Book Description

All papers in this proceedings volume were peer reviewed. The purview of this third conference was shifted toward biology and medicine. Among the topics covered were: the constructive role of noise in the central nervous system, neuronal networks, and sensory transduction (hearing in humans, photo- and electroreception in marine animals), encoding of information into nerve pulse trains, single molecules and noise (including single molecule detection and characterization by nanopores - molecular "Coulter counting"), concepts of noise in neurophysiology (randomness and order in brain and heart electrical activities under normal conditions and in pathology), the role of noise in genetic regulation and gene expression, biosensors, etc.

Book Description
Featuring Original Contributions from Dr. Stephen Hawking

Unfold the mysteries that vex the greatest minds in science

Gain extensive knowledge of the most challenging scientific problems and learn from more than 60 of the world's foremost scientists—among them, 40 Nobel laureates! Expand your horizons with a wide range of advanced scientific theories and techniques on problems concerning:

Permanently storing nuclear waste or eliminating it altogether
Harvesting energy from a reaction similar to that of the sun
Earthquake prediction
The creation of the universe
Comprehension of free will
The mystery of dark matter
The cosmological constant problem
The construction of a consistent quantum theory of gravity
And much more

Science has reached dazzling heights of discovery, transforming civilization in the process. And yet, some of the most fundamental questions remain unsolved! In The World's 20 Greatest Unsolved Problems, John Vacca—together with more than 60 of the world's most highly respected scientists—explains these problems in detail and describes the intellectual and technological hurdles to be overcome in order to solve them.

This book is indispensable for science buffs, teachers, students, and scientists who want to keep pace with the latest developments. The World's 20 Greatest Unsolved Problems delves deep into mysteries such as the creation of the universe, dark matter, the quantum theory of gravity, protein folding, free will, consciousness, earthquake prediction, Fullerenes, the quantum mechanical vacuum, storing or eliminating nuclear waste, and more. No other resource explains science's most compelling dilemmas with such clarity and authority, and nowhere else can you share the expertise of so many brilliant minds! You'll find

Complex topics made intelligible, as only experts in their fields can
Coverage of the key problems expected to dominate the next 40 years of scientific research
The World's 20 Greatest Unsolved Problems is must reading for anyone teaching science or performing scientific research. It also will fascinate the moderately technical reader or scientific novice. ... Read more

Customer Reviews (29)

Wide Range of Topics/Wide range of explanations
I really enjoyed reading this book.I am not very proficient in some of the areas covered.I thought the book did a great job of explaining the topics and then giving me some of the explanations that are possible and then finally coming to a conclusion.The book will definitely make you think and understand how little human beings really know about the universe they reside in.I feel after reading this book I know what questions should be asked and I now I have a better grasp of some of the world's greatest mysteries.

Error prone, biased and uninteresting
This book was a very disappointing gift.

I found Vacca's explanations of science's current puzzles to be tedious and frustrating.Most of the topics are explained in tedious detail, yet not enough detail to permit real understanding.As an example, it is perhaps impossible to explain the first few milliseconds after the Big Bang without heavy use of mathematics - yet Vacca tries, using extensive, vague and (to me) uninteresting prose.He repeats this performance, attempting to deal with issues ranging from quantum gravity to protein folding to free will.

The frustration is made worse by the inclusion of many grammatical and technical errors.There were more of these errors than I've seen in any comparable volume. He also makes excessive use of boldly-highlighted mid-page "Notes" boxes - which really should be relegated to footnotes.

Finally, Vacca's self-conferred mantle of scientific objectivity is destroyed in the chapter on "Free Energy."He seems to believe it's possible to suck all the energy the earth needs from the cosmic ether.So much for the laws of thermodynamics!His "Notes" in this chapter consist of jabs at the "naysayers and skeptics" with little clue that these "naysayers" comprise over 99% of the world's scientists.

If a layman wants a technical, yet non-mathematical explanation of cosmology and quantum physics, Hawking is much, much better.If a layman wants a well written and wide-ranging overview of science, Bill Bryson's _A Short History of Nearly Everything_ is much more engaging.

Fascinating despite some repetition and some fuzziness
The problems range from dark matter and dark energy through attempts to reconcile gravity and quantum mechanics to problems associated with DNA and proteins, to neuroscientific concerns about free will and consciousness to what to do about nuclear fusion and its waste.

There are other books on cutting edge problems in science that I have read, e.g., John Malone's Unsolved Mysteries of Science: A Mind-Expanding Journey through a Universe of Big Bangs, Particle Waves, and Other Perplexing Concepts (2001) or The Next Fifty Years: Science in the First Half of the Twenty-First Century (2002) edited by John Brockman; but there is only one other that is anywhere near as ambitious as this work.That book would beMagic Universe: The Oxford Guide to Modern Science (2003) by Nigel Calder.

