Duke Math News - April, 1997


Duke Wins the Putnam!

For the second time in four years, a team of three Duke students, Andrew O. Dittmer '99, Robert R. Schneck '97 and Noam M. Shazeer '98, won the William Lowell Putnam Mathematical Competition. Dittmer was also a member of Duke's first winning team in 1993. Princeton placed second in this competition while Harvard, which won 10 of the past 12 events, dropped to third.

In addition to the three team members, thirteen other Duke undergraduates participated in this competition. Johanna Miller '99 ranked among the top 3%, Chris Beasley '99, John Hyde '99 and Jeff Mermin '00 scored in the top 10% and Sarah Dean '00 and James Harrington '98 finished in the top quarter.

As with the 1993 money, the $7500 prize for the first place team will enable the math department to send majors to various math conferences and to continue the Undergraduate Math Lecture series. For their individual performances, Shazeer will receive $1000, Dittmer $750 and Schneck will take home $500. During the December 7, 1996, contest, 2407 students from over 300 schools in the United States and Canada labored for six hours on 12 challenging mathematical problems in the 57th edition of this longest running academic team competition. Although the maximum possible score on the 1996 competition was 120, less than half of the participants earned more than 2 points and less than a quarter correctly solved even one of the 12 problems. A sample of some of the problems appeared in the January 1997 Duke Math News. The geometric solution to one of these problems is now featured on the new Duke Math T-shirts and printed in the Problem Corner below together with another of the Putnam problems.

Wear the Putnam!

While they might not be quite as popular as the Duke NCAA tournament shirts of previous years, the Duke Math Putnam Champs shirt, with is distinctive geometrical design, will give you the opportunity to talk about the one national championship that a team of Duke undergraduates did win this year. You may purchase these shirts in room Physics 121 for $8. The purchase price includes the solution to the Putnam problem that inspired the geometric design.

Meritorious Modelling Team

This year's team for the Mathematical Contest in Modeling spent a gruelling yet entertaining weekend in the basement of the West Duke building, formulating their solution to the Velociraptor Problem. The team, consisting of Robert Schneck, Steve Wolfman, and Garrett Mitchener, earned a ranking of Meritorious, placing in the top 15% of all papers submitted.

The two problems for this year were to find a way to fairly schedule comittee meetings, and to discover the optimum hunting strategy for velociraptors, a species of predator dinosaurs. Unlike most math competitions, these problems have no single right answer. Solutions, which take the form of written papers, are judged based on creativity, clarity, and completeness.


Schneck Wins Churchill

Robert R. Schneck '97 was one of about 10 students from the United States to be selected to attend Cambridge University on an all-expense paid Winston Churchill Foundation Scholarship. Schneck, a North Carolina Mathematics Scholar, has captured many other prizes and awards during his four years at Duke. Prominent among these are his selection as Duke Faculty Scholar, two time winner of the Karl Menger award and co-winner of the Julia Dale prize for excellence in mathematics.

See http://www.dukenews.duke.edu/Dialogue/dialogue.328 for more information.

Two Students Earn Goldwater Scholarships

Christopher Beasley '99 and Johanna Miller '99 were named Barry M. Goldwater Scholars this year. These scholarships, worth up to $7500 per year, were given to 290 students who have demonstrated outstanding ability in mathematics, natural science or engineering. Since 1989, when the program was founded, 29 Duke University students, including 12 math majors, have won this distinguished award. For more information, see http://www.act.org/goldwater/

Other Awards

Julia Dale Prize Winners

The winners of the 1997 Julia Dale Prize for excellence in undergraduate mathematics are Robert Schneck '97 and Tung Tran '97. These students have taken graduate level courses in mathematics, performed at a high level in mathematics competitions, and provided leadership for the Duke University Mathematics Union. Both Mr. Schneck and Mr. Tran have presented research papers at national mathematical society meetings. In addition, both students are completing papers to vie for Latin Honors by honors project: Mr. Schneck's paper is titled ``Set Theory and Cardinal Arithmetic''; Mr. Tran's Paper is titled ``Counting Independent Subsets in Nearly-Regular Graphs.''

The Julia Dale Prize ($250 for each student) will be awarded at the Math Department ``Celebration'' in the Math Lounge on Thursday, April 24 at 5:00 p.m.

Julia Dale was an Assistant Professor of Mathematics at Duke University who died early in her career on January 13, 1936. Friends, relatives, and colleagues of Professor Dale established the Julia Dale Memorial Fund in 1938. It was the first memorial fund established in honor of a woman member of the Duke University faculty. The fund provides income to support the Julia Dale Prize.

