Duke University News May 10, 2005

MathDuke University News May 10, 2005

Mathematics since 1978, she has been a major Events participant in Calculus reform, and has been very active in the Women in Science and Engineering Math Department Party (WISE) program.This program o ers academic, nancial and social support to female undergrad- A large group of undergraduate and gradu- uate students in those elds. ate students mixed with the faculty at the annual mathematics department party on April 28. Jordan Ellenberg On the day after classes ended, members of the Duke mathematical community packed the de- In a March 1 talk scheduled by DUMU, Jordan partment lounge to enjoy sandwiches and con- Ellenberg of Princeton University and the Uni- versation in an informal setting. This party is the versity of Wisconsin gave an enjoyable talk relat- traditional occasion for the faculty to honor the ing the card game Set to questions in combinato- graduating students, contest participants and re- rial geometry. A dozen DUMU students enjoyed search students for their hard work and accom- chatting with him at dinner after his talk. plishments. Many math majors received the Set is a simple but addictive card game played 2005 Duke Math shirt, some received certi cates with a special 81-card deck. A standard "folk- and a few took home generous cash prizes. A lore question" among players of this game is: good time was had by all. what is the largest number of cards that can be on the table which do not allow a legal play?He Graduation Luncheon explained how this question, which seems to be At noon, immediately after Graduation Exer- about cards, is actually about geometry over a - cises on Sunday May 15, senior math majors nite eld.He presented several results and ended and their families will meet in the LSRC dining with a number of open mathematical problems room. In a short ceremony after lunch, those arising from this study. whose rst major is mathematics will receive Jordan Ellenberg received his bachelor's and their diplomas. doctorate at Harvard University. He won two Undergrad Math Colloquia gold medals in the International Mathematics Olympiad and was twice a Putnam Fellow.More recently, he has received a Sloan Fellowship and Dusa McDu an NSF-Career grant. On January 31, Dusa McDu , a mathematics His research concerns arithmetic algebraic ge- professor at SUNY Stony Brook, gave a collo- ometry. He writes an occasional column on quium aimed at the level of math majors entitled mathematics for Slate magazine and has pub- "4-dimensional polytopes and Symplectic Topol- lished the novel "The Grasshopper King." ogy".After this talk, several math majors took High School Math Meet her out to a leisurely dinner. Dusa McDu earned her Ph.D.in 1971 at Cam- On Saturday, October 30, the Gross Chemistry bridge University. She then studied with Israel Building was the site of the annual Duke Univer- M. Gel'fand in Moscow, who taught her to view sity High School Mathematics Meet.At double math as a kind of poetry and whose work greatly the attendance of ve years ago, 164 students in uenced her own. from schools as far as 300 miles away came to At the State University of New York at Stony compete for the prizes that were given to the top Brook, where McDu has been Professor of individuals and schools.

