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Comment by doone 27 minutes ago

AN UNHERALDED BREAKTHROUGH: THE ROSETTA STONE OF MATHEMATICS

From Scientific American:

There is no Nobel Prize in mathematics, but in 2001 the Norwegian government established a million-dollar Abel Prize, which is widely considered as an equivalent of the Nobel for mathematicians. This year’s prize was awarded to Pierre Deligne, professor emeritus at the Institute for Advanced Study in Princeton, N.J. Today, he is honored at a ceremony held in Oslo. Deligne’s most spectacular results are on the interface of two areas of mathematics: number theory and geometry. At first glance, the two subjects appear to be light-years apart. As the name suggests, number theory is the study of numbers, such as the familiar natural numbers (1, 2, 3, and so on) and fractions, or more exotic ones, such as the square root of two. Geometry, on the other hand, studies shapes, such as the sphere or the surface of a donut. But French mathematician André Weil had a penetrating insight that the two subjects are in fact closely related. In 1940, while Weil was imprisoned for refusing to serve in the army during World War II, he sent a letter to his sister Simone Weil, a noted philosopher, in which he articulated his vision of a mathematical Rosetta stone. Weil suggested that sentences written in the language of number theory could be translated into the language of geometry, and vice versa. “Nothing is more fertile than these illicit liaisons,” he wrote to his sister about the unexpected links he uncovered between the two subjects; “nothing gives more pleasure to the connoisseur.” And the key to his groundbreaking idea was something we encounter everyday when we look at the clock.

If we start working at 10:00 in the morning and work for eight hours, when do we finish? Well, 10 + 8 = 18, so a natural thing to say would be: “We finish at 18 o’clock.” This would be perfectly fine to say in France, where hours are recorded as numbers from zero to 24 (actually, not so fine, because a workday in France is usually limited to seven hours). But in the U.S. we say: “We finish at 6:00 pm.” How do we get six out of 18? We subtract 12: 18 – 12 = 6. Mathematicians call this “addition modulo 12.” Likewise, we can do addition modulo any whole number N. Just imagine a clock in which there are N hours instead of 12. For each N, we then obtain an esoteric-looking numerical system, in which we can do addition and multiplication, just like with ordinary numbers. For many years these systems looked, even to math practitioners, like something that would never have any real-world applications. In fact, English mathematician G.H. Hardy wrote, with defiance and pride, of the “uselessness” of number theory. But the joke was on him: these numerical systems are now ubiquitous in the encryption algorithms used in online banking. Every time we make a purchase online, arithmetic modulo N springs into action!

More here.

Posted by Azra Raza at 05:51 PM | Permalink |

Comment by doone on May 8, 2013 at 5:44pm

THE MATHEMATICS OF ROUGHNESS

Jim Holt reviews Benoit B. Mandelbrot's The Fractalist: Memoir of a Scientific Maverick, in the NYRB:

Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior. The key to it is a simple yet elusive idea, that of self-similarity.

To see what self-similarity means, consider a homely example: the cauliflower. Take a head of this vegetable and observe its form—the way it is composed of florets. Pull off one of those florets. What does it look like? It looks like a little head of cauliflower, with its own subflorets. Now pull off one of those subflorets. What does that look like? A still tinier cauliflower. If you continue this process—and you may soon need a magnifying glass—you’ll find that the smaller and smaller pieces all resemble the head you started with. The cauliflower is thus said to be self-similar. Each of its parts echoes the whole.

Other self-similar phenomena, each with its distinctive form, include clouds, coastlines, bolts of lightning, clusters of galaxies, the network of blood vessels in our bodies, and, quite possibly, the pattern of ups and downs in financial markets. The closer you look at a coastline, the more you find it is jagged, not smooth, and each jagged segment contains smaller, similarly jagged segments that can be described by Mandelbrot’s methods. Because of the essential roughness of self-similar forms, classical mathematics is ill-equipped to deal with them. Its methods, from the Greeks on down to the last century, have been better suited to smooth forms, like circles. (Note that a circle is not self-similar: if you cut it up into smaller and smaller segments, those segments become nearly straight.)

Only in the last few decades has a mathematics of roughness emerged, one that can get a grip on self-similarity and kindred matters like turbulence, noise, clustering, and chaos. And Mandelbrot was the prime mover behind it. 

Posted by Robin Varghese at 12:51 PM | Permalink

Comment by doone on May 4, 2013 at 8:05am

A MOST PROFOUND MATH PROBLEM

Alexander Nazaryan in The New Yorker:

On August 6, 2010, a computer scientist named Vinay Deolalikar published a paper with a name as concise as it was audacious: “P ≠ NP.” If Deolalikar was right, he had cut one of mathematics’ most tightly tied Gordian knots. In 2000, the P = NP problem was designated by the Clay Mathematics Institute as one of seven Millennium Problems—“important classic questions that have resisted solution for many years”—only one of which has been solved since. (The Poincaré Conjecture was vanquished in 2003 by the reclusive Russian mathematician Grigory Perelman, who refused the attached million-dollar prize.)

A few of the Clay problems are long-standing head-scratchers. The Reimann hypothesis, for example, made its debut in 1859. By contrast, P versus NP is relatively young, having been introduced by the University of Toronto mathematical theorist Stephen Cook in 1971, in a paper titled “The complexity of theorem-proving procedures,” though it had been touched upon two decades earlier in a letter by Kurt Gödel, whom David Foster Wallace branded “modern math’s absolute Prince of Darkness.” The question inherent in those three letters is a devilish one: Does P (problems that we can easily solve) equal NP (problems that we can easily check)?

Take your e-mail password as an analogy. Its veracity is checked within a nanosecond of your hitting the return key. But for someone to solve your password would probably be a fruitless pursuit, involving a near-infinite number of letter-number permutations—a trial and error lasting centuries upon centuries.

More here.

Posted by S. Abbas Raza at 09:21 AM | Permalink |

Comment by doone on May 2, 2013 at 5:37pm

Math Lawns

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Comment by doone on April 23, 2013 at 6:06pm

THE PYTHAGOREAN TEMPTATION

In his Degrees of Knowledge, Jacques Maritain argues that one central fault of the modern mind has been its propensity to think of mathematics rather than metaphysics as first philosophy. If we take number for the foundation of all things, then we deprive ourselves of the capacity to think of being; we truncate reality, and fail to see the elegant assent the human mind can make from the immediacy of sense experience, by way of abstraction, to the conception of being and, at last, – by the grace of God — to the immediate experience of Being. Too assiduous a delight in the quantitative may conceal our intellectual natures from us, and disfigure our lives. And yet, from Galileo through the contemporary Physicist, number has seemed to be something like the language of God (a phrase sometimes used to describe DNA, but that utterly misunderstands what DNA actually tells us about reality, as if God only spoke living things into being). As someone concerned with the way in which art, and poetry in particular, reveals being to us, the way in which it clarifies the vision and initiates us into a richer way of dwelling in the Real, I have always appreciated the admonitions of Maritain and others who would draw us, with St. Thomas Aquinas, to think being first, last, and always.

more from James Matthew Wilson at Front Porch Republic here.

Posted by Morgan Meis at 08:30 AM | Permalink

Comment by Michel on April 23, 2013 at 10:52am

Fractals give me vertigo.

Yes, music is math in time.

Comment by doone on April 23, 2013 at 9:57am

Yes it is, Michel

Fractal Acid Trip

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Comment by Michel on April 23, 2013 at 9:54am

Music is Math

Comment by doone on April 16, 2013 at 3:54pm

How Adorable

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Comment by Michel on April 16, 2013 at 10:25am

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