To compare these two books for the reader I would say that Calder's book is not only longer but covers more ground, is better edited and relies on a greater range of scientific authority.But Vacca's book has the virtue of narrowing in on just where the scientific action is while he does a good job of presenting the various opinions.That is, insofar as I, personally, can tell.To be honest, much of the material in all these books is above my level of expertise.Consequently I take most of what I read at face value.Clearly I cannot choose between cosmological models of inflation and quintessence.Nor do I have any firsthand experience with the complications of protein folding, etc.But neither will most readers.However we needn't be critical readers.It is enough to read appreciably about the wonders of science and how such wonders inform our beliefs and enrich our lives.

As for the repetition in the book and the typos and the other errors pointed out by other readers, it is good to understand that Vacca wrote this book by himself (although he interviewed and relied on the work of many scientists) and probably did so in a first draft/correct it mode (judging again from the repetition and some of the unpolished prose).Let's face it, life is short and a book like this needs to be written fast or it will become outdated before it hits the book stores.Furthermore, although he had editors to check for technical errors, editors to check his spelling and such, and had the benefit of the professionals at Prentice Hall, it is in the nature of a book like this that no single person with the exception of the author can really be close enough to the content to adequately edit it.

Now I want to look at a couple of the problems that Vacca discusses.

He talks about traveling back into the past and asserts that the usual paradoxes relating to killing your grandfather before your parents were born, etc. can be overcome by having you go back to a past in a parallel universe.Relying on the many worlds interpretation of quantum mechanics (which he seems to favor), Vacca finds this reasonable.The problem, however, is that Vacca has already in a previous chapter made it clear that there is no interaction between postulated parallel universes, so he ends up justifying travel to the past by making it doubly difficult: not only do you have to violate causality but you have to go to a parallel universe to do so!I imagine he would say in response that by going to another universe you actually avoid the violation of causality since you do not in any way affect the universe you are in.However to go back in time in the other universe you have to be in that universe.

Vacca suggests that the dark matter that cosmologists are now utterly convinced exists because of its gravitation presence is perhaps an example of a parallel universe.Actually he takes the opposite perspective and asks if a parallel universe exists (parallel to the dark matter) and answers that it does.It is us. (p.115)

Since gravity that makes us aware of the existence of that dark universe (and remember there is no evidence of any information about the dark matter via the electromagnetic force or the weak or strong nuclear forces) could it not make them aware of us?(Assuming there is somebody there to be aware.)Perhaps some day we will communicate with other universes through some type of gravitational mechanism.(Huh?--Well, maybe.)

On free will Vacca uses basically three authorities, Timothy O'Connor, Miroslav Backonja, and Paul J. Bertics, and from them constructs what he sees as the current understanding by neuroscience.I wasn't even aware that neuroscientists had a position on free will.I thought it was a purely philosophic or religious question.The opposing camps of naturalism (no such thing as free will) and libertarians (humans have free will) are reconciled in the neuroscientific community through the idea of "compatibilism," a word I encountered here for the first time.What it means is that the lack of free will (which most neuroscientists, Buddhists and myself, among others see as obvious) is made compatible with the societal and human psychological need to believe in free will (for punishment and criminal deterrence) by realizing that in an Orwellian way we can say that free will does not exist, but in order for society to run smoothly we must pretend that it does.Vacca discusses the ramifications from this doublethink and concludes that whether free will is an illusion or not depends on your point of view.Your free will is obvious, but that of others has to be taken on their say so.

Here's an example of Vacca's sometimes strikingly expressive prose: "As much as free will exposes humans to the threat of unlimited retaliation for wrong-doing, it nevertheless compensates them by making them the lords of their little domains, the micro-gods of their minds." (p. 394)

The worst science book that I have read recently
The book coveres big topics, e.g., free energy (Chapter 19) and nuclear fusion (Chapter 20). The author, however, did not even get simple facts correct. W (watt) is incorrectly identified as "Tungsten" on p.623 and 625. JET is said to be "Joint European Toms" (p.626). It should be "Torus" instead of "Toms". I wonder how many Toms are there in Europe under this scheme. It is a waste of tree to print this book.

The World's 20 Greatest Unsolved Problems
This textbook is a comprehensive treaty on the 20 unsolved problems of this century. The unsolved problems are well laid out in a logical sequence. The style and the presentation of the text makes it easy reading for non-technical readers.

The 20 unsloved problems are thought provoking, and reminds us that we are far from understanding this world we live in.

Astronomy: The mystery of dark matter - is there any other state of matter!!! The creation of this universe - a puzzle?

