Source: Slideshare, May 2009
Source: IHES website, Dec 2010
Source: Nature.com, Mar 2017
To explore the mathematical possibilities of alternative geometries, mathematicians imagine such ‘non-Euclidean’ spaces, where parallel lines can intersect or veer apart. Now, with the help of relatively affordable VR devices, researchers are making curved spaces — a counter-intuitive concept with implications for Einstein’s theory underlying gravity and also for seismology — more accessible. They may even uncover new mathematics in the process.
Visualizing such geometries could be especially useful as a mathematical tool, she says, because “very few people have thought of visualizing them at all”.
Source: Mike Dash history website, Oct 2011
June 7, 1890, when—for the first and only time—a woman ranked first in the mathematical examinations held at the University of Cambridge. It was the day that Philippa Fawcett placed “above the Senior Wrangler.”
The most serious candidates invariably hired tutors and worked more or less round the clock for months. The historian Alex Craik notes that C.T. Simpson, who ranked as Second Wrangler in 1841, topped off his efforts by studying for 20 hours a day in the week before the exams and “almost broke down from over-exertion…
G.F. Browne, the secretary of the Cambridge exam board, was also concerned—because he feared that the women entered in the 1890 math exams might be so far below par that they would disgrace themselves. He worried that one might even place last, a position known at Cambridge as “the Wooden Spoon.” Late on the evening of June 6, the day before the results were to be announced, Browne received a visit from the senior examiner, W. Rouse Ball, who confided that he had come to discuss “an unforeseen situation” concerning the women’s rankings. Notes Siklos, citing Browne’s own account:
After a moment’s thought, I said: ‘Do you mean one of them is the Wooden Spoon?’
‘No, it’s the other end!’
‘Then you will have to say, when you read out the women’s list, “Above the Senior Wrangler”; and you won’t get beyond the word ‘above.’ “
By morning, word that something extraordinary was about to occur had electrified Cambridge. Newnham students made their way to the Senate House en masse, and Fawcett’s elderly grandfather drove a horse-drawn buggy 60 miles from the Suffolk coast with her cousins Marion and Christina. Marion reported what happened next:
It was a most exciting scene in the Senate… Christina and I got seats in the gallery and grandpapa remained below. The gallery was crowded with girls and a few men, and the floor of the building was thronged with undergraduates as tightly packed as they could be. The lists were read out from the gallery and we heard splendidly. All the men’s names were read first, the Senior Wrangler [G.T. Bennett of St John’s College] was much cheered.
At last the man who had been reading shouted “Women.”… A fearfully agitating moment for Philippa it must have been…. He signalled with his hand for the men to keep quiet, but had to wait some time. At last he read Philippa’s name, and announced that she was “above the Senior Wrangler.”
The male undergraduates responded to the announcement with loud cheers and repeated calls to “Read Miss Fawcett’s name again.” Back at the college, “all the bells and gongs which could be found were rung,” there was an impromptu feast, bonfires were lit on the field hockey pitch, and Philippa was carried shoulder-high into the main hall—”with characteristic calmness,” Siklos notes, “marking herself ‘in’ on the board” as she swayed past. The men’s reaction was generous, particularly considering that when Cambridge voted against allowing women to become members of the university in 1921, the undergraduates of the day celebrated by battering down Newnham’s college gates.
The triumph was international news for days afterwards, the New York Times running a full column, headlined “Miss Fawcett’s honor: the kind of girl this lady Senior Wrangler is.” It soon emerged that Fawcett had scored 13 percent more points than had Bennett, the leading male, and a friendly examiner confided that “she was ahead on all the papers but two … her place had no element of accident in it.”
Source: Simons Foundation, Dec 2014
Gromov’s first bombshell was the homotopy principle, or “h-principle,” a general way of solving partial differential equations. “The geometric intuition behind the h-principle is something like this,” explained Larry Guth of the Massachusetts Institute of Technology. “If you had a sweater and you wanted to put it into a box, then because the sweater is soft, it is easy to put it into the box, and there are lots of ways to do it. But if you had to write a list of totally precise instructions about exactly how to put the sweater into the box, it would actually be hard and kind of complicated.”
In mathematics, the question was whether some high-dimensional object could be embedded into a given space. “And the only way to deal with high-dimensional objects, at least traditionally,” said Guth, “was to write down equations that say precisely where everything goes, and it’s hard to do that. Like the situation with the sweater, the only way that we could describe how to put the sweater into the box was to write a completely precise list of instructions about exactly how to do it, and it makes it look as if it is complicated. But Gromov found a good way of capturing the idea that the sweater is very soft, hence you can do almost anything and it will fit into the box.”
