Math News May 10, 2002♦Graduationedition♦

MathDuke University News May 10, 2002 ♦GraduationEdition♦

In 1968, David Schae er received his PhD Events in mathematics at MIT under the direction of Takeshi Kotake. He came to Duke after two Math Department Gala years at Brandeis University and eight years at MIT. His two volume work Singularities and Members of the Duke math community en- Groups in Bifurcation Theory (with Golubitsky joyed plentiful food and conversation in the and Stewart) is considered a classic reference informal atmosphere of the math lounge on in bifurcation theory. He has written more Thursday afternoon, April 18. In addition than 80 other publications and received numer- to a large group of math students and fac- ous honors, including an Alfred P. Sloan fellow- ulty, the party was attended by President Nan ship and the Max Planck Research Award. For Keohane, Provost Peter Lang, Deans Berndt more information, see http://www.math.duke. Mueller, Robert Thompson, Mary Nijhout and edu/applied/NDEMB/index.html Kay Singer. The faculty and administrators rec- ognized the many accomplishments of math majors this year. Undergraduate News Graduation Luncheon Graduating students who major in mathemat- Putnam Team ics and their families are cordially invited to a Duke collected $15000 for its third place n- luncheon in the Levine Science Research Cen- ish in the 62nd William Lowell Putnam Mathe- ter dining room after the Graduation Exercises matical Competition taken on December 1, 2001, on Sunday May 12. In a brief ceremony at the by nearly 3000 students at more than 400 col- end of the luncheon, Director of Undergraduate leges and universities throughout the United Studies, Stephanos Venakides, will present the States and Canada.The team members, sopho- diplomas to those with math as their rst ma- more David Arthur and seniors Nathan Curtis jor. and Kevin Lacker, each received a cash award. Di erential Equations Conference For the second time, Lacker was named Putnam Fellow for placing among the top 5 in the com- A conference in honor of James B. Duke Pro- petition. He received an award of $2500.For her fessor David G. Schae er will be held at Duke top ten individual ranking, junior Melanie Wood from May 20 through May 22, 2002. The con- won the $1000 Elizabeth Lowell Putnam Award ference will bring together a large body of dis- for the highest score among the participating fe- tinguished speakers in a meeting spanning pure males. Arthur also received a cash award for his and applied mathematics, the physical sciences, ranking among the top 25. Curtis and Oaz Nir and engineering disciplines. '05 were named Honorable Mention and Matt During his career, Schae er has made funda- Atwood '03 was in the top 5%.All of the 13 Duke mental contributions to bifurcation theory and students who participated ranked among the top the study of systems of hyperbolic conservation third of these most talented of math undergrad- laws. More recently, he has also turned his fo- uates in North America. cus to mathematical problems in biology and the Since 1990, Duke teams have won the com- nonlinear dynamics of granular materials. petition three times, placed second twice and

