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Introduction to Circle Packing: the Theory of Discrete Analytic Functions

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Introduction to Circle Packing: the Theory of Discrete Analytic Functions BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 3, July 2009, Pages 511–525 S 0273-0979(09)01245-2 Article electronically published on February 19, 2009 Introduction to circle packing: The theory of discrete analytic functions, by Kenneth Stephenson, Cambridge University Press, Cambridge, 2005, xii+356 pp., ISBN 13: 978-0-521-82356-2, £42 1. The context: A personal reminiscence Two important stories in the recent history of mathematics are those of the ge- ometrization of topology and the discretization of geometry. Having come of age during the unfolding of these stories as both observer and practitioner, this reviewer does not hold the detachment of the historian and, perhaps, can be forgiven the personal accounting that follows, along with its idiosyncratic telling. The first story begins at a time when the mathematical world is entrapped by abstraction. Bour- baki reigns, and generalization is the cry of the day. Coxeter is a curious doddering uncle, at best tolerated, at worst vilified as a practitioner of the unsophisticated mathematics of the nineteenth century. 1.1. The geometrization of topology. It is 1978 and I have just begun my graduate studies in mathematics. There is some excitement in the air over ideas of Bill Thurston that purport to offer a way to resolve the Poincar´e Conjecture by using nineteenth century mathematics—specifically, the noneuclidean geometry of Lobachevski and Bolyai—to classify all 3-manifolds. These ideas finally appear in a set of notes from Princeton a couple of years later, and the notes are both fas- cinating and infuriating—theorems are left unstated and often unproved, chapters are missing never to be seen, the particular dominates—but the notes are bulging with beautiful and exciting ideas, often with but sketches of intricate arguments to support the landscape that Thurston sees as he surveys the topology of 3-manifolds. Thurston’s vision is a throwback to the previous century, having much in common with the highly geometric, highly particular landscape that inspired Felix Klein and Max Dehn. These geometers walked around and within Riemann surfaces, one of the hot topics of the day, knew them intimately, and understood them in their particularity, not from the rarified heights that captured the mathematical world in general, and topology in particular, in the period from the 1930’s until the 1970’s. The influence of Thurston’s Princeton notes on the development of topology over the next 30 years would be pervasive, not only in its mathematical content, but even more so in its vision of how to do mathematics. It gave a generation of topolo- gists permission to get their collective hands dirty with the particular and to delve deeply into the study of specific structures on specific examples. What has geometry to do with topology? Thurston reminded us what Klein had known—that the topology of manifolds is closely related to the geometric structures they support. Just as surfaces may be classified and categorized using the mundane geometry of triangles and lines, Thurston suggested that the infinitely richer, more intricate world of 3-manifolds could, just possibly, be classified using the natural 2000 Mathematics Subject Classification. Primary 52C26. This review is dedicated to the memory of Oded Schramm, who worked in circle packing before his discovery of stochastic Loewner evolution and its applications to critical phenomena. This extraordinary mathematician’s untimely death on 01 September 2008 in a hiking accident was a great loss for our community. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 511 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 512 BOOK REVIEWS 3-dimensional geometries, which he classified and of which there are eight. And if he was right, the resolution of the most celebrated puzzle of topology—the Poincar´e Conjecture—would be but a corollary to this geometric classification. The Thurston Geometrization Conjecture dominated the discipline of geometric topology over the next three decades. Even after its recent resolution by Hamil- ton and Perelman, its imprint remains embedded in the working methodology of topologists, who have geometrized not only the topology of manifolds, but the fun- damental groups attached to these manifolds. Thus we have as legacy the young and very active field of geometric group theory that avers that the algebraic and combinatorial properties of groups are closely related to the geometries on which they act. This seems to be a candidate for the next organizing principle in topology. The decade of the 1980’s was an especially exciting and fertile time for topology as the geometric influence seemed to permeate everything. In the early part of the decade, Jim Cannon, inspired by Thurston, took up a careful study of the combi- natorial structure of fundamental groups of surfaces and 3-manifolds, principally