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A View from the Top: Analysis, Combinatorics and Number Theory

American Mathematical Society
Number of Pages: 

This book is full of interesting things. The flavor is a little like Aigner and Ziegler's Proofs from the Book: disparate elements are brought together in unexpected ways to make beautiful proofs. Most of the problems have a geometric flavor. The techniques are mostly from analysis even though they are applied primarily to discrete problems.

Several problems are part of extremal combinatorics, e.g., finding bounds on the size of a set in n-space based on the sizes of its projections, or on the number of different slopes of lines through two points of the set (the Besicovitch-Kakeya conjecture). There is a little bit of probability, in particular looking at the probability that two random integers are relatively prime, which leads into the Riemann zeta function, the distribution of primes, and the Riemann hypothesis. There's a moderate amount of finite fields work (actually only the integers modulo a prime, not general finite fields) that is presented primarily to formulate problems simple enough to tackle here. There's a very interesting chapter on van der Corput's theorems on oscillatory integrals, which leads into asymptotics for Fourier transforms. These in turn are used as a back door into Fourier analysis by developing some asymptotics for the number of lattice points inside a circle (Gauss's Circle Problem). All this in only 136 pages!

The book is intended as a text for a capstone course, that is, a final undergraduate course that ties together many areas of mathematics. It is also intended to teach problem solving. The structure of the book is to work out one problem in detail, then pose a number of related problems, some still unsolved. The student is continually urged to stretch out and try things, and to read the original literature.

I like this book, and I think it will be successful in a capstone course. I'm not as optimistic about its value for teaching problem solving: there are too many rabbits being pulled out of hats; I suspect the proofs, although clever and beautiful, are too unmotivated and may intimidate students.

There are an alarming number of errors (in this first printing). Many are typographical, but others are lapses in logic. In every case that I could check the final result was correct, but some of the intermediate steps were incorrect. There's also no index, a Very Bad Feature. Even though the book is very short, I often found myself wishing I could look up some earlier results in the index.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Date Received: 
Friday, November 9, 2007
Include In BLL Rating: 
Alex Iosevich
Student Mathematical Library 39
Publication Date: 
Allen Stenger
  • The Cauchy-Schwarz inequality
  • Projections in $\mathbb{R}^3$--The elephant makes an appearance
  • Projections in four dimensions
  • Projections and cubes
  • Incidences and matrices
  • Basics of grids over finite fields
  • Besicovitch-Kakeya conjecture in two dimensions
  • A gentle entry into higher dimensions
  • Some basic counting, probability and a few twists
  • A more involved taste of probability
  • Oscillatory integrals and fun that lies beyond
  • Integer points and a crash course on Fourier analysis
  • Return of the Fourier transform
  • It is time to say goodbye
  • Bibliography
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Modify Date: 
Saturday, December 22, 2007