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New Directions at the IAS

The Institute for Advanced Study (IAS), with its methods of equilibrium statistical mechanics serene, sloping lawns and dignified buildings, is do not work, and one needs to develop new ap- a shrine to intellectual achievement. Some uni- proaches. versities—like the one across town in Prince- The previous year the Institute hosted a re- ton—might rival the Institute in the distinction search program that examined properties of of their faculties, but for sheer focus on higher semiconductors and lubricants, substances that intellectual pursuits the IAS wins out. The In- have some fluidity but are not well described by stitute has no degree programs, no courses, and fluid mechanics. The main tools are kinetic equa- very little bureaucracy to distract its thinkers tions, the best known among them being the from their thoughts. Long known as the ulti- Boltzmann equations. “Kinetic theory is an area mate ivory tower, the Institute, under the di- where American are not so ac- rectorship of Phillip Griffiths, is applying its tive, but there is a lot of activity in Europe,” high standards to a broader range of goals. Spencer notes, “so we had a lot of participants Nowhere is this more evident than in the pro- from Europe.” Although the researchers worked grams of the IAS School of and in on some problems coming from industrial ap- the IAS/Park City Mathematics Institute, for plications, the thrust was nevertheless theoret- which the IAS served as the host site this sum- ical. The work contrasts with that of, say, the In- mer. stitute for Mathematics and its Applications at the University of Minnesota, where there are di- Applications a New Focus rect contacts with people from industry and the For many years the the School of Mathematics work is much closer to the specifics of given ap- at the Institute was known primarily as a center plications. for research in pure mathematics, especially Another major push in applied directions, areas such as representation theory and num- headed by IAS faculty member Enrico Bombieri, ber theory. More recently the school has broad- has centered on combinatorics and theoretical ened its agenda to include what Griffiths calls computer science. Many of the activities in this “theoretical applied mathematics”. The shift area have been organized in conjunction with began about ten years ago, when the Institute DIMACS, the NSF-sponsored center on discrete hired and Thomas Spencer. Caf- mathematics and computer science which is a farelli is an expert in partial differential equa- consortium of Rutgers and Princeton Universi- tions and is well known for his work on free ties and AT&T Research, Bellcore, and Lucent boundary problems and fluids (he recently Technologies. This coming year there will be a moved to Courant Institute). Spencer, a math- program on those areas of discrete mathemat- ematical physicst specializing in statistical me- ics that are close to statistical mechanics, and chanics and random media, has recently become efforts will be made to connect people who work interested in the statistical description of solu- in combinatorics with those working in math- tions to partial differential equations. For the ematical physics. past few years the Alfred P. Sloan Foundation has Robert MacPherson, who joined the IAS math- provided crucial finanacial support to support ematics faculty two years ago, has become in- the Institute’s initiatives in several aspects of ap- terested in some problems in theoretical com- plied mathematics. puter science and combinatorics since his This past year, the research program focused contacts with the researchers in these areas. In on wave (or weak) turbulence, such as one might fact, last year he solved a combinatorics con- find on the surface of a calm ocean. The prob- jecture posed by Gil Kalai, who was visiting the lem “sounds simple, but it’s not,” Spencer notes. IAS from the Hebrew University of Jerusalem, While there are relations to the problem of full and they ran a seminar on the solution. “Some turbulence and the usual descriptions given by would worry if this would fit into the Institute the Navier-Stokes equations, the mathematics is because we’re very theoretical,” says MacPher- very different. In particular, the conventional son. “But these people have talked to others in