Physics: A very concise and an excellent discussion on the nature of fundamental elementary particles and can there be a consistent theory of Quantum Gravity? If so, then we would know the mind of GOD.

Biology: DNA and multi-celluar life. Is there a computer program hidden in the DNA!

Paleontology: How present day microbiological information can be used to construct the ancient tree of life.

Neuroscience: Are we responsible for out actions? Lawyers would have a field day on this topic. Now for the ultimate mystery of all the mysteries - Consciouness.

Geology: This should make an interesting reading for those of us who live in California.

Chemistry: It is the essnece of formation of chemical bonds, without which there would be no DNA, hence no life as we understand.

Free Energy: Is there a such a thing as Free Lunch, I put it?

John has done an excellent work in presenting the most fascinating problems of this century. This should make an interesting reading for any inquiring mind.
Highly recommend. ... Read more

Book Description

Mathematics is kept alive by the appearance of new unsolved problems, problems posed from within mathematics itself, and also from the increasing number of disciplines where mathematics is applied. This book provides a steady supply of easily understood, if not easily solved, problems which can be considered in varying depths by mathematicians at all levels of mathematical maturity.

Book Description
Mathematicians and non-mathematicians alike have long been fascinated by geometrical problems, particularly those that are intuitive in the sense of being easy to state, perhaps with the aid of a simple diagram. Each section in the book describes a problem or a group of related problems. Usually the problems are capable of generalization of variation in many directions. The book can be appreciated at many levels and is intended for everyone from amateurs to research mathematicians. ... Read more

Customer Reviews (1)

A wide expanse of problems in geometry
After the counting numbers, geometry is the oldest branch of mathematics and no doubt the first one that required abstract thinking. Even so, thereis always a certain "concreteness" about it in the sense that diagrams canalmost always be constructed. The range of problems that fall under thegeometric umbrella is extremely wide and some even have practical uses.
This book is a testament to the wide range of problems that are geometricin nature. One of my favorites is known as the "worm problem." To be moreprecise, the question is, "find the convex set of least area where anycontinuous curve of length one can be placed in it." This type of problemhas ramifications in optimal packings, where a single type of containerneeds to be constructed for all possible ways an object can fold. Otherproblems such as tiling and dissection; packing and covering andcombinatorial geometry are also covered.
However, the best part ofthe book may be the extensive references. Every problem is followed by alist of references, so if you wish to take a crack at it, you will havelittle difficulty in locating the work done to the date of publication.
This is one of those books that always seems to beckon me when it lieson my bookshelf. Every once in awhile I pull it off and browse through it,admiring the skill and breadth of mathematicians in their pursuit of truth.It should be in every academic library.

Published in Journal ofRecreational Mathematics, reprinted with permission ... Read more

Book Description

Product Description
The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment.This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers. ... Read more

Customer Reviews (2)

pankajmath
an alternate proof to show that combination coeffiecient
C(p^n,r)is divisible by p; where p a prime n & r +ve
integers.
idea first we write the expression of c(p^n,r)
& then wecan conclude from the expression that this is
divisible by p (how ?).

Different Approch to Number theory
The author develops the premise that modern number theory evolved from the ancient Greek preoccupation with two mathematical problems.Searching for the esoteric Perfect Numbers (i.e., whole numbers whose proper divisors sumto the number itself1+2+3=6) and Diophantine Equations (i.e., findingintegral solutions to certain algebraic equations, for example, z^2 = x^2 +y^2).The author calls the later Pythagorianism.The book does a good jobof showing how Fermat's Little Theorem, Euler's generalization, and thefamous Law of Quadradic Reciprocity developed out of the search forMersenne Primes, and consequently Perfect Numbers.Again, it isinteresting to see how Pythagorianism led to the development of algebraicnumbers and eventually to the solution of Fermats "Big" Theorem. Along the way the author elaborates on some of the still unresolvedconjectures within number theory.The writing can be a little"dense" at times, so that some parts require a second reading. Overall the book is enjoyable to read and you will gain some insight thatwon't be gleaned from more standard texts. ... Read more

Book Description
Examines the legacy of Paul Erdos and the open problems he left to theworld of mathematics after his death in 1996. Text catalogues the unsolved problems of Erdos in one comprehensive and well- documented volume,designed to help continue the work of the absolute monarch of problemsolvers. Softcover. DLC: Graph theory. ... Read more

Customer Reviews (2)

Immediate throw away for non-professional Mathematicians
I got a book by Ron Graham and Fan Chung.
It is amazingly bad.
It is a book on graphs without one graph diagram
or graph matrix.
I buy these books for learning.
I'm left with not being able to give an honest review
because it would be politically suicide in Mathematics to
cross Ron Graham who has been president of both Mathematics societies.
The book is a very badly written one, too,
with a few mostly ancient( from as far back as the 1930's )
problems in graph theory.
I have to go to MathWorld to get an idea of what the graphs might look like!