… a list of questions: “What is mathematics and how has it originated? Where does the stream of mathematical ideas flow from? What is the ultimate source of mathematics in the brain? Gromov rebuffed as absurd his own question about what mathematics is. But, he continued, “you can say, ‘How is it being created, how is it being studied, how is it being learned?’”
“The way I think of mathematics,” he said, “is as a physical and psychological process, not some abstraction.” And herein arises his notion of a “bug on a leaf,” which gives a nice sampling of his inquisitive excursions away from research mathematics proper and into biology, evolution, the structure of the brain, and the question of how scientific ideas evolve.
“A bug on a leaf displays two simple phenomena,” he said. “It always does the same thing, one leg, another leg, one leg, another leg. Just movement. Many, many things are done this way, including Euclidean geometry. That’s one point. The other point is information theoretic. And that is that the bug spends more time on the edge of the leaf compared to the interior, and more time on the tip of a leaf compared to the interior. And your eye does the same thing when looking at an image, your eye will spend more time around the image.”
Both of these processes, with the bug and the eye, in Gromov’s opinion, are run by universal mechanisms. “The logic of the world forces the way we think,” he said. “This is what I have been thinking about for the last 10 years: the basic principle underlying thinking, and specifically underlying thinking about mathematics. It’s very different from logic, it’s not logic.” He calls it “ergologic” — a reconsideration of traditional logic, encompassing the “ergo system” and the “ergo brain” and “ergo thinking.”
“Life is similar to how mathematics is organized in our brain. If you don’t accept it, life is impossible,” he said. And then one final Gromovian snippet I managed to catch was this: “If it is impossible, you try to do it anyway.”
Mathematics is just a way of connecting ideas that are really far apart from each other and sort of building a path of stepping stone to get from Point A to Point B
… math is not just about formulas. It is about ideas.
Source: Quanta, Mar 2017
Unique prime factorization ensures that each number in a number system can be built up from prime numbers in exactly one way. In a number ring that includes a √-5 (in practice, mathematicians often employ number systems that use the square roots of negative numbers), duplicity creeps in: 6 is both 2 x 3 and also (1 + √-5) x (1 – √-5). All four of those factors are prime in the new number ring, giving 6 a dual existence that just won’t do when you’re trying to nail things down mathematically.
Unique prime factorization is a way of constructing a number system from fundamental building blocks. Without it, proofs can turn leaky. Mixing roots with the regular numbers failed as an attack on Fermat’s Last Theorem, but as often happens in math, the way in which it failed was provocative. It launched an area of inquiry unto itself called algebraic number theory.
Today mathematicians are actively engaged in the study of “class numbers” of number systems. In their crudest form, they’re a rating of how badly a number system fails the test of unique prime factorization, depending on which roots get mixed in: A number system that gets a “1” has unique prime factorization; a system that gets a “2” misses unique prime factorization by a little; a system that gets a “7” misses it by a lot more.
On their face, you’d expect class numbers to be randomly distributed — that class number 5 occurs with the same frequency as class number 6, or that half of all class numbers are even. That’s not the case, though, and current research in the subject aims to understand why. Today mathematicians are circling in on the structure that underlies class numbers and inching closer to establishing the truth about long-conjectured values. It’s an effort that has generated insights about how numbers behave that go far beyond a proof of any one problem.
Today, research on class numbers remains inspired by Gauss, but much of it takes place in a context established in the late 1970s by the mathematicians Henri Cohen, emeritus professor of mathematics at the University of Bordeaux, and Hendrik Lenstra, who recently retired from Leiden University in the Netherlands. Together they formulated the Cohen-Lenstra heuristics, which are a series of predictions about how frequently particular kinds of class numbers should appear. For example, the heuristics predict that 43 percent of class numbers are divisible by 3 in situations where you’re adjoining square roots of negative numbers.
By the time Cohen and Lenstra made their predictions, computers made it possible to rapidly calculate class numbers for billions of different number rings. As a result, there is good experimental evidence to support the Cohen-Lenstra heuristics. However, knowing something with confidence is entirely different from proving it.
“Probably in other sciences this is where you’d be done. However, in math that’s just the beginning. Now we want to know for sure,” said Melanie Wood, a mathematician at the University of Wisconsin-Madison.