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August 25, 2006

Faces of mathematics

The Millennium Prize Problems, edited by J. Carlson, A. Jaffe, and A. Wiles. Clay Mathematics Institute and American Mathematical Society, 2006. Link to MnCat Record
In sketching the history of work on the Millenium Prize Problems, the editors made the interesting choice to include more than a hundred photos of the mathematicians involved. Great pictures and commendable variety: different pictures for those who are mentioned multiple times because of their work on more than one problem, such as John Tate (Birch--Swinnerton-Dyer Conjecture and Hodge Conjecture) and André Weil (Hodge Conjecture and Riemann Hypothesis). But they had to repeat the portrait of Euler a few times. . .

August 23, 2006

What is 'Straight'? How Can We Draw a Straight Line?

Two Cornell professors use history to develop students' understanding of geometry concepts, as in this example:

"When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are a fixed distance from a center. Or we can say we use Euclid's definition of a circle. So, now what about drawing a straight line: Is there a tool (serving the role of a compass) that will draw a straight line? One could say: We can use a straightedge for constructing a straight line. Well, how do you know that your straightedge is straight? How can you check that something is straight? What does 'straight' mean? . . .
"An exact straight-line linkage in a plane was not known until 1864-1871 when a French army officer, Charles Nicolas Peaucellier (1832-1913), and a Russian graduate student, Lipmann I. Lipkin (1851-1875), independently developed a linkage that draws an exact straight line. . . .The drawing in Figure 7 depicts the working parts of the Peaucellier-Lipkin linkage. The linkage works because the point P is inverted to the point Q through a circle with center at C and a radius squared equal to s2 - d2. The point P is constrained to move on a circle that has center at D and that passes through C and thus Q must move along a straight line. If the distance between C and D is changed to g (not equal to f), then Q instead of moving along a straight line will move along an arc of a circle of radius (s2 - d2)f/(g2 - f2). This allows one to draw an arc of a large circle without using its center. Note that the definition of a straight line used here is: 'A straight line is a circle of infinite radius.' For a learning module on these topics, see []."

--from Daina Taimina and David W. Henderson, "How to Use History to Clarify Common Confusions in Geometry," in From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, MAA Notes #68 (2005): 57-73. Link to MnCat Record

August 3, 2006

"Letters to a Young Mathematician"

Couching them in the form of letters is a little awkward, but Ian Stewart's observations on mathematics and advice on becoming a mathematician are insightful and entertainingly presented. Samples:
"What is mathematics? In despair, some have proposed the definition 'Mathematics is what mathematicians do.' And what are mathematicians? 'People who do mathematics.' This argument is almost Platonic in its perfect circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who sees opportunities for doing business when others might miss them. A mathematician is someone who sees opportunities for doing mathematics. I'm pretty sure that's right, and it pins down an important difference between mathematicians and everybody else." (p. 32)
"When you study any subject, the rate at which you can understand new material tends to accelerate the more you already know. You've learned the rules of the game, you've gotten good and playing it, so learning the next level is easier. At least it would be, except that at higher levels you set yourself higher standards. Math is like that. To perhaps an extreme degree, it builds new concepts on top of old ones. If math were a building, it would resemble a pyramid erected upside down. Built on a narrow base, the structure would tower into the clouds, each floor larger than the one below. The taller the building becomes, the more space there is to build more. That's perhaps a little too simple a description. There would be funny little excrescences protruding all over the place, twisting and turning; decorations like minarets and domes and gargoyles; stairways and secret passageways unexpectedly connecting distant rooms; diving boards suspended over dizzying voids. But the inverted pyramid would dominate. All subjects are like that to some extent, but their pyramids do not widen so rapidly, and new buildings are often put up beside existing ones. These subjects resemble cities, and if you don't like the building you are in, you can always move to another one and start afresh. Mathematics is all one thing, and moving house is not an option." (pp. 38-39)
--Stewart, Ian. Letters to a Young Mathematician. New York: Basic Books, 2006.
Link to MnCat Record

Finite Element Exterior Calculus

IMA Director Doug Arnold co-authored a chapter in the latest Acta Numerica:
"In this paper we survey and develop the finite element exterior calculus, a new theoretical approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. . . Thus the main theme of the paper is the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them, and their implications and applications in numerical PDEs."
Douglas N. Arnold, Richard S. Falk and Ragnar Winther. "Finite element exterior calculus, homological techniques, and applications." Acta Numerica (2006), pp. 1–155.