Chaos ... What chaos?

"The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does."
(Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

God of Chaos

A Fractal by Ken Keller

15cents 4 your thoughts

15cents 4 yours thought

After his death in 1955, Einstein became the epitomy of intelligence in general, not just of scientific genius. His image appeared on postage stamps and magazine covers. He became the subject of poetry, sculpture and painting. "Einstein advertisements" implied that buyers were smart to buy the product, or likely to become smart if they did. Conversely, the simplicity and ease of use of a product was endorsed by the assurance that you did not need to be an Einstein to use it. The most elaborate advertising campaign that exploited Einstein's image was the 1979 Pennaco Hosiery catalogue, intriguingly entitled "The Theory of Relativity — Fall-Winter 79". Women were exhorted to "Believe in the theory of relativity". "Fashion relates beautifully — making each part relative to the next — and thus creating fantastic total outfits. Thank you, Mr. Einstein, for the theory... and Happy 100th Birthday!"

A tour de force

... do you recognize this formula?

Fomenko pictures

"I do not really think of myself as an artist. I am a mathematician. To me, my drawings are photographs of some strange and interesting mathematical world" - Professor A.T. Fomenko of the Moscow State Unversity.

The history of the Fields Medal

The history of the Fields Medal begins in the Committee of the International Congress set up by the University of Toronto in November of 1923, with the purpose of organizing the 1924 Congress to be held in Toronto.

Fields was its chairman, and his colleague J.L.Synge the secretary. Although Fields probably conceived of the medal at some earlier time, the first recorded mention of it is in the minutes of a meeting of that committee on February 24, 1931 where it is “resolved that the sum of $2,500 should be set apart for two medals to be awarded in connection with successive International Mathematical Congresses through an international committee appointed for such purpose initially by the executive of the International Mathematical Congress, but later by the International Mathematical Union”. The $2,500 was evidently the balance on hand after all expenses of the 1924 Congress had been met.

At the next meeting of the committee, in January 1932, Fields indicated that the idea of the medal had the support of the major mathematical societies of France, Germany, Italy, Switzerland and the United States, and he also outlined the principles behind the proposed medal. The genesis of the rule that it be awarded only to mathematicians no older than forty is evidently the statement that “… while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others”. Then he continued “In commenting on the work of the medalists it might be well to be conservative in one’s statements, to avoid invidious comparisons explicit or implied. The Committee might ease matters by saying that they had decided to make the awards along certain lines not alone because of the outstanding character of the achievement but also with a view to encouraging further development along these lines”. And mindful of the turmoil of ten years earlier (over the exclusion of the “Central Powers” from the 1924 Congress), he added “the medals should be of a character as purely international and impersonal as possible. There should not be attached to them in any way the name of any country, institution or person”.

Of course, in spite of Fields’s intentions, the medal became known as the Fields Medal when it was awarded for the first time in Oslo in 1936. It is interesting to note that, at the same meeting, it was decided that “the Chairman should see the Prime Minister of Canada to arrange if possible how permanence of capital and of interest of the fund might be assured”. Such an arrangement was apparently never made, and the monetary value of the Fields Prize is presently $15,000Can (about $9500US), hardly commensurate with its stature as the “Nobel Prize in Mathematics”.

It is not known why Nobel chose not to establish a prize in Mathematics, although there are several theories about the lack of one. Fields then proceeded with the planning of the award of the first medals, but fell ill in May of 1932 and died 3 months later. Just before his death, with Synge at his bedside, he made his will. It included an amount of $47,000 to be added to the funds for the medal. Synge carried Fields’s proposal to the Congress in Zürich in September of that year. It was accepted, and a committee consisting of G.D.Birkhoff, Carathéodory, E.Cartan, Severi and Takagi was formed to make the first awards at the Oslo Congress in 1936. They chose Lars Ahlfors of Finland and Jesse Douglas of the U.S.A.

Unfortunately, war again intervened, and the next ICM was not held until 1950, in Cambridge, Massachusetts, when Laurent Schwartz and Atle Selberg were selected as the Fields Medalists. A list of all Fields Medal winners (with a short description of their work) can be found here. An analysis by Michael Monastyrsky of the effect of Fields Medalists on 20th century mathematics and physics, delivered in a lecture at the Fields symposium “The legacy of John Charles Fields” held in Toronto in June, 2000, is available here. The medal itself Fields specified that the medals should “each contain at least 200 dollars worth of gold and be of a fair size, probably 7.5 centimetres in diameter. Because of their international character the language to be employed it would seem should be Latin or Greek”.
The medal does in fact meet these specifications (in 1933 dollars!). Its monetary value has at least on one occasion been of critical importance: in the turmoil at the end of World War II, Ahlfors became separated from his wife, and was allowed to leave Finland with only 10 crowns. He smuggled out his Fields Medal and pawned it, enabling him to reach his wife in Zürich. (He later retrieved it with the help of some Swiss friends). The medal, struck every four years in the Royal Canadian Mint, was designed by the Canadian sculptor R.Tait McKenzie. For the obverse, he chose a picture of Archimedes from a collection at Columbia University. The Latin inscription from the Roman poet Manilius surrounding his image may be translated “to pass beyond your understanding and make yourself master of the universe”. The phrase comes from Manilius’s Astronomica 4.392 from the first century A.D

Einstein and his blindfriend

Albert Einstein (1879-1955)

This story shows how complex Einstein could be. Not long after his arrival in Princeton he was invited, by the wife of one of the professors of mathematics at Princeton, to be guest of honor at a tea.-Reluctantly, Einstein consented. After the tea had progressed for a time, the excited hostess, thrilled to have such an eminent guest of honor, fluttered out into the center of activity and with raised arms silenced the group. Bubbling out some words expressing her thrill and pleasure, she turned to Einstein and said: "I wonder, Dr. Einstein, if you would be so kind as to explain to my guests in a few words, just what is relativity theory ? "
Without any hesitation Einstein rose to his feet and told a story. He said he was reminded of a walk he one day had with his blind friend. The day was hot and he turned to the blind friend and said, "I wish I had a glass of milk."
"Glass," replied the blind friend, "I know what that is. But what do you mean by milk?"
"Why, milk is a white fluid," explained Einstein.
"Now fluid, I know what that is," said the blind man. "but what is white ? "
" Oh, white is the color of a swan's feathers."
" Feathers, now I know what they are, but what is a swan ? "
"A swan is a bird with a crooked neck."
" Neck, I know what that is, but what do you mean by crooked ? "
At this point Einstein said he lost his patience. He seized his blind friend's arm and pulled it straight. "There, now your arm is straight," he said. Then he bent the blind friend's arm at the elbow. "Now it is crooked."
"Ah," said the blind friend. "Now I know what milk is."
And Einstein, at the tea, sat down.