mario Bonk. We gave an affirmative answer to a recent question by N. Curien and showed that two independent samples of the Continuum Random Tree (CRT) are almost surely homeomorphic. I will discuss some background and give an outline of the proof.

Professor Mattti Vuorinen, University of Turku, Finland

Title What is a hyperbolic type metric?
Abstract The basic idea of Euclidean geometry is that the space is unlimited and its local structure is similar everywhere. In geometric function theory one usually studies maps defined on proper subdomains of the Euclidean space ${\mathbb R}^n, n \ge 2,$ in which case the Euclidean geometry is no longer adequate because of the existence of boundary points. What is needed is "relative geometry" of the domain. During the past forty years many authors have introduced geometries which to some extent are similar to the hyperbolic geometry. These geometries are based on what we call hyperbolic type metrics. Some examples are M\"obius invariant metrics such as the Apollonian, Ferrand, Seittenranta metrics as well as metrics defined in terms of conformal capacity or similarity invariant metrics such as the distance ratio, quasihyperbolic, visual angle, and triangular ratio metrics. A survey of this research is given.

Professor Marshall Williams, Kansas State University, KS

Title On the branch set of quasiregular maps in metric spaces
Abstract An important fact in the classical theory of quasiregular mappings, as developed by Reshetnyak and others in the 1960s, is that the branch set and its image have measure $0$. Later results of Sarvas, Bonk-Heinonen, Onninen-Rajala improved this to show that the branch set is a union of porous sets. In this talk, I will discuss joint work with Changyu Guo, in which we extend these results to a large class of metric spaces.

Professor Jang-Mei Wu, University of Illinois at Urbana-Champaign, IL

Title Julia sets and wild Cantor sets
Abstract There is a Fatou-Julia type theory associated with the iteration of uniformly quasiregular mappings on $\overline{\mathbb{R}^n}$. A map $f\colon \overline{\mathbb{R}^n} \to \overline{\mathbb{R}^n}$ is uniformly quasiregular (uqr) if $f$ and all of its iterates $f^k$ are quasiregular with a uniform bound on the dilatation.
It is known that certain structures can arise as Julia sets of uqr mappings: We construct a uniformly quasiregular map $f:\mathbb{R}^3 \to \mathbb{R}^3$ whose Julia set is an Antoine necklace (joint work with Alastair Fletcher).