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The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time
Paperback
Published January 1, 2004
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About the author
Keith Devlin
85 books165 followersDr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.
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December 14, 2024
Chapter 2 on Yang-Mills theory and the mass gap hypothesis was basically a thirty-page summary of the entire history of physics. I understand the desire to make the point that physicists assume the universe operates according to mathematical laws, but by focusing so much on the pre-history (e.g. Plato’s belief that matter was composed of the five regular solids), the author was forced to give a very superficial skim through the modern physics that needed to be described in order to set the stage for the actual Millennium Problem. Even the problem and its implications were glossed over. This is a shame because gauge theory and group theory are such rich topics.
In the discussion in Chapter 3 of P vs. NP, the author inexplicably conflates the traveling salesman optimization problem with the traveling salesman decision problem. This sounds like a small point but TSP was used as a scaffolding for the discussion of the various time complexity classes, so when he then made comments about NP using TSP as an example, it was extremely confusing. I sat there for minutes just trying to figure out how a solution could possibly be verified in polynomial time (it probably can’t be, since for the optimization problem we don’t know that it is in NP — whereas the decision problem is NP-complete so this is almost surely what he was referring to). I kept looking for a point in the chapter I must have missed where the author explicitly switched from one definition to the other, but never found one. It’s a bit bizarre — almost as if he expected the reader not to think of the implications of the statements he made.
In the last two chapters on the Birch and Swinnerton-Dyer Conjecture and the Hodge Conjecture, the author developed an annoying habit of interjecting to say how difficult the upcoming math would be. Some may find it comforting to know that if they struggle it’s not because of a lack of basic competency, but I mentally tense up whenever someone gives this warning, and it makes me assume I won’t be able to understand what follows and shut down. And I assume most readers picking up a book on some of the most difficult math problems ever formulated will be ready for some difficulties and don’t need constant warnings.
One complaint I have with the structure of the book is that all chapters are roughly the same length at 30 pages. Some problems (especially the Hodge conjecture, which was the shortest by far despite being the most abstract) deserve more explanation, and I wish the necessary space had been devoted to them.
Overall, the book falls in a bit of an awkward middle ground between being optimal for a general audience and optimal for a mathematically inclined (but not professional) audience. By explaining the basic concepts in detail, not enough time is left to explain the higher-level concepts and the problems themselves, so both groups will be unhappy.
In the discussion in Chapter 3 of P vs. NP, the author inexplicably conflates the traveling salesman optimization problem with the traveling salesman decision problem. This sounds like a small point but TSP was used as a scaffolding for the discussion of the various time complexity classes, so when he then made comments about NP using TSP as an example, it was extremely confusing. I sat there for minutes just trying to figure out how a solution could possibly be verified in polynomial time (it probably can’t be, since for the optimization problem we don’t know that it is in NP — whereas the decision problem is NP-complete so this is almost surely what he was referring to). I kept looking for a point in the chapter I must have missed where the author explicitly switched from one definition to the other, but never found one. It’s a bit bizarre — almost as if he expected the reader not to think of the implications of the statements he made.
In the last two chapters on the Birch and Swinnerton-Dyer Conjecture and the Hodge Conjecture, the author developed an annoying habit of interjecting to say how difficult the upcoming math would be. Some may find it comforting to know that if they struggle it’s not because of a lack of basic competency, but I mentally tense up whenever someone gives this warning, and it makes me assume I won’t be able to understand what follows and shut down. And I assume most readers picking up a book on some of the most difficult math problems ever formulated will be ready for some difficulties and don’t need constant warnings.
One complaint I have with the structure of the book is that all chapters are roughly the same length at 30 pages. Some problems (especially the Hodge conjecture, which was the shortest by far despite being the most abstract) deserve more explanation, and I wish the necessary space had been devoted to them.
Overall, the book falls in a bit of an awkward middle ground between being optimal for a general audience and optimal for a mathematically inclined (but not professional) audience. By explaining the basic concepts in detail, not enough time is left to explain the higher-level concepts and the problems themselves, so both groups will be unhappy.
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