1 Duke Math News May 10, 2002♦GraduationEdition♦

third twice. They have ranked in the top This was one of the series of Undergraduate 10 for all but one of the past 10 years.For Mathematics Lectures sponsored and arranged more information, see http://www.math.duke. by members of DUMU. edu/news/awards/competitions.html#putnam. Julia Dale Prize Math Modeling The 2002 Julia Dale Prize for excellence in un- A Duke team has been named Outstanding for dergraduate mathematics has been awarded to the FIFTH year in a row in the math contest in seniors Kevin D. Lacker and Samuel W. Malone. modeling. The team has been invited to present Lacker shared this award for his excellence their results at the 50th annual meeting of the in many advanced mathematics courses and his Society for Industrial and Applied Mathematics success in mathematical contests.In the national in July. Each team member will receive a $300 Putnam competition, Lacker ranked among the award from SIAM. top ten of 3,000 competitors for all four of his In a four day and four night stretch last Febru- years at Duke, twice being named a Putnam Fel- ary, the team of David Arthur, Sam Malone low for nishing among the top ve individual and Oaz Nir wrote a 42 page paper to estimate participants.He will attend graduate school at the optimal level of \overbooking" for airlines. the University of California, Berkeley, on an NSF They incorporated sophisticated probability the- Graduate Fellowship and plans to work in the ory and an "auction" system and supported their area of Arti cial Intelligence. Lacker is a Presi- algorithm with nine pages of computer code. Us- dential Research Fellow and Goldwater Scholar. ing an average value for the price of tickets, num- Malone has been honored for his many re- ber of seats, and \no-show" rate, their algorithm search projects, his unprecedented four wins suggests that by selling 177 seats for a 150 seat in the Mathematics Contest in Modeling and aircraft, airlines would maximize revenues. his many special scholarship awards including The Outstanding papers will be published A. B. Duke Scholar, Goldwater Scholar, Faculty in the journal Undergraduate Mathemat- Scholar and Rhodes Scholar. Malone has pub- ics and its Applications. Malone has been lished six research papers during his four years on an Outstanding MCM team each of at Duke and was one of the rst PRUV Fellows his four years at Duke. See http://www. at Duke. As a Rhodes Scholar, he will study eco- comap.com/undergraduate/contests/mcm/ nomics at Oxford University next year with a fo- 2002Results.html and http://www.math.duke. cus on nance and development. For his senior edu/news/awards/competitions.html#modeling thesis, Black-Scholes Revised: Alternative Price for more information. Processes and Stochastic Volatility, Malone will Graduate with Highest Distinction. Undergraduate Math Lecture Julia Dale joined the Duke mathematics de- Walter Mientka, the former director of the partment in 1930 as an Assistant Professor and USA Mathematics Olympiad program and pro- died of renal failure just six years later at the age fessor at the University of Nebraska, presented of 43. Friends and relatives established the Ju- a lecture entitled Approximations of Arithmetic lia Dale Memorial Fund to honor Duke under- Sums and their Applications to Number Theo- graduates for their mathematical achievements. retic Functions on January 29, 2002, to a large For more information about Dale and for a list group of math majors. of previous winners, see http://www.www.math. Mientka addressed the following ques- duke.edu/news/awards/dale/index.html. tion.How does one nd a useful approximation Alice T. Schafer Prize for the sum of an in nite series whose argu- ments are the natural and real numbers?The At the meetings of the mathematical societies results have implications for the Prime Number in San Diego in January, the Association for Theorem and other questions in number theory. Women in Mathematics (AWM) presented the Twelfth Annual Alice T. Schafer Prize to Melanie

2 Duke Math News May 10, 2002♦GraduationEdition♦

Wood '03. The AWM cited her for several re- Duke Math Meet search projects, including a paper accepted for Over 100 high school students from Georgia to publication in a major mathematics journal, her Virginia participated in the annual Duke Math outstanding success in graduate level mathemat- Meet on Saturday, November 17, 2001. Mem- ics courses starting in her rst year at Duke bers of the Duke University Math Union sent and her excellence in mathematical competi- out invitations, wrote and graded the problems tions. Wood is an A. B. Duke Scholar and a B. and presented the awards to the winners of this M. Goldwater Scholar.She expects to pursue a ARML style team/individual competition. The doctorate in mathematics. winner was the A team from Thomas Je erson This prize for excellence in mathematics by High School of Science and Technology. NC an undergraduate woman was established in School of Science and Math and the B team from honor of Alice T. Schafer, Professor Emerita from TJHSST tied for second place and the Charlotte Wellesley College, who contributed signi cantly Math Club and Chapel Hill HS nished fourth to women in mathematics throughout her ca- and fth among the 18 teams of six. Congratu- reer. The rst Schafer prize was won by Duke lations to DUMU president Mike Miller and to graduate Jeanne Nielson Clelland in 1991. Other David Arthur, Nathan Curtis, Melanie Wood and Duke winners include Jennifer Slimowitz, Hon- the other DUMU members for a job well done! orable Mention, and Sarah Dean '00, Runner Up. For more information, see http://www. Karl Menger award awm-math.org/schaferprize.html. The Menger award is given in recognition of Virginia Tech Math Contest superior performance on the William Lowell Put- nam Mathematical Competition. The winners of Duke students continued their domination of the $250 prize are David Arthur '04, Kevin Lacker the Virginia Tech Regional Math Contest. On '02 and Melanie Wood '03. These students each Saturday November 3, 2001, 269 students from ranked among the top 25 among the 3000 par- 35 schools throughout the southeast, includ- ticipants. This is the fourth Menger award for ing 16 from Duke, competed in this two and Lacker. a half hour contest of mathematical ingenuity Born in Vienna 100 years ago this year, Karl and problem solving ability. For placing second, Menger exerted a strong in uence on many David Arthur '04 won $150. He was followed by elds of mathematics in Europe and Amer- third and fourth place contestants Kevin Lacker ica throughout his long life.He was a member '02 and Oaz Nir '05 who each received $50. Other of the Vienna Circle, a group of philosophers Duke students nishing among the top 15% were and mathematicians that also included Carnap, Matt Atwood '03, Albert Chu '05, Noah Dean G odel, Hahn and Reidemeister. His daugh- '02, David Mermin '04, Mike Miller '03, Lori ter contributed his papers to the Duke library Peacock '04, Adam Siegel '04, Mayank Varia '05, and established the Menger Award in 1990.For and Charles Wang '05. more information, see http://www.math.duke. ACM International edu/news/awards/menger/index.html The Duke team of David Arthur '04, Andrew PRUV Research Fellows Chatham '03 and Kevin Lacker '02 tied for 8th The third class of PRUV fellows will be- among the 64 teams competing at the World Fi- gin their research in mathematics this summer. nals of the 26th annual ACM International Col- Those selected for this program sponsored by legiate Programming Contest held in Honolulu the National Science Foundation are rising se- on March 23, 2002. These teams were selected niors Thomas Bringley, Ben Cook, Marie Guer- from regional competitions involving over 3,000 raty, Meredith Houlton, Steve Nicklas and Dane teams representing 1,300 universities in 70 coun- Voris and rising juniors Eric Fountain and David tries on six continents. See http://cpc.baylor. Marks. Each student will study under the direc- edu/cpc/ for more details. tion of a Duke professor for six weeks in May and

