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A monthly survey of math news
4 log 3 - a new cosmic constant? John Baez (UC Riverside) has a "news and views" piece in the February 13 2003 Nature entitled "The Quantum of Area?". We start by asking whether black holes have a discrete spectrum of energy levels. According to Baez, a complete answer would require an understanding of "how quantum mechanics and general relativity fit together -- one of the great unsolved problems in physics." But two completely different ways of guessing have recently come to the same answer: the spectrum of discrete energy levels is related to the surface area of the black hole, and the quantum of surface area is exactly 4 times the natural logarithm of 3 times the Planck area (which itself is about 10-70 m2). The "surface" is actually the event horizon - "the closest distance an object can approach a black hole before being sucked in," so it is an imaginary boundary, but nevertheless acts in many ways "like a flexible membrane," and has a geometry of its own: it is flat except at points where it is punctured by one of the "threads" postulated by loop quantum gravity theory. Recent work by Shahar Hod (Hebrew University), Olaf Dreyer (Penn State; available online) and Lubos Motl (Harvard; available online) relates to earlier research by Hawking, Ashketar and Baez himself.
Ethnomath in the Times Magazine. The context is the release last month of a new math curriculum for New York City schools. The conflict, nothing new, is between the "back-to-basics" advocates of "proved practices like memorization, repetition and the mastery of algorithm" and the advocates of constructivist teaching techniques, which the B2B folks dismiss as "fuzzy math." The report, by Dirk Olin in the February 23 2003 New York Times Magazine, leaves this news item behind and takes us to the debate on "Ethnomathematics." (The name is certainly unfortunate, because it suggests racial mathematics rather than the cultural mathematics which is being discussed.) In this case the Fuzzies do not have a precise position, but take up, with various degrees of rigor, the argument stated by Ubiratan d'Ambrosio, as quoted by Olin: "Mathematics is absolutely integrated with Western civilization, which conquered and dominated the entire world. The only possibilily of building up a planetary civilization depends on restoring the dignty of the losers." Olin also quotes from Marcia Asher (Ihaca College; author of the recent "Mathematics Elsewhere") and Ron Eglash (RPI). As spokesman for an opposing view he chooses David Klein (Cal State Northridge): "But ancient techniques and early discoveries in math will not take students very far who want to do something in the real world with mathematics." A reasonable middle road is taken by Judith Grabiner (Pitzer College): "Putting the math in its cultural context helps teach the mathematics and makes it more meaningful to students, since it has a human context."
Math reform in Russia. The Feburary 13 2003 Nature ran a review by Valery N. Soyfer (George Mason University) of the Russian book with translated title "The Education That We May Lose." The book, "by a group of leading mathematicians," was edited by Victor A. Sadovnichii, the president of Moscow State Universty and a mathematician himself. What is putting education at risk is a wave of reforms being implemented by the Russian Ministry of Education. The high school curriculum is shifting away from mathematics and natural sciences, and towards social sciences, information technology, physical education, etc. As Sadovnichii states: "The virtues of the Russian high school, which the entire world spoke of with real respect, have always depended first of all on basic science ... ." Two interesting points among many brought up in this review: First, the opinion of Igor F. Sharygin, a former high school teacher, on the civic importance of mathematics. (In Soyfer's words) "people who are mathematically literate and understand what proof means cannot easliy be manipulated." Next, the book's devoting nearly 100 pages to "a full translation of the US National Commission on Science Teaching (the John Glenn Commission), along with the text of a programme of educational reforms proposed by US President George W. Bush. According to these documents ... the US leadership is determined to counter the decline in the standard of mathematics and science teaching in US public schools."
Quantum computing in The Economist. "Heads and Tails" is the clever title of a piece in The Economist for January 2 2003. The subtitle, "Practical quantum computers are another step closer," refers to work published that day in Nature by a team at the Institut für Experimentalphysik in Innsbruck led by Stefan Gulde. Their summary begins: "Determining classically whether a coin is fair (head on one side, tail on the other) or fake (heads or tails on both sides) requires an examination of each side. However, the analogous quantum procedure (the Deutsch-Jozsa algorithm) requires just one examination step." The article explains how this quantum algorithm can be implemented "on a quantum processor based on a single trapped [Calcium] ion," essentially a one-atom quantum CPU. According to The Economist, this experiment shows how to raise from 10 to 100 the number of possible entangled qubits in a device. "And, since the power of a quantum computer should rise exponentially with the number of entangled qubits, that tenfold leap would have enormous consequences. Many obstacles remain, but computing's quantum mechanics may one day be more concerned with counting coins than with flipping them." For more about the Deutsch-Jozsa algorithm, see Three Lectures on Quantum Computing by Zdzislaw (Gustav) Meglicki, Indiana University.