Among the connected components of the interior of the Mandel-brot set are those that are hyperbolic. These components consist of parameters c E C for which the critical point z0 = 0 of fc : z 7 -> z2 + c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z0 is periodic. For a given n > 1, the Gleason polynomial for period n is the monic polynomial Gam, E Z[c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if Gam, factors over Z. In this article, we factor Gam, modulo 2. We prove the following remarkable fact: the number of irreducible factors of Gam, modulo 2 is equal to the number of real roots of Gam,.

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at most $C$, the problem of determining whether $F$ is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of $C$ and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality.

It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps.

An orientation-preserving branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. Inspired by classical, non-dynamical notions such as Hurwitz equivalence of branched covers of surfaces, we develop invariants for such maps. We then apply these notions to the classification and enumeration of NET maps. As an application, we obtain a complete classification of the dynamic critical orbit portraits of NET maps.

A branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of simple affine data. This data can then be used as input for algorithms developed for the computation of fundamental invariants, now systematically tabulated in a large census.

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion—combinatorial, dynamical, algebraic, and coarse-geometric.

We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.

This paper is concerned with growth series for expansion complexes for finite subdivision rules. Suppose X is an expansion complex for a finite subdivision rule with bounded valence and mesh approaching 0, and let S be a seed for X. One can define a growth series for (X,S) by giving the tiles in the seed norm 0 and then using either the skinny path norm or the fat path norm to recursively define norms for the other tiles. The main theorem is that, with respect to either of these norms, the growth series for (X,S) has polynomial growth. Furthermore, the degrees of the growth rates of hyperbolic expansion complexes are dense in the ray [2,∞).

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bitwisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bitwist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers.

In this paper, we introduce a new class of epidemics on networks which we call SI(S/I). SI(S/I) networks differ from SIS networks in allowing an infected individual to become reinfected without first passing to the susceptible state. We use a covering-graph construction to compare SIR, SIS, and SI(S/I) networks. Like the SIR networks that cover them, SI(S/I) networks exhibit infection probabilities that are monotone with respect to both transmission probabilities and the initial set of infectives. The same covering-graph construction allows us to characterize the recurrent states in an SIS or SI(S/I) network with reinfection.

Erling Folner proved that the amenability or nonamenability of a countable group depends on the complexity of its finite subsets. Complexity has three measures: maximum Folner ratio, optimal cooling function, and minimum cooling norm. Our first aim is to show that, for a fixed finite subset, these three measures are tightly bound to one another. We then explore their algorithmic calculation. Our intent is to provide a theoretical background for algorithmically exploring the amenability and nonamenability of discrete groups.

We introduce and study a class of Thurston maps from the 2-sphere to itself which we call nearly Euclidean Thurston (NET) maps. These are simple generalizations of Euclidean Thurston maps.

{1/8 < 1/4 < 3/8 < 1/2 < 3/4} one rooted binary tree to another. Such trees can be encoded by sequences of rational numbers: The root of the tree is represented by the dyadic fraction 1/2. Each dyadic fraction x·(1/2)k ∈ (0,1) (x odd) is the unique parent of two dyadic children, namely, (2x± 1) · (1/2)k+1. A sequence S = {a1 < · · · < an} ⊂ (0,1) of dyadic fractions is called a tree sequence of length n ≥ 0 if the parent of each ai 6= 1/2 is also in S.

This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite subdivision rules with one tile type. An example is given of a Lattes map which is not the subdivision map of a finite subdivision rule with either i) two tile types and 1-skeleton of the subdivision complex a circle or ii) one tile type.

Suppose f : S 2 → S 2 is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting. This is a first step in a program to clarify the relationships among various notions of expansion for noninvertible dynamical systems with branching behavior.

Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3-ball in a mechanical way so that the quotient is automatically a closed, orientable 3-manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original face-pairing. This allowed us to find very simple face-pairing descriptions of many, though presumably not all, 3-manifolds.Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twisted-face-pairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3-manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold.

We consider epidemics on social networks and address the question of whether administering a safe vaccine to one or more individuals can raise another individual's chances of becoming infected. Surprisingly, this can happen if transmission probabilities vary over time. If transmission probabilities do not vary with time, we show that in the discrete SIR model vaccination cannot cause collateral damage. We phrase this question in terms of monotonicity properties and answer it using bond percolation methods. By passing to a covering graph we are able to extend these results to models with more complicated latent and infective states.

J.R.R. Tolkien, author of The Hobbit and The Lord of the Rings, despaired of interruptions that took him away from his work. It’s not so much the interruptions alone, he said, as the fear of interruptions. We all have our tricks to find uninterrupted hours for mathematics. When I was in Wisconsin it got really cold during the winter and it stayed really cold. The way I found uninterrupted time was to buy some shoes with heavy soles and a coat with a tunnel hood and walk to and from school. I felt guilty about avoiding people, because surely our relationships with others are the most important part of our lives. But then I remembered that many of our most valued relationships are those that we have with people that we have never seen and people that we’ve never talked with. Because of writing, those of us who live far from the centers of mathematics have our chance to spend our time with those who are truly great. This was exactly my relationship with Lars Ahlfors. I spent many, many hours learning complex variables from his book Complex Analysis [1], and later I read very carefully his book Conformal Invariants [2]. This was how I came to know Ahlfors and appreciate him. In 1912/1913, in two consecutive issues of the Mathematische Annalen, Carathéodory published three very interesting papers. The first of these papers [13] was about the dependence of the Riemann mapping on boundary data. The second paper [14] was on continuous extensions to the boundary if the conformal mapping was defined on a Jordan domain. In the third [15], he took care of the problem when the boundary was not a simple closed curve and he defined the notion of prime end. These three papers set the stage for the work of Lars Ahlfors in the following way. These results were published in 1922 in the textbook Funktionentheorie by Hurwitz and Courant [19]. But the proof that Hurwitz and Courant gave of the continuous extension to the boundary was different from the proof given by Carathéodory. They used the method which has come to be known as the length-area method. They say this about the source. “The explanation of the questions treated in §7 is due to Osgood and above all to Carathéodory, who in particular recognized the significance of prime ends. The presentation given here follows work of the author.”[19, p. 353] That’s as far as we’ve been able to trace the length-area method

The theorem on squaring a rectangle (see Schramm [Israel J. Math. 84 (1993)] and Cannon-Floyd-Parry [Contemp. Math. 169, AMS, 1994]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemann mapping.