Since it appears that the graphs are the dog that wags the matrices,
I thought I might get further with some better understanding.
It appears this book is the wrong place to get it.

One reason as I see it that Fan Chung and Ron Graham
don't answer emails much is that they really aren't all that nice, good or smart:
their "meal ticket" was Erdos and he is dead now.
That's not a "pretty" conclusion or one I wanted, you know?
Tattle tale stories about the great man aren't going to save this one.

Even for me this may take years to get my money's worth out of.
I looked at the review by Charles Ashbacher and I had to set the record straight.
He is a friend of a friend who is also into integer sequences.
Somebody has to be honest here.

Only one of many such books that could be written
What is incredible about this book is not that there are over one hundred pages of unsolved problems posed all or in part by Paul Erdos. The amazing thing is that the word graphs could be replaced by several other mathematical words or phrases and a similar book could be written. Erdos was an expert in so many areas of mathematics and perhaps his greatest ability was in putting forward just the right questions to just the right people. There is very little explanation of the problem proposals, the authors rely on a great deal of the listing of references to fill in the details. Therefore, anyone interested in exploring the problems in greater detail should be prepared to spend some time and effort in tracking down the relevant articles. Fortunately, the authors themselves did a great deal of that, as there are complete references for every problem that appears.
The range of problems is a demonstration of the depth of his understanding of graph theory, and also a demonstration of how little is still unresolved. I put forward no pretense to understanding any more than a few of the problems in this book. However, that did not alter my interest in the problems, as I was able to understand the fundamentals of almost all of them. Reading this book is one of the most educational experiences that I have had in the past year and I encourage all mathematicians at the level of slightest interest in graph theory and above to read it.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description
An in-depth look at the theories behind the most intriguing puzzles in physics, chemistry, biology, earth science, and astronomy
In The Five Biggest Ideas in Science, authors Arthur W. Wiggins and Charles M. Wynn discussed science's most important current ideas. Now, they tackle the questions that science has been unable to answer-so far. Choosing one unsolved problem from each discipline, they explore the current scientific thinking behind these questions: How are particle masses determined? How did simple atoms first combine to form complex molecules? What role does the genome play in the development of life? Why is it so difficult to predict the weather? And what is the future of the universe? Featuring cartoons by Sidney Harris, the book includes discussions of recent theories such as the God particle, string theory, "brane" theories, and the Theory of Everything and also explores other science questions.
Arthur W. Wiggins (Farmington Hills, MI) is a Professor of Physics at Oakland Community College in Michigan. Charles M. Wynn (Willimantic, CT) is a Professor of Chemistry at Eastern Connecticut State College. They collaborated on The Five Biggest Ideas in Science (0-471-13812-6). ... Read more

Customer Reviews (4)

5 Unsolved Problems - Or is it 100's

This book is a GREAT introduction to anyone interested in science, wishing to become better informed about scientific issues, or wanting to know what many of our best scientists are working on right now.It is not particularly deep and is therefore a good primer for the non-scientifically oriented reader.

The authors open with a careful discussion of the scientific process.Most of us make our daily life decisions on the basis of tradition, word of mouth, and limited anecdotal experience - and refer more important decisions to a professional.How enlightening it is to be reminded of the rigorous testing, the meticulous standards and procedures, the relentless re-examination in the face of new data, and the necessary grilling by peers that is part of the day to day activity of scientists.Science uses a different language from that used in everyday speech - that language being mathematics."Because mathematical skills require a great deal of effort to acquire, explaining scientific hypotheses to people not trained in mathematics requires translation of mathematical concepts into conversational language.Unfortunately, the meaning of the hypothesis may suffer in the process."

The five unsolved problems are then discussed, one each from the fields of physics, chemistry, biology, geology, and astronomy.Then 27 more are asked in an "appendix" type chapter and given only half-page answers.I like these sorts of books because they help me to maintain general scientific literacy.

For every bit of knowledge we gain, new questions invariably arise - that never-ending quest being the nature of science.I personally think we'll still be looking for the TOE (Theory of Everything) in 100 years, but reading about unsolved problems will notify me if string theory is abandoned, if parallel universes are found, or if the complete structure and function of the proteome is ascertained (fat chance).I might add that the running cartoon commentary by Sidney Harris is superb!

Fine introduction for the general reader
This is written by two guys who teach physics (Wiggins) and chemistry (Wynn) at the college level in a reader-friendly manner in which each of the five unsolved problems are presented, explained, and critiqued.Each section is then concluded with indications of how these problems might be solved.There are some nice cartoons by Sidney Harris to augment the text.

