Mathematics Unlimited - 2001 and Beyond [ハードカバー]

Maybe you like the idea of this book but you are wondering, does it live up to the promise, will you actually read much of it, is it just too big? The answers are yes, probably yes, and no.

I've browsed maybe half of the chapters. Each one is clear, easy to skim through, with a lot to dig into if you want. So far I've found not one "book report" just summarizing recent work. I've found fascinating helpful interpretations of subjects I don't know -- and challenging positions on subjects that I have my own view of.

Of course you can find fault. The book leans more to applied math than I'd like. And what about functional analysis?

Overall, I am stunned to think I wavered on buying this. I almost passed it up. It would have been a big mistake.

This book is a look to the future of mathematics based on the trends in mathematical thinking at the present time. I did not read all the articles in the book, so my review will be limited to those I did. The article "Experimental Mathematics" by D. Bailey and J. Borwein is an overview of a somewhat controversial activity in mathematics. This activity, characterized as "experimental" mathematics, has, the authors argue, enabled very interesting mathematical problems to be eventually solved. They outline in the article the recent discovery of how to calculate the the nth digit of Pi without computing any of the first n-1 digits without multiple-precision arithmetic and needing only low memory. The calculation scheme was based on a formula that was discovered by a computer, the first time this has happened.according to the authors. Experimental mathematics can be viewed as "real-time" discovery of mathematics, as well as letting us visualize the mathematical structures involved using computer graphics. Mathematicians interested in network modeling will appreciate the article by F. Kelly entitled "Mathematical Modeling of the Internet". Interestingly, his approach makes use of dynamical systems, with the goal of studying the behavior and stability of the TCP/IP protocol. The most interesting section of the article is the section on packet marking strategies. One can find surprising connections between strategies for packet marking, packet shaping, and network QoS, with techniques in option pricing from financial engineering. This is particularly true for frame relay networks. This connection was not discussed in Kelly's article, but I have found these connections in developing my own network models. Kelly gives good insight on how to apply techniques from optimization theory and dynamical systems to study the behavior of modern networks. The network modeling of the 21st century will have to contend with wireless, DWDM, and other more exotic technologies. By far the most interesting articles in the book were the two articles "Geometric Aspects of Mirror Symmetry" by D. Morrison and "A Chapter in Physical Mathematics" by K. Marathe. The constructions that take place in the areas discussed in these two articles have to rank as the most fascinating in all of mathematics. And most interestingly, the ideas had a powerful influence from theoretical physics. One can say without question that physical ideas coming from quantum field theory/high energy physics justify a rephrasing of the words of the famous physicist Eugene Wigner. One could now speak of "the reasonable influence of physics in mathematics". Physical ideas have permeated many different areas of mathematics and will continue to do so. Some mathematicians have classified this influence as "physical mathematics" because some of the mathematical constructions have not been justified rigorously. Several brilliant mathematical developments have occurred in the last two decades resulting from ideas from high energy physics, such as quantum invariants of knots and three-dimensional manifolds, Seiberg-Witten theory, mirror symmetry in algebraic geometry, and supersymmetry and index theorems. These exciting results could be described best as kind of a "quantization of mathematics", and the future will hold more of the this line of thinking. Every construction in mathematics will have a quantum analog, with a correspondence between mappings/structures in "ordinary" or "classical" mathematics and unitary transformations/noncommutative structures in the "quantized" version. An example of this kind of development is occurring today in the field of non-commutative geometry. "Mathematics Unlimited-2001 And Beyond" is a brief glimpse of what will be an exciting century for mathematics. Quantum computation will no doubt become a reality soon, and its computational power, coupled with the needs of the information age, will push mathematics to new dizzying heights. What was called experimental and physical mathematics in the book will continue to have their niches; but "pure" mathematics will also hold its ground and continue to solidify and advance. The mathematical adventure is just beginning......