3 Duke Math News May 10, 2002♦GraduationEdition♦

June in areas related to mathematical biology, - nancial mathematics, network theory, game the- New Courses ory and theoretical physics. The following year, they will continue their research with the aim of Math 236/Physics 292 graduation with distinction. General Relativity The members of the rst PRUV class, Michael Colsher, Sam Malone, Carl Miller and Daniel Introduction to the basic concepts and tech- Neill, have each won major scholarships to con- niques of General Relativity.The course will tinue their studies. Members of the second cover tensor calculus, Lorentzian geometry, Ein- class, Matt Atwood, Tom Finley, Carl Pearson, stein's equations, weak eld limits, cosmology, Tristan Tager and Melanie Wood, have each car- black holes, and gravitational lensing.Each stu- ried out signi cant research projects. dent will write a paper selected from a list of topics tailored to the student's mathematics Goldwater Scholars and/or physics background (e.g., Morse inequal- Math/CS major, David Arthur '04 was one of ities and its applications, Lagrangian and Hamil- four Duke students and the only sophomore to tonian methods, cosmic censorship, large-scale be named B. M. Goldwater Scholar for 2002.This structure of the universe, detection of extra- prestigious honor and its $7500 scholarship is solar planets, dark matter reconstruction, grav- given to over 300 students annually for out- itational waves, etc.). standing achievement in mathematics, science |Instructor Arlie Petters and engineering. The total number of Goldwa- ter Scholars from Duke since 1989 is now 49 of whom 20 have been math majors. Nationally Problem Corner about 10% of the Goldwater Scholars and math majors. See http://www.act.org/goldwater/ Solutions From Last Issue 2002pressr.html for more information. 1. Problem. Find all solutions, if any, in positive integers to

Graduate Student News a + b = c where c is the least common multiple of Receiving Ph.D. a and b. David Ambrose, student of J. Thomas Beale, Solution. Consider any solution to the given will be taking a postdoctoral position at the equation. Since a = c − b, where b and Courant Institute. His thesis is titled Well- c are both multiples of a, it follows that posedness of vortex sheets with surface tension. a is also a multiple of b. Similarly, b Anne Collins, student of John Harer and is a multiple of a, and hence a = b. Pankaj Agarwal, will be taking a postdoctoral po- Thus, the least common multiple of a sition at Stanford University. Her thesis is ti- and b is a, so the equation reduces to tled Con guration Spaces in Robotic Manipula- a + a = a, which has no solutions in tion and Motion Planning. positive integers. Marianty Ionel, student of Robert Bryant, ac- 2. Problem. Let i be the square root of −1 and cepted a postdoctoral position at McMaster Uni- let n be a positive integer. Prove that versity in Canada. Her thesis is titled Second or- the polynomial der families of Special Lagrangian 4-folds in C4. f(x) = (x + i)n − (x − i)n has only real roots. Must they all be distinct?