1 Duke Math News May 10, 2005

The top two individual winners were Thomas a Princeton graduate student, nished second. Mildorf and Menyoung Lee, both of Thomas Jef- In 2003, Ms. Wood became the rst U.S. woman ferson High School in Alexandria, VA. Their high and only the second woman in the world to be school nished rst in the team competition as named a Lowell Putnam fellow after nishing in well.Third place individual prize went to Arnav the top ve in the competition. Tripathy of East Chapel Hill High School with On Saturday December 4, 2004, a record 3733 fourth place to Je Tang of NCSSM and fth place participants from 515 colleges and universities to John Pardon of Durham Academy. in the United States and Canada spent six hours The meet is an annual event sponsored by the working on 12 challenging math problems. Less Duke Math Union (DUMU), currently presided than half of the participants were credited with over by Duke senior Oaz Nir.Many thanks go to even one point out of 120. him and meet organizers Paul Wrayno, Michelle For completing 7 of the 12 problems, Niki- Hu, Nikifor Bliznashki, Keigo Kawaji, and Jackie for Bliznashki '07 ranked 17th and Oaz Nir '05 Ou for their outstanding contribution to the and Lingren Zhang '08 were named honorable Duke community. mention for nishing among the top two per- cent. Freshmen Kshipra Bhawalkar and Tirasan Duke Summer Workship on Undergraduate Khandhawit were among the top 100 and sopho- Research in Mathematics mores Brandon Levin and James Zou ranked among the next 100. Morgan Brown '07, Abhijit The summer workshop, which will be held May Mehta '06 and Jason Shapiro '06 were also cited 18-27, 2005, is intended for rising seniors who for strong showing. Of the 22 Duke students who are interested in graduate study in applied math- took the competition, 17 ranked among the top ematics. The topic for this year's workshop is third of all participants. "Mathematical models of granular ows." The The Math Department will receive $15,000 for models will describe the dynamics of granular this third place nish. A Duke team has nished material (e.g., sand, powders, grains, rocks, soil, rst, second or third 10 times since 1990 and etc.) in terms of algebraic or di erential equa- has nished among the top ten for all but two tions for the mechanics of colliding particles. of those years. Over the course of the workshop, the organizers will develop models involving ordinary di erential equations for the motion, analyze models to Mathematical Contest in Modeling understand their mathematical and physical im- In an international competition, held Febru- plications, and conduct numerical and physical ary 4 through 7, 2005, a team of three Duke stu- experiments to validate the models. The students won an Outstanding rating in the Math- dents will work on small research projects re- ematical Contest in Modeling (MCM). The con- lated to the topic, and will present their results test is held annually, drawing entries from about near the end of the program. 750 colleges, universities, and high schools from around the world.This was one of two teams in- vited by the Mathematics Association of America Undergraduate News present their results at the summer meetings of the MAA in Albuquerque NM this August. Putnam Competition Success Juniors Adam Chandler, Pradeep Baliga, and Matthew Mian worked all weekend to construct For the fth time in six years, a team of three mathematical models of trac ow on a toll Duke students placed third in the annual William road, and to determine the optimal number of Lowell Putnam mathematical competition. In lanes for a toll plaza. In their 50 page pa- 2000 the Duke team came in rst. MIT won per, "The Booth Tolls for Thee," they describe this year's competition for the second year in a and analyze sophisticated simulations based on row. Princeton, coached by Melanie Wood T'03, queuing theory and cellular automata. After

2 Duke Math News May 10, 2005 subjecting their simulations to rush hour condi- 2005.He plays cello in the Duke symphony or- tions on I-95, they concluded that the number of chestra and has been a Duke tour guide. booths should be approximately 1.65 times the For more information, see www.act.org/ number of travel lanes plus 0.9, so as to mini- goldwater/ mize the time cost to impatient drivers and the Karl Menger Award expense of operating the booths. Four other teams from Duke competed in the Nikifor Bliznashki '07, Oaz Nir '05 and Lingren contest.Three of those teams earned the sec- Zhang '08 have won the 2005 Menger Award for ond highest rating of Meritorious in the MCM: their excellent ranking in the 2004 the Putnam Brandon Levin '07, Matthew Fischer '06, and competition.Each student gets $250 in addition Nikifor Bliznashki '07;Abhijit Mehta '06, Ben- to the $600 for their third place nish. Bliznashki jamin Mickle '06, and Ibraheem Mohammed '07; and Nir received this award in 2004 as well. As James Zou '07, Oaz Nir '05, and Rahul Satija top Putnam scorer in the Southeast region, Bliz- '06.The remaining team competed in the Inter- nashki has been awarded $100 from the South- disciplinary Contest in Modeling (ICM), a new east Section of the Mathematical Association of division in the contest.The team of Vyacheslav America. Zhang, a rst year student from Shang- Kungurtsev '07, Qinzheng Tian '06, and Aaron hai, will receive an additional $250 for winning Wise '08 was Duke's rst to compete in the ICM, the Virginia Tech Math Contest in October 2004 and they earned a rating of Honorable Mention. and Nir will collect $200 for nishing third. Duke has a long record of success in the MCM, Karl Menger (1902-1985), a member of the Vi- which requires teams of three students to solve enna Circle of Logical Positivism that ourished an open-ended problem in applied mathemat- around 1930, was a renowned mathematician ics. Teams have about four days to research the who made signi cant contributions to areas of problem, come up with a solution, and write a logic, philosophy and economics as well as math- paper. Duke teams earned Outstanding ratings ematics. Among his students in Vienna was Kurt in 1998, 1999, 2000, 2001, and 2002, and Meri- G odel. After leaving Europe in 1937, Menger torious ratings in all other years since 1993. taught at Notre Dame and then the Illinois Insti- Goldwater Scholarships tute of Technology until he retired in 1971. For more information see www.math.duke.edu/news/ Physics and math major, Peter Blair, and math awards/menger/index.html and chemistry major, Adam Chandler, have been named 2005 B.M.Goldwater Scholars.The $7500 Julia Dale Prize award is presented annually to about 300 sopho- The 2005 Julia Dale Award for excellence in mores and juniors for outstanding achievement mathematics is shared by Oaz Nir and Mayank in mathematics, science and engineering.Since H. Varia. this program began in 1989, 58 Duke students Nir is an A. B. Duke Scholar, PRUV Fellow, including 25 math majors have been granted this Putnam team member for three years and two- prestigious honor. term president of DUMU. His senior thesis, un- Blair, a Mellon Mays Scholar from Nassau, der the direction of John Harer, is entitled "Me- Bahamas, is working with Professor Arlie Pet- chanical arms and algebraic topology." He will ters studying how light bends as it travels around be attending graduate school in applied mathe- black holes.He is president of the Society of matics at MIT. Physics Students at Duke, mentors in the AGAPE Varia is a NC Math Scholar who has excelled Corner Boys Home and sings in a gospel choir. in his course work at Duke. His senior thesis, Chandler, a PRUV Fellow from Burlington, under the direction of Jonathan Hanke, is enti- NC, is working with Dr.Garrett Mitchener on tled "Explicit computation of the L function of a modelling language changes.In the Mathemat- Kummer surface". He will be attending graduate ics Contests in Modelling, his teams were desig- school in pure mathematics at MIT. nated as Meritorious in 2004 and Outstanding in