Menger Award Winners

The Karl Menger Award is given in recognition of superior performance in the William Lowell Putnam Mathematical Competition. This year's awards go to the members of Duke's First-Prize-Winning Putnam team, Andrew Dittmer '99, Noam Shazeer '98, and Robert Schneck '97. The Menger Award ($250 for each student) will be given at the Math Department ``Celebration'' in the Math Lounge on Thursday, April 24 at 5:00 p.m.

Karl Menger was a well-known twentieth century mathematician who held academic positions in Europe and the United States and was widely published. The Karl Menger Award was established by a gift to Duke University from George and Eva Menger-Hammond. Karl Menger was Eva Menger-Hammond's father.

Math Department Celebration

On Thursday, April 24, there will be a Math Department ``Celebration'' honoring the Class of '97. The event, which will be held at 5:00 p.m. in the Math Department Lounge, will feature a picnic-style buffet and an awards ceremony.

Scholarship winners, participants in the Virginia Tech Mathematics Contest, The Mathematical Contest in Modeling, and the Putnam Competition will be recognized. In addition, the Julia Dale Prize and the Menger Award will be conferred.

Please join this celebration. For some students, this will be the last opportunity to meet and talk informally with math faculty and friends; and for some faculty and staff, this will be a last opportunity to wish graduating students farewell.

All math faculty, graduate students, and staff, and all mathematics majors and minors are invited to this special end-of-school-year event.


Four Students Go to REU's

A record high of four math majors will be conducting research at universities around the country. These Research Experiences for Undergraduate programs are sponsored by the National Science Foundation. Students receive a stipend and living expenses of $2000 to $3000 for the six to eight week program. Johanna Miller '98 will attend the University of Minnesota in Duluth, the longest running such program. Last summer, Miller attended the REU program at Hope College in Michigan. She delivered a paper on her research at the American Mathematical Society convention in San Diego in January. Sarah Dean '00 will research mathematical aspects of medical imaging at the University of Kentucky in Lexington, Robert McNerney '99 will study problems in differential geometry and topology at Tulane University in New Orleans, and Spencer Shepard '00 will investigate inverse problems and circuit theory at the University of Washington in Seattle.

Tran and Schneck Earn Honors

Undergraduate Robert R. Schneck is currently pursuing an honors thesis with the help of Dr. Richard E. Hodel. His thesis concerns set theory and in particular cardinal arithmetic, with applications to topology and other fields. Much of the work centers on what one can prove with respect to the continuum hypothesis. The cardinality or ``size'' of the set of natural numbers is denoted , and is the cardinality of its power set and of the real numbers. The next largest cardinal number is , and the continuum hypothesis states that . Godel and Cohen have proven that if the normal axioms of set theory are consistent, it is impossible to prove or disprove the continuum hypothesis.

Tung Tran is likewise working on his honors thesis in the area of set theory and how to place objects so that no two are ``close'' to each other. A subset of vertices in a graph is independent if no two vertices are connected by an edge. For grids, it turns out that the number of such placements grows exponentially with the size of the graph. This can be proven with techniques from linear algebra, and estimated with Markov Chain Monte Carlo methods. Tran's research includes graphs for which the same method applies, and others for which it does not.

Robert and Tung will be presenting part of their work on Wednesday, April 23 as follows:

3:30 p.m.
120 Physics
Robert Schneck presents ``Singular Cardinals and the Generalized Continuum Hypothesis''
4:10 p.m.
135 Pyhsics
4:30 p.m.
120 Physics
Tung Tran presents ``Counting Independent Subsets in Nearly-Regular Graphs''

All are invited to attend.

In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.

-=-Hugo Rossi


Fermat's Last Theorem

by Chad Schoen

David Jones, who received an undergraduate degree from Duke in mathematics and is now completing his graduate work with Professor Ribet at UC Berkeley, will give a talk in the graduate seminar on Friday, April 25 at 4 p.m. on Fermat's Last Theorem. The talk should be accessible to first year graduate students and to undergraduates with similar preparation in algebra and number theory.

Undergraduate Lecture Series

by Robert Schneck and Tung Tran

On March 25, Lisa Fauci of the mathematics department at Tulane University came to Duke as part of the Undergraduate Mathematics Lecture series. Her talk was entitled ``Modelling Biofilm Processes in a Moving Fluid.'' It concerned her recent work in computational modelling of bacterial ``clumping'' in moving fluids, which has applications to E.P.A. efforts to use bacteria in pollution cleanup.

The Undergraduate Lecture Series concluded on April 9 with a presentation of ``Solved and Unsolved Problems in Grade School Math.'' After acknowledging the nation's top Putnam team, major expositor of modern mathematics Barry Cipra stumped them with questions second graders have worked on (although not solved). His problems stem from what grade school students do every day: fold paper, play games, and fill in boxes with numbers so that everything adds up right. Cipra's questions demonstrated the subtle details of mathematics, and the difference between intuition and understanding.

