By Michael Atiyah

Even if the Fields medal doesn't have an identical public attractiveness because the Nobel prizes, they proportion the same highbrow status. it really is limited to 1 box - that of arithmetic - and an age restrict of forty has turn into an authorised culture. arithmetic has in general been interpreted as natural arithmetic, and this isn't so unreasonable considering that significant contributions in a few utilized parts can (and were) well-known with Nobel prizes. The restrict to forty years is of marginal value, in view that such a lot mathematicians have made their mark lengthy sooner than this age. a listing of Fields medallists and their contributions supply an outline of arithmetic during the last 60 years. It highlights the parts during which, at a number of occasions, maximum growth has been made. This quantity doesn't faux to be complete, neither is it an historic rfile. nonetheless, it offers contributions from 22 Fields medallists and so presents a hugely fascinating and sundry photo. The contributions themselves characterize the alternative of the person medallists. at times the articles relate on to the paintings for which the Fields medals have been presented. In different instances new articles were produced which relate to extra present pursuits of the medallists. this means that whereas Fields medallists needs to be less than forty on the time of the award, their mathematical improvement is going way past this age. in truth the age restrict of forty used to be selected in order that younger mathematicians will be inspired of their destiny paintings.

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**Extra resources for Fields Medallists' Lectures (World Scientific Series in 20th Century Mathematics, Vol 5)**

**Example text**

Prove the real numbers are not countable. [Hint: they are. an ai2 an. a2i a22 α 2 3·. a3i a32 a 3 3.. /th digit in the decimal expansion of the ¿th number (arranged for counting) between 0 and 1. än ä22 (X33. . ] 7. Define a Cauchy sequence in a metric space. ) 8. Show directly from the definition that {l/n} is a Cauchy sequence. 9. Let , Xl = 1 , 1 X2 = - , 2' 3 Xz = - , * Z 4 4' "* = 5 8> · · · > - » Xn_2 + Xn-l 2 Show directly from the definition that {xn} is a Cauchy sequence. 48 III. Completeness Properties 10.

To verify this directly, consider I I / » - / I l = sup|/ n (aO -f(x) 0

2. 10. 3. 11. 4. 12. 5. 9 to show that 6. 7. 12 in terms of {1/nJ 6 lv. (a) Show geometrically that ···. hChChC 1 / ■ * + 1l 1 dx X k+1 and hence 1 1 1 k~ " k + 1 ^ for every k = i, 2, . . , n, (b) Set 1 - - fk+l dx / — = c* and show l i m ^ « , ] ^ ^ ck exists (finite). 4. >o, 26 II. Sequence Spaces and Infinite (c) Find Ci + c2 + c3 + · · · + cn-i = Σ / L i c¿ (d) Show m Series closed form. lim I ( 1 + \ + I + - - - + - ) - In n 1 = lim Σ ck. n->ooL\ 2 3 nj J n^ k=1 This number is called Euler's constant or Mascheroni's constant.