The five "biggest" problems are (from different disciplines and not without controversy):

1. The nature and origin of mass.(Why do some particles have mass while others do not?)

It is obvious that we really do not understand the nature of mass from our inability to form a unified theory involving gravity, a theory that would unify quantum mechanics and relativity.Indeed I think physicists are just whistling in the dark when they talk about particles and fields.It's clear to even this casual observer that the real nature of particles/waves, particles/fields is not really understood, and perhaps cannot be understood in anything other than a once or twice removed mathematical sense.We can write equations that describe what we observe, but the intrinsic nature of all phenomena remains veiled.We avoid infinities in the mathematics of physics as a long-observed and much beloved rule (something like Occam's Razor) with the result that we (necessarily) "construct" limits on the physical world like those named after Max Planck.Beyond (or "below," or "under" or "smaller than," etc.) those limits is potentially a whole universe of physics much like what might be beyond the Big Bang in cosmology.

Authors Wiggins and Wynn acknowledge that the Standard Model of physics has a "dark side" (p. 30) and that the long-sought Higgs field particle may be a "mathematic convenience." (p. 31)From my point of view everything in physics (and this includes all of string theory) that has not met with experimental proof is possibly a "mathematical convenience."This is not to denigrate physics or physicists.On the contrary.What physicists have accomplished toward an understanding of the world in which we live stands as one of humankind's most glorious achievements.The problem is that (as quoted from J.B.S. Haldane on page 159) "the universe is not only queerer than we suppose, but queerer than we can suppose."In other words don't hold your breath for the dreamed-of "Theory of Everything."And if it arrives, don't imagine that "everything" really is "everything."

2.How did lifeless chemical reactions become life?

The authors present some of the history (Stanley Miller's primordial soup experiment; panspermia, etc.), outline the problems, tell us a little about DNA and RNA, and finish with how the puzzle might be solved and by whom.I would observe that imbedded within this question is a theoretical bugaboo that first needs to be resolved.We have to agreeably define what "life" is before we can hope to make a distinction between very complex but "lifeless" self-replicating molecules and molecules forming living organisms.As such, the problem is one of definition as much as anything else.Clearly if we left out our notion of things living as opposed to things not living, we might discover a step-by-step continuum without a clear demarcation point.

3. How do proteins control cells and tissues?("What is the complete structure and function of the proteome?")

The authors note that since the genome has been mapped and sequenced, "the unsolved problem" in biology "has shifted" to "How do protein molecules built from directions provided by [the]...genomes contribute to the structure and function of organisms?" (p. 71)A very complex problem indeed, but at least it is a practical problem and not a theoretical one, and as such (unlike some others in this book) is one that conceivably can be solved through a whole lot of hard work.

This is about cells and how they function.The authors reprise the genome mapping and sequencing story, and then point to "Protemics: The Next Frontier."

4. Can we predict the weather?(Or, how accurate can our weather predictions be?)

This of course is about complexity theory and why that famous butterfly in the Sahara continues to influence the formation of hurricanes in the Carribean.

Quick answer: accuracy will continue to fall off as the square of the distance in time.Just joking, but clearly the more lengthy the forecast, the more uncertain it will continue to be.

5.Why is the universe expanding faster and faster?

Ah, yes.What IS the nature of Einstein's fudge factor that has recently returned?I love this one.The real question is what IS all that dark energy and dark matter out there?As the authors point out only 4%(!) (see page 129) of all the matter in the universe is accounted for in terms of things seen.Seventy-three percent is in the form of dark energy and another 23% in the form of dark matter.It is amazing to realize that over 90% of what exists is stuff we know next to nothing about!

A nice part of the book are the "folders" at the back in which many other interesting issues are briefly presented.The problems in the "Problem Folder" are organized according to disciplines, "Physics Problems," Chemistry Problems," etc.There are sixteen ideas in the "Ideas Folder," including such things as anti-matter, protein folding, chaos theory, global warming, and so on.

Bottom line: Wiggings and Wynn do a good job of introducing the general reader to what scientists at the horizon are working on.It's really amazing to realize how far we've come as knowledge-seeking creatures, and then to get a glimpse of how incredibly much there is we don't know.