3 Duke Math News May 10, 2005

Julia Dale (b. 1893) was a professor of equations, boundary layers, dimensional analy- mathematics at Duke from 1930 until her un- sis, and stability theory. timely death in 1936. Her friends and rel- This course will be co-taught by Professor Tom atives established the Julia Dale Memorial Witelski and Dr. Rachel Levy. The prerequisites Fund in her honor.For more information, see are Math 103, physics (mechanics) and di eren- www.math.duke.edu/news/awards/dale/ tial equations (Math 107/108 or 111 or 131). For more information, see ACES or PRUV Research Fellows www.math.duke.edu/ witelski/196/. This The PRUV Research program, now in its sixth course is partially supported by research grants year, supports undergraduates for six weeks of from the National Science Foundation and will intensive math research under the direction of o er continuing opportunities for summer un- a Duke professor. Each of these students is ex- dergraduate research projects in mathematical pected to write a senior thesis worthy of gradua- analysis and experimental physics. tion with distinction. For more information, see www.math.duke.edu/vigre/pruv/index.html. ACM World Finals The following students have been selected as A team of three Duke students have distin- PRUV Fellows for the summer of 2005. guished themselves in the 2005 ACM World Fi- nals. Garrett Casto '05, Ben Mickle '06 and Matt • Nikifor Bliznashki [with W. Pardon] Edwards '08 traveled to Shanghai in early April to compete with 77 other teams culled in prelim- • Vyacheslav Kungurtsev [with M. Huber] inary rounds from over 1500 teams from 71 coun- • Brandon Levin [with L. Saper] tries. Four of 22 teams from the United States nished among the top 40 in what is claimed Peter Merkx [with H.Bray] • to be the oldest, largest and most prestigious • Ibraheem Mohammed[with A. Petters] programming contest in the world. Duke tied with MIT and Cal Tech and among US universi- • Qinzheng Tian [with T. Witelski] ties, only the University of Illinois ranked strictly higher. Casto will graduate with a major in Gabriel Williams [with A. Petters] • BME and CS. Mickle is a CS and Math major • Yee Lok Wong [with J. Mattingly] who who teamed with other math majors ear- lier this year to gain a Meritorious ranking in the Undergraduate Courses for Fall 2005 Math Contest in Modelling. Freshman Edwards also expects to major in CS and Math.For more This fall, Math 196S (Seminar in Mathemati- information, see http://icpc.baylor.edu/icpc/. cal Modeling) will focus on applications of mathematics to solving models of physical uid dynamics. Graduate Program News This course provides an introduction to the mathematical modeling of modern problems in uid dynamics. Beginning with the derivation of Graduating Ph.D Students the fundamental equations of uid motion, we will construct models for waves and free-surface Robert Buckingham has accepted an Assistant ows, lubrication theory and the shallow water Professorship at the University of Michigan at equations. Techniques for solving these prob- Ann Arbor. His thesis "Long-Time Asymptotics lems analytically and numerically will be devel- of the Nonlinear Schr odinger Equation Shock oped with an emphasis on gaining an insight into Problem" was written under the direction of the underlying physics. We will cover topics in- Stephanos Venakides. cluding hyperbolic and parabolic partial di er- John Cain has accepted an Assistant Professor- ential equations, nonlinear ordinary di erential ship at Virginia Commonwealth Univesity. His

