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more exciting points of interaction between math and physics:
It also serves as a deep link, via spectral theory, between geometry(math), and quantum gravity(physics). While the subject has roots far back, this lovely book presents some of the more exciting developments in the past decade. One of the success stories in interdisciplinary theoretical science! It is well written, and will be a great source for grad students. This very nice book further points toward the research trends of the future. Moreover, the results presented in the book are timeless. The book will be of value also ten years from now. Being an acknowledged authority in the subject, this author is in a unique position to write a book on the central themes and theories in the subject.
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10,000 years ago the Mediterranean Basin, from the Atlantic ocean to the Euphrates and Tigris Rivers was covered by vast forests. To the North, the forests extended to the retreating glaciers in Scandinavia. To the South, they extended from the seashore deep into what is now the Sahara Desert. In the East,deep forests of giant trees spread from the sea coast of Israel north through Lebanon and Syria into Turkey and then down into Iraq.The climate was temperate, the rivers ran clear all year long, and the soil was fertile.
The original inhabitants were hunters and gatherers, and for them the forests were both a challenge and a rich source of food. With the coming of agriculture, however, the forests became the adversary: Quid est agricola, silvae adversarius. Agriculture required clearing the forests for fields. As population increased, villages were built using wood. From these grew towns and then cities. And as these grew the forest was cut for fields, for timber and for firewood. Civilization and deforestation grew together.
Thirgood shows how civilization and deforestation were symbiotically linked. The forests supplied wood for cooking, for heat and for building. Wood became the prime strategic resource for the cities and kingdoms that grew from them. Timber was necessary for the building of ships, which made possible long-distance trade. But wood was also the prime fuel, necessary for the making of pottery and metalsmithing. Thus the forests provided not only the vehicles to carry the trade, but the trade goods themselves. The great forests became the prizes of great empires.
Thirgood links the rise of civilization in the Mediterranean Basin to the decline of the forests. But he shows that kings and governors recognized early on that such a strategic resource should be protected. Thus reforestation, as well as deforestation,occurred in the ancient world. Trees were cut, but under powerful rulers, the forests were not despoiled.
Thirgood believes that three factors combined to destroy the Mediterranean forests: 1) War ( in which trees were cut for strategic purposes, but in which forests also were burned to deprive enemies of those very resources); 2) Weak governments ( who were unable to keep wholesale clearing or excessive exploitation from occuring); and 3) The triumph of pastoralization over agriculture ( which he dates from the conquest of the area by Islam [632-732AD]).
Of the three key factors of deforestation, the most damaging has been pastoralization, particularly the herding of goats. Goats not only climb trees to eat the leaves, they also eat all of the new trees, preventing reforestation. Togeher with sheep, they eat all of the grasses as well, thus denuding the land.
As deforestation became widespread the climate began to be affected. Rain that had been stored by the forests now ran off,carrying with it the soil. Streams silted up and then dried up. Fields were washed away. Cities that had been built surrounded by fertile fields in the midst of forests, now became ruins in the midst of sterile plains. Speaking of the hundreds of dead cities in northern Syria, Thirgood notes: "These ruins stand on rock. The archaeological evidence inidcates that, over the whole region, from three to six feet of soil has been removed." (p.103) In Turkey, which in Byzantine times (300AD-1000AD) had been known as a region of the " thousand and one churches", following the seventh century no new churches were built and those that existed fell into ruin.(p24) All along the southern shore of the Mediterranean, the forests were replaced with man-made desert. "Present-day Mediterranean land managers believe that the landscape decay and general desiccation of the Mediterranean region is not a consequence of adverse climatic conditions as [Ellsworth] Huntington claimed, but as a result of man's misuse of the land." (p.25)
Thirgood is a realist, not a romantic. He recognizes that civilization could not have developed without the use of the forest resources. But this also imposes a resonsibility to preserve and replenish those resources.
There are only two things wrong with this book: 1) It was published in 1981 and needs to be brought up to date; and 2) It is out of print and needs to be reprinted. It is an outstanding book, that covers a major subject in a brief compass, masterfully.
This is most definitely not a book to learn algebraic number theory from. Also, the development of the properties of Dedekind domains and in general a lot of the proofs in the first three chapters are rather weird, and I think unnecessarily complicated. You need comfort with basic commutative algebra(tensor products and localization, for example) to read the first part. The one good thing that can be said about the first three chapters is that they are very comprehensive. But if you're buying this book for the class field theory, and you've already studied algebraic number theory, then you'll quickly pass over the first three chapters and get to the good stuff in the last three. Chapter four develops some really pretty analytic theory, such as the class number formula and the frobenius density theorem. The fact that this theory is first developed and then used to prove the reciprocity law, rather than the more usual other way around, is the distinctive feature of this book's treatment, which allows it to dispense with ideles and most of the group cohomology.
Group cohomology is not absent from this treatment, however. Chapter five starts with some basic lemmas about it-which are presented with no motivation, and the treatment is almost unreadable, I think, if you haven't seen any homological algebra elsewhere. Seeing it elsewhere is not a logical prerequisite, as he doesn't assume any knowledge of it, but there's quite a bit of diagram chasing in this chapter, which he doesn't really do a great job of explaining. This whole chapter, which culminates in the proofs of the reciprocity theorem and existence theorem, is very complicated, but that is no fault of the author's, as this material is fundamentally difficult.
The sixth chapter covers applications of all the theory developed, primarily to quadratic extensions of Q. This chapter is perhaps best considered as a reward for all the hard work in the earlier sections, as the style of presentation changes considerably, being less abstract and more example-oriented.
This books whole approach to the reciprocity law is based closely on Artin's original proof, I think. That is an advantage and disadvantage. It reduces the prerequisites to a minimum, and lets us see the truly ingenious nature of Artin's original proof, but more modern treatments, which start with the local theory, and use ideles, are ultimately easier to understand, I think. But you can't really appreciate what makes these methods superior if you haven't struggled with the traditional ones.
So, this book fills a definite niche in providing a uniquely traditional and 'elementary' treatment of class field theory, but doesn't do it as well as it might. In my opinion and knowledge, the only really well-written treatment whose only difficulties are the mathematically necessary ones-which are substantial-is Jurgen Neukirch's in Algebraic Number Theory.
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