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almost every area of mathematics about the teachers to the IAS, but, according to MacPher- problems that are growing out of computer sci- son, the PCMI constitutes the largest organized ence.” contingent of high school teachers ever brought This year the IAS is launching an ambitious, to the Institute. three-year program in mathematical physics, The PCMI grew out of the Regional headed by IAS faculty members , Institute that started six years ago in Park City, from the School of Mathematics, and Edward with funding from the NSF. This is the third Witten, from the School of Natural Sciences. The year that the IAS has served as institutional goal of the program is to familiarize math- sponsor, and the first year that the PCMI has met ematicians with the latest ideas in theoretical in Princeton rather than in Park City. The PCMI physics, such as string theory and mirror sym- brings together high school teachers, under- metry. Over the years Witten’s intuition about graduates, graduate students, and researchers, physics has allowed him to come up with bold with parallel sessions for each group and in- conjectures that mathematicians seized upon formal discussions and social events for every- to prove some spectacular theorems. The most one. The unifying theme of the program is the recent example came in late 1994, when Witten spirit of research and discovery. For example, and his colleague Nathan Seiberg of Rutgers Uni- Gregory Lawler of and Emily versity proposed that certain equations con- Puckette of Occidental College, who served as in- tained much of the information of Donaldson structors in the PCMI program for undergradu- theory. Now known as the Seiberg-Witten equa- ates, say they covered material that is off the tions, they have sparked a revolution in low-di- beaten track of usual undergraduate courses in mensional topology. probability and that is closer to the frontier of The IAS program will not try to mine the research. The courses combined lectures and Seiberg-Witten equations for more mathemati- informal group discussions with the students. cal theorems. Rather, the program’s focus will In computer laboratory sessions, students got be on developing mathematicians’ knowledge to explore some open problems in random walks. of physics so that they too can tap into physi- The idea of integrating research and educa- cal intuition and eventually gain insight into tion has become a hot topic, particularly with the mathematical problems. “Mathematicians will NSF. In fact NSF Director Neal Lane has made it learn about path integrals, renormalization, a theme of his tenure. Lane visited the Institute gauge theory—some physics, basically,” Spencer during the PCMI and was the keynote speaker explains. “What they can get out of it is a new at a special event there entitled “A Celebration way of thinking about mathematics—and it won’t of Mathematics: Teachers and Researchers Work- be a rigorous way of thinking.” For this reason, ing Toward Excellence in Mathematics Educa- the participants will be people “who will not in- tion”. In an interview, Lane said that the idea of sist on proving theorems at every step,” says integrating research and education is based on Spencer. MacPherson calls it “unlike anything the assumption students benefit by learning in anyone has ever tried before.” an environment in which research is going on. This less rigorous approach has been the “In the classroom is a person who in other hours cause of some controversy in the mathematical of the day is probing fundamental, difficult, community. In the article “Theoretical Math- challenging questions” in science or mathemat- ematics”, which appeared in the Bulletin of the ics, he said. “This creates a certain attitude to- AMS in 1993, Arthur Jaffe and Frank Quinn laid ward inquiry and discovery, a frame of mind, a out the concerns that many have to taking such different take on the world. It’s a combination a nonrigorous approach in mathematics re- of enthusiasm, openness, and creative thought search. In setting up the new program, the In- processes.” Sometimes professors can bring into stitute seems to be taking to heart the views of the classroom things they have been doing in Sir Michael Atiyah, who, in a rebuttal to the Jaffe- their research. Said Lane, “There’s an element of Quinn article, advocated a “more buccaneering romance about it.” style” in mathematical research. In higher education, such integration can hap- pen fairly easily. The PCMI tries to siphon some Linking Research and Education of the excitement off to the high schools by The late , who was a professor linking teachers and researchers. But do the var- at the IAS from 1952 until his retirement in ious groups at the PCMI really mix? According 1977, worked hard on problems of mathemat- to John Polking of Rice University, who serves ics education for the last twenty years of his life. as convener of the PCMI Steering Committee, the Following in Whitney’s tradition, the IAS this mixing of the groups is “the overriding pur- summer played host to thirty-seven high school pose” of the program. “Everyone in mathemat- teachers as part of the IAS/Park City Mathematics ics has their own job, things they are good at, Institute (PCMI). Whitney occasionally brought things they have to do,” he explained, so it is not

