By Honorary Professor Michael Atiyah Sir, Daniel Iagolnitzer
Even though the Fields medal doesn't have an analogous public acceptance because the Nobel prizes, they proportion an identical highbrow status. it truly is limited to 1 box - that of arithmetic - and an age restrict of forty has develop into an authorised culture. arithmetic has generally been interpreted as natural arithmetic, and this isn't so unreasonable on the grounds that significant contributions in a few utilized parts can (and were) well-known with Nobel prizes. The restrict to forty years is of marginal importance, on account that so much mathematicians have made their mark lengthy prior to this age. a listing of Fields medallists and their contributions supply an outline of arithmetic over the last 60 years. It highlights the parts within which, at a variety of instances, maximum development has been made. This quantity doesn't faux to be finished, neither is it an historic rfile. however, it offers contributions from 22 Fields medallists and so offers a hugely fascinating and sundry photograph. The contributions themselves signify the alternative of the person medallists. in certain cases the articles relate on to the paintings for which the Fields medals have been provided. In different circumstances new articles were produced which relate to extra present pursuits of the medallists. this means that whereas Fields medallists has to be lower than forty on the time of the award, their mathematical improvement is going well beyond this age. actually the age restrict of forty was once selected in order that younger mathematicians will be inspired of their destiny paintings.
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The recurrent part is the limit set, and it is of interest only if it has positive measure. But even if the area conjecture is true Sullivan’s work remains significant for groups whose limit set is the whole sphere. Sullivan has several theorems, but the one that has captured my special interest because I understand it best asserts that there is no invariant vector field supported on the limit set. If the limit set is the whole sphere there is no invariant vector field, period. In an equivalent formulation, the limit set carries no Beltrami differential.
But even with this principal limitation the task was not an easy one, and it was felt t o be very encouraging for the expectations we may entertain of the future development of our science that we had to choose among so many young and very talented mathematicians, each of whom should certainly have been worthy of an official appreciation of his work. Our choice fell on Professor Atle Selberg and 26 Fields Medallists’ Lectures Professor Laurent Schwartz, and I feel sure that all members of the Congress will agree with the committee that these two young mathematicians not only are most promising as to their future work but have already given contributions of the uttermost importance and originality to our science; indeed they have already written their names in the history of mathematics of our century.
For the general case it seems that one would need a better topological classification of three-manifolds with constant negative curvature. It is therefore not suprising that the problem has come to the attention of the topologists, and I am happy to report that at least two leading topologists are actively engaged in research on this problem. I believe that this pooling of resources will be very fruitful, and it would of course not be the first time that analysis topology, and vice versa. Some time ago W.