Department News

Faculty Changes for Next Year

Professor John Harer, chair of the department, will be on leave next year in Italy. Professor Bill Pardon will serve as acting chair of mathematics for the year. Also, Professors David Morrison and Xin Zhou will return from leave at the Institute for Advanced Study in Princeton. Professor Johnathan Wahl, currently of the University of North Carolina, will assume the editorship of the Duke Math Journal.

Four new faces will be added to the department next fall. Lori Carmack and Radu Constantienscu will join the faculty as assistant research professors, and Paul Aspinwall and Ronen Plesser have been hired jointly by the math and physics departments.

Math Modeling Seminar on Self-Similarity and Nonlinearity

In the fall semester, Mathematics 196S, Seminar in Mathematical Modeling, will be organized around the topic of ``Self similarity and scaling in nonlinear systems.'' The course will be taught by Professor Andrea Bertozzi. The course will provide an introduction to the use of scaling, self-similarity, and dimensional analysis to understand complex physical processes and to interpret results obtained from engineering and scientific experiments and numerical calculations. The mathematical tools that will be used in the course will include similarity variables and separation of variables, fractal sets, dynamical systems, transition to chaos, and scientific computing. The prerequisites for the course are Math 103 and 104, and at least one course in differential equations.

Students will work on individual projects in lieu of homework and exams. All students will be required to complete a written report, in an expository research style, as well as an oral presentation. In addition to the mathematical emphasis, the course will provide experience in researching original sources in the scientific literature, scientific computing, technical writing and presentation.

While students will be required to write expository papers, the subject matter will allow for students interested in research to have the opportunity to get involved in a project stemming from this course.

More information is available on the department's web-site at http://www.math.duke.edu/ faculty/bertozzi/Math196s.html.

Math Degree Candidates, Academic Year 1996-97

First Majors

Mr. Peter Dunn Ahlin* Mr. Evan Scott Bauman* Mr. Christopher Scott Blackburn* Ms. Jaclyn Bova* Mr. Murray Clifton Briggs* Mr. Neil Patrick Brown* Mr. Chad Michael Cannon* Mr. Arnaldo Nelson Cavazos III* Ms. Darla Annette Defrance* Ms. Christy Lynn Dover* Ms. Tamara Lavette Gordon* Ms. Kirsten Susanna Green* Ms. Whitney Whelan Greene* Mr. Jason Ries Helbraun* Mr. Lee Ryan Insinga* Mr. Vinish Jain* Ms. Linda Maureen Kane* Mr. Paul Wookjong Kim* Mr. Min Ce Li* Ms. Bridget Catherine Lloyd* Mr. Xavier Ignacia Marquez* Mr. Wilburn Brent McLamb* Ms. Shannon Leigh Meeks* Mr. Lenny Raphael Monsour* Ms. Misty Rose Morris* Ms. Mary Elizabeth Morrison* Ms. Stacy Ella Nelson* Ms. Stephanie Adams Newhall* Ms. Elizabeth Jean Osterwise* Mr. Gordon Lee Rashman III* Ms. Marnie Tara Russotto* Mr. Robert Richard Schneck* Mr. Joseph Leon Sparks* Ms. Karin K. Stoskopf* Ms. Dawn Michelle Strickland* Mr. Tung Thanh Tran* Mr. Anthony Paul Volpe

Second Majors

Mr. Kazuhito Adachi* Mr. John Hardy Vannort Gieske* Mr. Sheldon Franklin Hall III* Mr. Derek Anthony Jerina* Mr. Christopher Wallace Loveland* Mr. Jason Vincent Morse* Mr. Benjamin Jon Nieporent* Mr. Scott Francis Powell* Mr. Jeremy Thomas Salmon* Mr. Jason Asher Smith* Mr. Jason Wayne Smith* Mr. Jason George Zeibel


Mr. Ali Zain Banatwala* Ms. Irem Kizilviranlioglu* Ms. Hamra Ogelman* Mr. Jason Lee Sender* Mr. Andrey Iossifovich Starobin

Master of Arts

Mr. Aaron Christopher Wesley Ashih* Mr. James Blaine Ott* Mr. Andrew Rhea Schretter* Mr. Edward Wilder Welsh


Ms. Kayne Marie Arthurs* Mr. Po-Jen Cheng* Ms. Anne-Marie Filip

We arrive at truth, not by reason only, but also by the heart.

-=-Blaise Pascal

Problem Corner

Solutions from Last Issue

He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.


New Problems

Faculty Sponsor - David Kraines, dkrain@math.duke.edu
Student Editor - Garrett Mitchener, garrett@math.duke.edu

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