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possible that all of the groups will be interested he was clearly enthusiastic: “This is such a win- in the same things. However, “they gain mutual ner. How could anybody not like this?” In the respect because they are all doing what they do class, the teachers enjoyed the discussion even as well as possible.” A random sampling of PCMI when a few giggles indicated that the presenta- participants report that the amount of mixing is tion was sailing over some heads. Questions variable. There was plenty of interaction be- about the mathematics mingled easily with sug- tween the graduate students and researchers gestions about how to use models to demon- and some between the graduate students and the strate notions of curvature in the classroom.1 undergraduates, but less between the high school The program also includes sessions that focus teachers and the other groups. more exclusively on classroom practice; these are One reason the teachers are more isolated is led by Naomi Fisher of the University of Illinois that their program is more structured than the at Chicago and Cindy Hays of McCallum High programs for the other groups, so they become School in Austin, Texas. As Fisher points out, a more cohesive clan. Groups of teachers are these sessions are very different from the usual drawn from specific geographic sites and at- “in-service” program for training teachers in the tend the program for two consecutive summers. use of a new kind of teaching module or tech- Each site has a director, usually a nique. Indeed, much of what they do in the PCMI in a local university or college. During the aca- program is not immediately transferrable to the demic year, the teachers meet as a group and high school classroom. “What we are aiming for with site directors. By contrast, the topic for the is to encourage teachers to take a fresh look at PCMI research program, and hence the partici- familiar mathematics and standard high school pants, changes each year; this year the topic was topics in order to consider ways to rearrange probability and next year it will be symplectic their own curricula, and introduce new ways to geometry. The graduate and undergraduate pro- present standard topics to make their teaching grams follow suit and therefore attract a different more coherent and absorbing for their students,” group of students each year. Fisher explained. In particular, they try to get the In addition, the teachers’ activities in the PCMI teachers to make connections between separate provide a sharper contrast with their day-to-day parts of the mathematics curriculum, especially work than those of the other groups. For many between geometry and other areas. She observes of the teachers, the PCMI is an oasis from the that some of the teachers make slow, evolu- pressures of teaching high school. Jo Vaccaro, a tionary changes shifts, while others make bold, teacher at Stafford High School in Houston and radical changes. Said Fisher, “What is exciting to a participant in the Rice University PCMI Site, said me is to see teachers gain the confidence to that at least twice during the school year am- make these decisions for themselves and carry bulances visited her school to rescue students them out successfully.” who had overdosed on drugs or attempted sui- cide, and several other students had suicidal Clashing Views tendencies or severe behavioral problems. “Many One thing that comes out in some of the PCMI of the students have terrible home lives,” she activities is that often the researchers and the noted. “Many have little motivation to put a lot teachers have very different views on high school of effort into their schoolwork.” For the teach- mathematics education. In a lunchtime conver- ers, “It is sometimes difficult to keep ourselves sation with Elton Pei Hsu of Northwestern Uni- motivated.” Programs like the PCMI “help me versity, who led a graduate course at the PCMI, recharge my enthusiasm and refocus my love for Michael Cranston of the University of Rochester, teaching,” she said. who was in the research program, and under- What the PCMI teachers’ program tries to do graduate instructors Gregory Lawler and Emily is get the teachers thinking deeply about and ex- Puckette, there was general agreement that it is ploring mathematics. Polking taught a course on essential that high school students have a solid geometry which included such things as Girard’s grasp of basic techniques such as factoring poly- theorem about the sum of the angles of a spher- nomials. There was also considerable wariness ical triangle. After the teachers had experimented of student dependence on calculators and sym- with drawing triangles on inflatable beach balls, pathy for the notion that calculators should not they took a crack at discovering the formula in be allowed on exams. (And it’s easy to see why: Girard’s theorem, which says that the sum of the Lawler said that he had seen on an e-mail dis- 2 angles is π+ (area of triangle)/r . As Polking was cussion group the comment that now that one planning to discuss the generalization to the has calculators there is no longer a need for Gauss-Bonnet theorem (in which the triangle is such formulas as sin 2x = 2 sin x cos x.) a geodesic triangle on a more general surface, 2 and the term (area of triangle)/r is replaced by 1Notes from this class may be found at the web site the integral of the curvature over the triangle), http://math.rice.edu/~pcmi/sphere.

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Such views make many teachers shake their heads and cluck their tongues. Having tasted a richer and more exciting menu of mathematics than the one they were accustomed to, they are skeptical of views that smack of “back to basics”. To those wary of the trend toward nontradi- tional approaches to teaching mathematics, Shirley Hill has this to say: “We certainly didn’t do it awfully well the other way.” Hill is a retired mathematics professor from the University of Missouri at Kansas City who serves on the PCMI Oversight Board. If the goal is to produce a few top-notch mathematicians, perhaps no changes are needed in the teaching of mathematics, she said. “But I think you’ve got to look at a broader population that’s got to understand a lot more mathematics. And now you have to ask, Would the past programs produce that? I don’t think so. What’s happening now may not be fully evolved, but we don’t have a great record of suc- cess to fall back on.” Tim Giesbrecht is a teacher in Franklin Junior High School in Pocatello who is a member of the Idaho State University PCMI site. He said that in order to capture the attention of his students he has to present mathematics in such a way that it relates to their lives in some fashion. Visual- ization helps a great deal, he said. For example, he worked with his students on the mathemat- ics behind a simple harmonic oscillator. Being able to model the motion on a graphing calcu- lator helped to spark the students’ interest. Gies- brecht compared students’ interest in math- ematics to their interest in cars. “It’s difficult to get students interested in the details of learn- ing how to fix cars,” he said. “What they want to do is drive cars. In the same way, we want to show them how to ‘drive’ mathematics, not ‘fix’ it.” For teachers to be able to take that kind of ex- ploratory approach, they need to be confident about their mathematical knowledge—and to be comfortable knowing that they don’t have all the answers. Cindy Hays from McCallum High School in Austin, Texas, was on a panel, held in con- junction with “A Celebration of Mathematics”. which focused on connecting mathematics re- searchers and teachers. One of the best things about the PCMI, Hays noted, was that it “got me in touch with how fun it is not to know the an- swers.” In talking to the research mathematicians attending the PCMI, she found that they don’t know all the answers either. “I have much more freedom to talk about different topics with the students,” she declared. “I won’t necessarily un- derstand all of hyperbolic geometry or spheri- cal geometry, but I can give my kids a glimpse of the magic and the power of mathematics with- out knowing all the answers.”

—Allyn Jackson

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