4 Duke Math News May 10, 2005 thesis "Issues in the One-Dimensional Dynam- Math News. He was the author or co-author of ics of a Paced Cardiac Fiber" was written under 38 professional articles in the elds of combina- the direction of David Schae er. torics and ergodic theory, many written jointly Carina Curto has accepted a Post-Doc Position with the late professor Leonard Carlitz. at Rutgers University. Her thesis "Matrix Model After retiring from Duke, he volunteered his Superpotentials and Calabi-Yau Spaces: an ADE time to help elementary school students with Classi cation" was written under the direction of reading and to instruct GED classes. He was in- David Morrison. terested in, talented in, or fond of: Japanese and Andrew Feist wrote his thesis on "Two Prob- Spanish language and culture, the game of Go, lems in Delay Di erential Equations" under the classical music, stamp collecting, photography, direction of Thomas Witelski. sports cars, gardening, astronomy, entomology Daniel Fox has accepted a Visiting Assistant and etymology, computers and robotics, Bon- Professorship at the University of California at sai, cheesecake, butterscotch pudding and maple Irvine. His thesis "Second Order Families of Co- sugar candy. associative 4-Folds" was written under the di- Surviving Scoville are his son Steven and rection of Robert Bryant. daughter Satsuki (from his marriage with Takae Guoqiang Yang wrote his thesis "Quantitative Scoville), their spouses, three grandchildren, a Models for Dorsal closure in Drosophila Em- sister, nieces and nephews, and his long-time bryos" under the direction of Stephanos Ve- companion, Alice Davidson. Friends, neighbors, nakides. relatives and former colleagues gathered to re- Dan Yasaki has accepted a Visiting Assis- member him at a memorial service in January. tant Professorship at the University of Mas- sachusetts. His thesis "On the Existence of Promotions Spines for Q-rank 1 Groups" was written under Leslie Saper will be promoted to professor the direction of Les Saper. and Thomas Witelski to associate professor ef- Wenjun Ying will be a Research Associate at fective this fall.Saper's research involves locally Duke University. His thesis "A Multilevel Adap- symmetric varieties, automorphic forms, L2- tive Approach for Computational Cardiology" cohomology and intersection cohomology, and was written under the direction of John Trangen- geometrical analysis of singularities.Witelski's stein. research centers on use of nonlinear di erential equations to describe the complicated dynamics of uids in various physical situations. Faculty News

Richard Scoville Richard Scoville, associate professor emeritus, died on November 26, 2004, of lung cancer. He had lived for 25,488 days. Scoville was the rst of his family to attend col- lege, receiving a full-tuition scholarship to Yale University. There he earned his bachelors, mas- ters and Ph.D. degrees, all in mathematics. Scoville joined the Duke Mathematics Depart- ment in 1961 and continued as a faculty member until he retired in 1998. He served as director of undergraduate studies from 1992 until 1995. In that role, he edited the rst issues of this Duke

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Solution From Gauss's Lemma, it is su- Problem Corner cient to show that

n n−1 n−2 Solutions from Last Issue n!f(x) = x + nx + n(n − 1)x + ... + n! is irreducible over Z[x]. From Chebyshev's 1. Problem 1. Prove that if 7|a2 + b2, then 7|a theorem, there is a prime number p such and 7|b. that m < p ≤ 2m for every natural number m. If n is even, n = 2k for some natural . Then between and there is a prime Solution One can easily check that a2 gives k k 2k number and , therefore but remainders {0, 1, 2, 4} when divided by 7, q q < n < 2q q|n 2 . If is odd, and take therefore a2 + b2 can be divisible by 7 if and q 6 |n! n n = 2k + 1 again a prime number between and . only if a2 and b2 give remainders 0. Since 7 q k 2k Then Since is even and is prime, it follows that 7|a and 7|b. q ≤ n − 1 < 2q n − 1 2q is even, it follows that n − 1 < 2q − 1 and therefore q < n < 2q. Again, we can 2. Problem 2. Let a, b, m and n be natural conclude that q|n and q2 6 |n!. Now, if we numbers with a and b relatively prime. apply Eisenstein's criteria to n!f(x) for q, we Show that (an − bn)|(am − bm) if and only if obtain that n!f(x) is irreducible in Z[x] and n|m. therefore f(x) is irreducible in Q[x].

