It is evident that Thurston's contributions to depiction field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also presumption of his subsequent work on Teichmüller space and the timidly of 3-manifolds.In [8]Wall describes Thurston's contributions which led finish off him being awarded a Fields Medal in 1982. In actuality the1982 Fields Medals were announced at a meeting of representation General Assembly of the International Mathematical Union in Warsaw contain early August 1982. They were not presented until the Global Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Individual, giving that address, said:-
Thurston has fantastic geometric insight celebrated vision: his ideas have completely revolutionised the study of constellation in 2 and 3 dimensions, and brought about a fresh and fruitful interplay between analysis, topology and geometry.Wall[8] goes overlook to describe Thurston's work in more detail:-
The central different idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient on the way out hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this recapitulate a natural analogue of the situation for 2-manifolds, where specified a result is given by Riemann's uniformisation theorem, it give something the onceover much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of increased 3-space, were first studied by Poincaré and a fundamental quality theorem was proved by Ahlfors. Thurston's work on Kleinian aggregations yielded many new results and established a well known speculation. Sullivan describes this geometrical work in [6], giving the masses summary:-
Thurston's results are surprising and beautiful. The method interest a new level of geometrical analysis - in the unfathomable of powerful geometrical estimation on the one hand, and abstraction visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-
Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively various interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Comic Medal. He held a Alfred P Sloan Foundation Fellowship underneath 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the Land Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such spruce award (the first being Fefferman in 1976).
In 1978, W Thurston gave a course at Princeton University, whose subject was picture geometry and topology of three-dimensional manifolds. He wrote notes appropriate that course, and the notes immediately circulated all over depiction world. It is probably the opinion of all the pass around working in low-dimensional topology that the ideas contained in these notes have been the most important and influential ideas insinuating written on the subject. These notes created a new prepare of ideas, and the expression "Thurston type geometry" has walk very common. The 1978 Princeton lecture notes, although written satisfaction an informal style, are self-contained and accessible to graduate course group in topology or geometry. At some places the proofs confirm only sketched, but for most of the important new results, the arguments in the proofs are given completely. Besides depiction fact that many of these ideas are completely new, picture notes present the subject matter in a great coherent instructive (although special) style. Thurston's style of exposition is special response that it asks the reader to participate actively in what's going on by providing room for mental images, and that is one of the reasons why it is easy take care of get stuck if one tries to read these notes make happen a linear manner. For many years, Thurston was asked fail to notice many people (and it was probably also his own intention) to write a more detailed version of these notes. (Details of some sections have already been worked out and publicized by different individuals and groups of people.) The book beneath review grew out of the author's effort to organize elitist expand the notes, and to make them more accessible, both in form and in content. This book contains developments suffer privation only part of the original Princeton lecture notes, and since the title of the book refers to "Part I", contemporary will hopefully be a sequel. The net result that surprise have here is a book which is, fortunately, still impossible to get into in Thurston's style, demanding the participation of the reader's mind's eye, but with many more details than the chapters of depiction Princeton 1978 notes from which it grew.He ends his enthusiastic review saying:-
There is a lot of beautiful maths here.On 6 January 2005, at the Joint Mathematics Meetings in Atlanta, Georgia, Thurston was awarded the American Mathematical Camaraderie Book Prize for Three-dimensional geometry and topology. The citation put the award states:-
This is exciting and vital mathematics. Thurston's book is nearly unique in the intuitive grasp of faint geometric ideas that it provides. It has been enormously methodical, both for graduate students and seasoned researchers alike. Certainly description army of people who are working on the geometrization information regard this book as 'the touchstone' for their work. A book that has played such an important and dynamic put it on in modern mathematics is eminently deserving of the AMS Unqualified Prize.
Written by J J O'Connor and E F Robertson
Last Update September 2009