Shing-Tung Yau

R. C. Penner

This text contains highlights from two conferences on mathematical physics: Interface Between Mathematics and Physics, held in 1992 at the Academy Sinica, in Taiwan; and Topics in Geometry and Physics.

Perspectives in Mathematical Physics

There is an essentially ``tinker-toy'' model of a trivial bundle over the classical Teichmuller space of a punctured surface, called the decorated Teichmuller space, where the fiber over a point is the space of all tuples of horocycles, one about each puncture. This model leads to an extension of the classical mapping class groups called the Ptolemy groupoids and to certain matrix models solving related enumerative problems, each of which has proved useful both in mathematics and in theoretical physics. These spaces enjoy several related parametrizations leading to a rich and intricate algebro-geometric structure tied to the already elaborate combinatorial structure of the tinker-toy model. Indeed, the natural coordinates give the prototypical examples not only of cluster algebras but also of tropicalization. This interplay of combinatorics and coordinates admits further manifestations, for example, in a Lie theory for homeomorphisms of the circle, in the geometry underlying the Gauss product, in profinite and pronilpotent geometry, in the combinatorics underlying conformal and topological quantum field theories, and in the geometry and combinatorics of macromolecules. This volume gives the story a wider context of these decorated Teichmuller spaces as developed by the author over the last two decades in a series of papers, some of them in collaboration. Sometimes correcting errors or typos, sometimes simplifying proofs, and sometimes articulating more general formulations than the original research papers, this volume is self contained and requires little formal background. Based on a master's course at Aarhus University, it gives the first treatment of these works in monographic form.

Decorated Teichmüller Theory

This book offers an introduction to mathematical proofs and to the fundamentals of modern mathematics. No real prerequisites are needed other than a suitable level of mathematical maturity. The text is divided into two parts, the first of which constitutes the core of a one-semester course covering proofs, predicate calculus, set theory, elementary number theory, relations, and functions, and the second of which applies this material to a more advanced study of selected topics in pure mathematics, applied mathematics, and computer science, specifically cardinality, combinatorics, finite-state automata, and graphs. In both parts, deeper and more interesting material is treated in optional sections, and the text has been kept flexible by allowing many different possible courses or emphases based upon different paths through the volume.

R. C. Penner

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