cocompact Fuchsian and Kleinian groups, constructing by hand on huge pieces of paper the Cayley graphs of example after example. He has relayed to me that the graphs of the groups associated to hyperbolic manifolds began to construct themselves, in the sense that he gained an immediate understanding of the rest of the graph, after he had constructed a large enough neighborhood of the identity. There was something automatic that took over in the construction and, after a visit with Thurston at Princeton, automatic group theory emerged as a new idea that has found currency among topologists studying fundamental groups. In this work, Cannon anticipated the thin triangle condition as the sine qua non of negative cur- vature, itself the principal organizing feature of Thurston’s classification scheme. He studied negatively curved groups, rather than negatively curved manifolds, and showed that the resulting geometric structure on the Cayley graphs of such groups provides combinatorial tools that make the structure of the group amenable to com- puter computations. This was a marriage of group theory with both geometry and computer science, and had immediate ramifications in the topology of manifolds. Ultimately, these ideas led to one of the most beautiful conjectures left still un- resolved by Perelman’s final resolution of the Geometrization Conjecture; namely, that a negatively curved group with 2-sphere boundary is, essentially, a cocompact Kleinian group, the fundamental group of a compact hyperbolic 3-manifold.1,2 Can- non’s attempt to resolve this conjecture in collaboration with his colleagues, Bill Floyd and Walter Parry, has produced some of the most elegant geometric results in recent memory, especially in their elucidation of finite subdivision rules and their use as combinatorial constructs in producing conformal structures on sur- faces; see Figure 1. Activity surrounding Cannon’s program led in the 1990’s to two discretizations of the classical Riemann Mapping Theorem, the Discrete Rie- mann Mapping Theorem of Cannon-Floyd-Parry [13] that offers a discrete analogue, proved independently by Schramm [22], and the Combinatorial Riemann Mapping Theorem of Cannon [12] that constructs a classical conformal structure on a space that carries an appropriate combinatorial superstructure. We will return to these developments later, but we first mention another strand of the story. 1Precisely, the group acts properly discontinuously, cocompactly, and isometrically on hyper- bolic 3-space. 2For a beautifully presented description of this work, see [14]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BOOK REVIEWS 513 Figure 1. Examples of circle packing–generated embeddings of expansion complexes of Cannon-Floyd-Parry that tile the complex plane and are generated by finite subdivision rules with pentagonal combinatorics. The upper left tiling is particularly interesting as it is regular in the sense that each of the pentagons the eye sees at every level of scaling is conformally regular—conformally equiva- lent to a regular pentagon preserving vertices. Moreover, the whole infinite tiling can be generated from any single elementary tile, just by repeated anti-conformal reflections across edges. See [10]. In 1987, Mikhail Gromov increased the excitement and accelerated this project in geometrizing topology and group theory with the publication of his essay Hy- perbolic Groups [15] in which he presents his grand, but rather sparsely argued, view of negative curvature in group theory. He followed this in 1993 with the influ- ential Asymptotic Invariants of Infinite Groups [16] accomplishing a similar task for nonpositive curvature. Almost immediately after the respective publications of these two works, groups of mathematicians offered detailed proofs elucidating the big ideas of Gromov, and graduate students in succeeding years wrote theses whose License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 514 BOOK REVIEWS geneses lay in these essays, sometimes explaining, with detailed proofs, a remark of a single sentence from one of the essays. Finally, it is interesting to mention, albeit briefly, that the resolution of the Ge- ometrization Conjecture in the past five years relies less on the spirit of nineteenth century hyperbolic geometry, which lies at the heart of Thurston’s original insights, and more on the modern theory of differential equations. Thurston had proved the conjecture for large classes of manifolds, including Haken manifolds, for which he received the Fields Medal in 1982. That same year, Richard Hamilton proposed a method for solving part of the conjecture using Riemannian geometry and the flow of Ricci curvature [17], and later generalized this to a proposal for proving the general conjecture. Hamilton pushed his program forward over the next two decades, but it took the insights of Grigori Perelman to overcome difficulties that arose from singularities of the Ricci flow, in which the principal ingredient is the modern theory of differential equations and global analysis.
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