A Fun Way to Discover the Intriguing
We teach subjects not related to the science Wiggins and Wynn address - except in the ways that really matter. For example, reviewer Winston teaches American Government; he finds this book profoundly "usable" in that its point includes the notion(disused in some high political places) that problems can be treated by thinking, and shouldn't be expected to be solved by wish fulfillment and received authority.In this sense the book is a continuation of the earlier "Quantum Leaps in The Wrong Direction," only this time the scientific problems are "harder," as they are particular (albeit big!) quests on which scientists have been working. The pleasure scientists take in the pursuit is communicated in a way that leaves no doubt by the authors, and it is marvelous, breath-taking and invigorating. This is just one of the delights of the book.Another is that there is an unsolved mystery for everyone! The chapter sub-headings are a great device and most welcome. The pictures are fascinating and the cartoons are right-on and funny. Reviewer Friedler, who teaches creative and expository writing, finds the book accessible and readable in its well-timed understatement and clever analogy, and in its style and conceptual clarity. It flows and the reader understands.This is not an easy thing, given the mighty topics. We highly recommend this delightful book for its captivating material and entertaining reading.

A survey course in Science
This is like a condensed survey course of various facets of Science. The authors cut to the chase by focusing on one major unsolved problem in each discipline. Their breadth of knowledge is impressive and their presentations are comprehensible for laymen like me. I was particularly attracted to the question "How was the first living thing formed?" and also "Why is the universe expanding faster and faster?". The accompanying cartoons are well done. The ideas folder at the end touches on many current news topics like greenhouse gases and genetic engineering.An interesting read for those who wonder "Why?". ... Read more

Product Description
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. The book contains definitions, unsolved problems, questions, theoremscorollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ). ... Read more

Book Description
The definitive account of the Everests of mathematics--the seven unsolved problems that define the state of the art in contemporary math.

In 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader. ... Read more

Customer Reviews (19)

Seven of the greatest mathematical problems
If you want to know about seven of the most difficult unsolved math problems for which the Clay Mathematics Institute offers 1 million dolars a piece to whoever can solve them, this is the right book. Actually, we might talk about six unsolved problems since Perelman apparently solved the famous Poincaré conjecture.

A quite readable account for someone who has some training in math.

The best that could be done in a linear medium
This compromise between the desire to be comprehensible to a wide audience, and to describe aspects of highly abstract mathematics, works better than one might expect, but it is still a compromise.(I skipped years of school and took my Math degree at London University too immature to be successful.)

In my opinion, the only way this (book)/project could have been successful would have been for the Clay Institute to have commissioned a website with home pages for each of the problems, and a large web of explanatory pages for the various mathematical concepts involved.

There was one place I thought the author (no doubt overwhelmed with the purely mathematical difficulties of the task he had set himself) missed an opportunity to be clear.His Navier-Stokes equations describe a perfectly incompressible fluid.Clearly this is a mathematical abstraction - the speed of sound in such a fluid would be greater than the speed of light, indeed infinite.I think the whole thing would have been clearer if he had noted that the real question to be answered is "are the equations for an incompressible fluid a useful approximation to the behaviour of a real fluid, or does the attempt to approximate inevitably lead to nonsense?".Attempts to simulate multi-body gravitational interactions on a PC screen, for example, seem easy to program, but simple programs that calculate forces at an instant and then step positions forward a finite time, inevitably lead to all the particles eventually shooting off the screen, simply because two particles very close together at the instant have mometary huge forces on them, and the approximation that the force is constant over a step is then nonsensical.So far, the Navier-Stokes equations seem to fail in the same way.The question is, can this be fixed?At least that's my understanding, but it does not come through in this book.

Good reading for non-experts
It is probably impossible to satisfy everyone when writing a book about modern mathematics: no matter how good the book, some readers are bound to find it too primitive, while others will be hopelessly lost. The author seems to have tried to find the middle ground, perhaps a little on the "simple" side. A professional mathematician would probably find this book far too elementary; as a chemist, I found it educational. In places, it goes on and on about elementary concepts instead of progressing quickly to something more advanced. But overall, it was a good and stimulating reading that provided a glimpse of contemporary mathematics. Recommended if you are a non-mathematician with an interest in mathematics.

hard math made interesting...
This week I finished reading The Millennium Problems, by Keith Devlin. It's a look into seven of the hardest, unsolved mathematics issues we have on our hands today. A prize for solving any of these puzzles has been offered by the Clay Mathematics Institute, offering one million dollars to anyone who solves or resolves any of them.

Devlin's book is a math populizer, and he does his best to illustrate the seven puzzles in question. They are:

1. The Riemann Hypothesis, which asks if there is a pattern to the distribution of the prime numbers, related to the zeta function.
2. Yang-Mills Theory and the Mass Gap Hypothesis, which would help us understand why the electron has mass.
3. The P vs. NP problem, which seeks to understand the types of problems that computers can analyze, by trying to determine whether problems can be broken up into two groups: easy to find an answer (P), vs. easy to check the answer (NP).
4. The Navier-Stokes Equations, which are differential equations governing fluid dynamics, but don't have known general solutions.
5. The Poincare Conjecture which is a toplogical problem for 4-dimensional objects, asking the question as to whether the surface of a four-dimensional sphere is simply connected.
6. The Birch and Swinnerton-Dyer Conjecture, which relates to whether a particular class of equations have solutions.
7. The Hodge Conjecture, which is a rather complicated piece of work in analytic geometry.


