Mathematics Unlimited - 2001 and Beyond

by Björn Engquist

Book Overview

This is a book guaranteed to delight the reader. It not only depicts the state of mathematics at the end of the century, but is also full of remarkable insights into its future de- velopment as we... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 3540669132
• ISBN13: 9783540669135
• Release Date: January 2001
• Publisher: Springer
• Length: 1236 Pages
• Weight: 6.20 lbs.
• Dimensions: 2.6" x 7.9" x 9.8"

Customer Reviews

If you are interested, buy it.

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.

An awesome look at the future of mathematics

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 fr