The Mathematical Century: The 30 Greatest Problems of the Last 100 Years

This is a survey for the layperson of some of the most important mathematical advances of the twentieth century, grouped into categories of pure math, applied math, and computer math. Although it is not possible for the layperson to truly understand any of this material, the author nonetheless does a reasonably good job of presenting it in an interesting and not-too-technical way. I would recommend this book to anyone who is curious about what mathematicians are up to and the kinds of discoveries they have made. Some are quite counter-intuitive and definitely stranger than fiction! The last chapter presents major unsolved problems for the 21st century, and is a little out of date in that the Poincare Conjecture was proved in 2002 by Grigori Perelman based on work by Richard Hamilton. This is quite an interesting story in its own right and is well presented by Szpiro's book "Poincare's Prize...", which I also recommend.

The Mathematical Century, by Piergiorgio Odifreddi, subtitled The 30 Greatest Problems of the Last 100 Years, is what I would call the best book of its kind that I have ever read. The meaning of the phrase "of its kind" comes from one simple fact, quite obvious if you would see a page or two, that this is a very technical, mathematics-oriented book. It is not like the average storybook of fictional characters going through an archetypal journey, but it does tell the story of how these 30 problems have been analyzed by real mathematicians and solved over the years. Considering the sheer size and original complexity of some of these ideas and problems, Odifreddi does an excellent job of explaining, although you may need to stop and really think abstract for a few minutes. For those of you who enjoy abstract thinking, interesting ideas, or just mathematical concepts, this book is a rare treat. Since this quality of a book is often a self-judging trait, a sample of the book is necessary. In this passage, Odifreddi's way of presenting and narrating the history of the problems and their very concept is captivating: "One of the great achievements of nineteenth-century mathematics was the classification of the 2-dimensional surfaces from the topological point of view, that is, by regarding them as rubber sheets that can be deformed at will provided they are not torn apart. From this abstract point of view, an inflated ball and a deflated one are the same surface, even if ..." (Odifreddi, 78) For each problem presented, there are about six pages about its origin, nature, history, attempts at solving, and, for all but the open problems (ones yet to be answered), the solution. The problems include many old questions of geometry, algebra, calculus, and number theory while several more modern ones as well. An interesting factor of many of these problems is how much computers show up in this storyline, as tools that themselves prove too incapable to display or solve some problems in full completeness. To me, this was a little surprising, considering I usually see computers able to do just about anything that programmers tell them to do. Often, these problems are solved with what seems at first the most irrelevant equation or concept that happens to tie in exactly to complete the puzzle. All of this Odifreddi packs efficiently into short sections whose length compares to pamphlets, keeping this brief but informative rather than something of a textbook. With the style of light reading, even the depth of these questions keeps reading this from being overly involved. In addition, the capacity this book has for stimulating abstract thinking far outweighs anything else. To begin with, the very nature of many of the ideas presented begs for abstraction, like presenting a geometric surface that intersects itself and thus cannot have constructed model to represent it. Take note that in this 180 page book there are thirty problems and even more conce

I thoroughly enjoyed this book. It is a fast blast through cutting edge math at a very high level. Some readers may want more in the way of explanation while others might be overwhelmed by the subject matter that is, admittedly, glossed over, but I found it to be a refreshing compromise between these extremes.