I have to be honest...while I followed the explanations given in the book for the last two, I know I have no hope of explaining them. The author does his best with the material, but he even admits that those two problems are rather obtuse.

All in all, it's a good book, for the mathematically inclined. The author provides good explanations for the problems, illustrating their histories, and the stories of those folks who originated the problems. Check it out...but only if math is your bag, baby.

An uneven account
As the author himself relays many times in the introduction, it is not the easiest task to explain to lay readers the forefront of mathematical research (or for that matter, the forefront of any academic discipline). However tried the author did, and in my opinion failed to convey the fundamental issues.

History is always a good place to start from when describing a problem. And the author excels in putting together the many strands of history leading up to the seven millennium problems. But history in itself cannot be sufficient in itself without describing the actuals. For instance in a chapter describing the Navier-Stokes theorem the author bluntly writes down the differential equation for users to read. Yes, specialists will relate to them, but for the lay readers? The chapter on Hodge's conjecture is even worse. The author acknowledges honestly there that he has no means to explain in lay terms the question in hand. In such case shouldn't the chapter be expanded to attempt to explain instead of just stating that there is no easy way? How else would writing a book on such topic be justified?

Even in the better chapters (the chapter on Riemann's hypothesis for e.g.) the disconnect between the good historical material and the lack of description for the real problem are evident. The book should either have been written for professional mathematicians or, if intended for general readers, should have limited its scope to things the author found clear-cut methods to explain. I admire the author's attempt, but as it is, the book appears uneven and incomplete. ... Read more

Book Description

This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science.

The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.

Book Description
UNSOLVED PROBLEMS OF SCIENCE UNSOLVED PROBLEMS OF SCIENCE by A. W. HASLETT Sometime Foundation Scholar of Kings College, Cambridge LONDON G. BELL AND SONS, LTD 1935 CONTENTS Authors Note page ix Chapter I. THE UNENDING QUEST The ignorance of science its cost the progress of science practical achieve ments knowledge not fully applied discoveries seldom revolutionary problems can be discovered as well as facts different types of ignorance of detail of principle of the nature of things the limitations of science their relation to religion a coming difficulty page I Chapter II. THE CREATION OF THE UNIVERSE Difficulty of creation from nothingness an alternative approach astro nomers survey of space one universe among many the scattering of the universes all the universes once one the Relativity interpretation space itself expanding the Concertina Universe the Explosive Crea tion Universe Creation by Toppling connection with the birth of the stars a non-Relativity picture leads to similar conclusions an open choice page i Chapter III. ARE THERE OTHER WORLDS THAN OURS Martian tales discovery of the canals the requirements of life on the earth, water and oxygen temperature limitations possibilities of adapta tion when the suns heat fails conditions on Mars it undoubtedly has an atmosphere the canals discussed evidence of plant life living on Mars the mystery of Venus planets of other stars their low birth-rate connection with the expanding universe life a rare accident or a de liberate creation page 40 Chapter IV. THIS CHANGING WORLD Mountains point the way our not-so-solid earth seas on the mountain tops Scandinavia tilting perhaps England too coastal erosion mountain growth to-day a radium furnace within the earth earthquakes and mountain chains the forecasting problem the worlds changing geography even continents may not be fixed the Wegener theory of drift evolution and the geologist attractiveness of the Wegener picture attempted measurements of drift a new synthesis page 60 VI CONTENTS Chapter V. OUR WEATHER CAULDRON Economic importance of weather the long-range problem the scale of natures operations the ocean boiler house anti-cyclones and depressions ten million million tons of air shifted annually country omens in perspec tive the quest for cycles sunspot changes world weather relationships a gigantic survey India-South America-Siberia-Honolulu building up a forecasting formula a 4-to-i chance many records to be examined ignorance above ground level the missing control factors page 91 Chapter VI. MESSAGES FROM SPACE Messages are always being received cosmic rays meteors space not empty our own atmosphere as a border land two lines of evidence point to warmth discovery of the cosmic cloud chalk , salt and ordinary air theory that the cosmic cloud is hot meteors from beyond the sun discovery of cosmic rays welcomed in balloons pursued to the bottom of lakes more penetrating X-rays recreation in space annihilation of matter an alternative picture the earth as a magnet effect on incoming rays two international surveys both explanations may be right page 119 Chapter VII. THE ORIGIN OF MAN Evolution proved man can scarcely be an exception yet his ancestry so far irrecoverable arguments from anatomy from blood sera from embryo development search for missing links Neanderthal man Java ape-man Piltdown man in England Pekin man no single line of ascent the pro blem of Africa our ape relations difficulties of parallel evolution hopes of further finds page 147 Chapter VIII... ... Read more