4 Duke Math News May 10, 2002♦GraduationEdition♦

Solution. Suppose r is a repeated root of the New Problems given equation. Then, f(r) = f0(r) = 0 Thus |r + i|n = |r − i|n so |r + i| = |r − i| 1. If the sum of the coecients of polynomials Letting and simplifying, we r = a + bi p(x) and q(x) are both 1, prove that the sum nd . Therefore must be real. b = 0 f of the coecients of p(q(x)) is also 1. Also n(r + i)n−1 = n(r − i)n−1. Since f must be real, r+i and r−i cannot 2. Find all real numbers, x and y, satisfying be 0. Divide the above equations to get 50y 1 (r + i) = 1 (r − i), which is impossible. n n x = 11 − 2 2 It follows that (r+i)n −(r−1)n = 0 has x + y 50x no repeated roots. y = 10 − x2 + y2 3. Problem. Let n be a positive integer. De ne the ordered set of positive integers, S, 3. Putnam 2001.In a triangle of area 1, each of to be quaint if the sum of the elements three lines from the vertices to the opposite in S plus the number of elements in S sides are bisected by another of these lines is n. For example, if n = 6, then the forming an interior triangle. Find the area quaint sets are < 5 >, < 3, 1 >, < 2, 2 >, of this smaller triangle. < 1, 3 >, and < 1, 1, 1 >. Let f(S) denote the product of the elements in S. Com- pute the sum of f(S) as S ranges over all quaint sets. Math Degree Candidates, Solution. Let S = (a1, a2, ..., am) be any ordered set of positive integers. Then, Academic Year 2001-2002 f(S) counts the number of (unordered) sets Tof distinct positive integers satis- First Majors fying: Wendell Bernard Barnwell III has precisely elements. T 2m Alison Rochlin Benton For each positive integer , the k Arman Bhalla th smallest element in is 2k T John Gwyn Campbell . a1 + a2 + ... + ak + k Francesco Nicholas Caruso Noah Barker Dean Thus, if we sum f(S) over all quaint Faith Anne Drickamer sets S, we are counting the number of Jennifer Rose Fiorita unordered sets T of distinct positive Sreelata Kintala integers satisfying: Katharine Nicholette Knight Kevin David Lacker T has an even number of elements. Olga Natalia Lipinska The largest element in T is n. Samuel Westmoreland Malone William Blair Searle It follows that the sum of f(S) as S Matthew George Semrad ranges over all quaint sets is given by Quinn Matthew Snyder n−1 + n−1 + n−1 + ... Taowei David Wang 1 3 5 Frederick Craig Zahrn 1 = (1 + 1)n−1 − (1 − 1)n−1 2 = 2n−2

5 Duke Math News May 10, 2002♦GraduationEdition♦

Second Majors Mark Andrew Ozaki Juliet Rachel Crowder Pulliam Justin Bledin Stacie Harper Rabinowitz Andrew Scott Brohl Dennis William Simon Patricia I-Pearn Chen Michael Joseph Smolow Benjamin Rudolf Crawford David Allen Stuebe Bristol James Crawford Sherwin Leu Su Benjamin Lovegren De Bivort Dana Catherine Upton Sitaramesh Emani David Patrick Williams Marco Antonio Garcia Enrico Arturo Zappi Richard Albert Huang John Christopher Zeok Jared Ian Strauss Klyman Jason Adam Koslofsky Master of Arts Edward James Moulton Neil Kamal Puri David Fredrick Anderson Bradley Williams Renuart Robert John Buckingham Kimberly Fayme Roitman John Wesley Cain Patrick Jay Rosenthal Benjamin Perry Cooke Timothy Douglas Roy Carina Pamela Curto Paola Sardi Daniel Fox Michael Joseph Smolow Daniel Abraham Goldstein Andrew David Tuttle Ryan Patrick Haskett Joel Jonathan Winegarden Robert Laszlo Karp Edith Yunching Yang Timothy Allen Lucas Phillip Mathey Minors Wenjun Ying

Eric Mark Aldrich Ph.D. Nicholas David Atchison Kyle Thomas Beauchamp David Michael Ambrose Christina Lynn Belmore Anne Collins Julie Paige Brown Marianty Lacramioara Ionel Hao Chen Kevin Brian Chen David Cheonho Chung Jonathan Greely Curtis Parima Damrithamanij Faculty Sponsor David Kraines ...... Joshua Rothschild Danziger [email protected] Production Manager John Ballard Davenport Shara M. Player ..... [email protected] Peter Matthew Fishman Christopher Brian Fleizach Department of Mathematics David Scott Frist Box 90320 Charles Jeremy Kelley Durham, NC 27708-0320 James Connor Kelly http://www.math.duke.edu/math news/ Francis Xavier Lane Ryan Maged Mostafa Michael Victor Nehme Kathryn Rita Nowicki