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Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space

Shmuel Weinberger
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
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The simultaneously intimidating and intriguing title is in fact a perfect summary of the book itself. We can slightly expand it here as: Dear Reader. In this book, we will talk about computability and decidability, and we will use these basic notions of modern mathematical logic and theoretical computer science in our search for an understanding of the Riemannian moduli space. Our methods (which we will collect together under the heading of the logical method or the method of eastern philosophy) will lead us to various interesting rigidity results. When viewed from such a perspective, we will see that the Riemannian moduli space has a complexity which can be seen to be somewhat self-similar, which we will note with a wink towards those who know and love fractals.

Now how does a hypothetical reader react to such a text? Clearly this depends on the basic disposition, tastes and capacities of the mentioned reader. The title calls almost everyone in this world, at least anyone reading the MAA reviews. In fact the first introductory chapter (Chapter I) is accessible to and should be exciting for any undergraduate with some interest in matters mathematical. This is due to the fact that because the book comprises of (expanded and annotated) notes from a series of five lectures (2000 Porter Lectures in Rice University), its organization reflects the decisions of a very experienced lecturer. Start with a first lecture that everyone who is intrigued by your title will get something out of. Then slowly move ahead towards your goal as you gradually increase the requirements from your audience and specialize slowly so that those who remain till the end are those most likely to understand and digest all your methods and ideas. So the second lecture (which happens to be Chapter 1 with its innocuous sounding title: Group Theory) can be appreciated by most mathematics undergraduates; even though many will not digest all that is presented (the author travels a very fast track mentioning presentations, word problems, Dehn functions, and homology among other things), there still are some gems closer to the surface. With the third lecture aka. Chapter 2 (titled Designer Homology Spheres) starts the heavy use of machinery. At this point, the remaining audience has most likely had a few years in grad school, with hopefully some exposure to basic differential geometry and some homological language. The fourth lecture which is Chapter 3 in the book is titled The Roles of Entropy. This is a short chapter, where notions like entropy and Kolmogorov complexity are connected to geometric questions. The level of sophistication required on the part of the audience is again comparable to the previous chapter. The fourth chapter, titled The Large Scale Fractal Geometry of Riemannian Moduli Space, is the longest and as its title suggests, brings together all the ideas developed so far and presents the complexity theory of the Riemannian moduli space. This is quite a new theory still in its infancy, and we get the news from one of its creators. However, this last chapter is quite complicated, one who is not comfortable with the homological language and has only a basic exposure to differential geometry will get lost in the arguments. The author is fully aware of these difficulties and so presents his results upfront in the first section. Then he feels free to roam about proving his results in whichever manner he sees fit. This attitude once again fits perfectly within the lecture format.

The book has extensive bibliographical notes at the end of each chapter which flow smoothly connecting the various threads started in the chapter to one another; these definitely add a lot to the reading experience. Overall one leaves the book with the feeling that it must have been a wonderfully entertaining series of lectures with tons of insight and exciting new mathematics. The informal language used in the text reflects the lecture style as well, and makes the reader feel as if she were chatting with Professor Weinberger during the departmental coffee hour. However, the chat eventually gets a bit too involved, and at some point the reader may prefer to skip certain details and follow only the main flow of the ideas involved.

For a more mathematical review check out MathSciNET (the MathSciNET review was still in preparation as of May 2005).

Gizem Karaali teaches at the University of California in Santa Barbara.

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