Book Description
This book contains discussions of hundreds of open questions in number theory, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, extensive new material has been added, and corrections have been included throughout the book. This volume is an invaluable supplement to any course in number theory. ... Read more

Customer Reviews (2)

An excursion into the labyrinths carved by numbers
Number theory is the most enigmatic of disciplines, in that the problems are so easy to state and understand and yet so hard to resolve. Furthermore, when solved, the proofs are sometimes surprisingly easy. Inthis collection, Guy has put together a truly fascinating survey of what iscurrently (un)known about numbers.
Each page is an excursion into theextensive labyrinths carved out by numbers. Approximately once a month, Iscan it looking for new avenues to explore. Invariably, I see something,sketch out some possible proof routes and then end in frustration. Atypical result of working in number theory.
Whether you are anamateur or professional, if you have an interest in number theory, you willlike this book. Perhaps you will be able to make some progress towardsresolving some of these problems. It is certainly possible, as no field hashad more positive contributions from amateurs than number theory. EvenFermat fit the definition of an amateur.

Outstanding source of intersting problems in number theory
Another of my 10 favorite books.A constant source of inspiration ... Read more

Book Description
Victor Klee and Stan Wagon discuss some of the unsolved problems in number theory and geometry, many of which can be understood by readers with a very modest mathematical background. The presentation is organized around 24 central problems, many of which are accompanied by other, related problems. The authors place each problem in its historical and mathematical context, and the discussion is at the level of undergraduate mathematics. Each problem section is presented in two parts. The first gives an elementary overview discussing the history and both the solved and unsolved variants of the problem. The second part contains more details, including a few proofs of related results, a wider and deeper survey of what is known about the problem and its relatives, and a large collection of references. Both parts contain exercises, with solutions. The book is aimed at both teachers and students of mathematics who want to know more about famous unsolved problems. ... Read more

Customer Reviews (2)

Were to Find a Math Problem
I admit I didn't read the entire book. But don't think that discredits my review. You see this is a special math book... One that you don't read cover to cover. You simply skim through the book and pick a unsolved problem that interest you. So you see it isn't read cover to cover like a textbook. And that is what makes this such an excellent math book. The book isn't about remembering rules it is about problem solving. And the organization of the book helps in gathering facts and understanding how others have approached the problem.

Unsolved problems is part of what mathematics are based on. Most of the content is easy to understand at undergraduate level. For fun I recommend only reading the problem's description and do your own research and later compare it to the second section of the book. I have worked on Prime numbers and have made some progress. (Just check my profile.) Math work does not get done without math problems. So if you are looking for a learning experience this is an excellent place to start.

Essential background information on old, unsolved problems
If progress towards the solution of the unsolved problems is to be continuous, then those in the field must be reminded on a regular basis. Furthermore, it is pretty clear that many of these problems will not be solved using "traditional" approaches, but by some sort of back or side door method. Therefore, those in other areas of mathematics need to see the unsolved problems presented. One never knows when someone working in an area that appears to be unrelated will read a problem and realize that what they are doing can be used to obtain a solution.
This work not only presents the problems, but also gives enough background to allow the non-specialist to understand them. Several leading theorems and their proofs are given, all in a clear, concise style. Many exercises are given at the end of sections, with hints toward solution at the back of the book.
The methods used would also allow this book to be used as a text in an advanced undergraduate colloquium or beginning graduate seminar in the golden oldies of mathematics. The competent undergraduate will have no difficulty comprehending the majority of the material.
An excellent addition to any collection, this book is a key. For it is clear that many of the unsolved problems will be defeated by one who never started with that intent, but saw the problem and had the correct leap of insight.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description
Philosophers and historians of the French Revolution have seen Rousseau's influence as the decisive link between the doctrines of the Enlightenment and the practice of its revolutionary disciples. Professor Leigh here addresses the bibliographical foundations of that question, without which all attempts to settle it in the past have lacked authority. Introducing the most advanced techniques to identify variant and pirate editions of Rousseau's writings, he establishes that there were at least 28 separate imprints and an additional 12 reprints of the Contrat Social in collective editions between 1762 and 1783. Professor Leigh shows also that Rousseau's life and thought excited a fascination and interest in the last years of the Ancien Regime which was nursed by the publishers of his Oeuvres, who sought to satisfy an apparently ceaseless demand by extending their editions, while at the same time attempting to ward off both their creditors and their imitators. ... Read more