Solution First, let us suppose that n|m and New Problems m = nk. Then am − bm = (an)k − (bn)k = n n n k−1 n k−1 which is (a − b )((a ) + ... (b ) ) 1. Problem 1 Find all n for which 10101 . . . 10101 clearly divisible by an − bn. Conversely, | {z } 2n−1digits suppose that (an − bn)|(am − bm). Then is prime. (an − bn)|[(am − bm) − am−n(an − bn)] = (am−nbn − bm) = bn(am−n − bm−n). 2. Problem 2 Let f(x) be a polynomial with in- Since (a, b) = 1 it follows that teger coecients, and let α1, α2,..., and αn (an − bn)|(am−n − bm−n). If we be its complex roots. Prove that if f(a) is do this q times, we obtain that prime for some integer a with |a| > max|αi|+ (an − bn)|(am−nq − bm−nq). From the 1, then f(x) is irreducible in Z[x]. divison theorem for m = nq + s, 0 ≤ s < n 3. Problem 3 Prove that some q, s and therefore (an − bn)|(as − bs). But an−bn n−1 n−2 n−1 a−b = (a + a b + ... + b ) > m µ ¶µ ¶ m µ ¶µ ¶ s s X m n + k X m n (as−1 + as−2b + ... + bs−1) = a −b and = 2k. a−b k m k k therefore as − bs = 0. Since (a, b) = 1 it k=0 k=0 follows that s = 0 and therefore n|m. Submit solutions or suggestions for new problems to Problem Editor Nikifor Bliznashki 3. Problem 3. Show that the polynomial [email protected] x2 x3 xn f(x) = 1 + x + + + ... + 2! 3! n! is irreducible in Q[x] for every natural number n.

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Minors Math Degree Candidates, Academic Year 2004-2005 Juan Pablo Bermudez Stephen Trent Corbin First Majors Christopher Lee Cox Danielle M Davidian John Russel Denton Jennifer Lynn Beall Grant Stephen Gilliam William E Bigner Claire Louise Grandadam Andrew Philip Card Rachel Merritt Heath Stephen Harrison Coit Megan Clare Leftwich Justin Kyle Davis Supanika Leurcharusmee Michael Faber Lauren A Lind Pauline Hong David Ross Martin Thomas William Horn Emily Jayne McDowell Paul George Jameson Michele E Pugh Emily Efrosini Khoury Jonathan Howard Rick David Lawrence Mermin Denise Gayle Rotatori Colin B Middleton Shayla Charelle Sanford Benjamin Andrew Morgan Isaac E Specter Ryan Chase Morton-Wurst Colleen Michelle Torke Oaz Nir Charles Pai Wang Emily Marie Page Jae-Hoon Yoon Lauren Ashleigh Price Jason Aaron Schanker Ashley Allison Siebert Master of Arts Tadena A Simpson Ann-Drea Marie Small Rann Bar-on Lauren Marie Smith Sergei Belov Mihaela Froehlich Second Majors Conrad Hengesbach Jenny Law Christ T Balich Abraham Smith Krupal Pravin Bhatt Dennis Sean Casey Ph.D Nicolette Chao Quan Cheng Robert Buckingham Michael N Economo John Cain Michelle Hu Carina Curto Pooja Nemichand Jain Timothy Deering Ethan Thomas Neil Andrew Feist Merrill J Roller Gregory Firestone Benjamin David Seidman Daniel Fox Mayank Harshad Varia Guoqiang Yang Dan Yasaki Third Major Wenjin Ying

Matthew M Engelhard

7 Duke Math News May 10, 2005

Duke Math News The Duke Math News is published several times a year and is distributed to those in the Duke mathematics community by campus mail. For previous editions and other news, see www.math.duke.edu/news/. We welcome items of interest for our next issue. Send them to [email protected] or [email protected] To read about other news, honors and events concerning mathematics at Duke, visit www. math.duke.edu/news/. The on-line calendar at www.math.duke.edu/mcal lists both regular and special seminars and colloquia for the upcom- ing weeks. The department maintains video archives of talks, lecture series and special con- ferences at Duke, many of which are available, on-line. See www.math.duke.edu/computing/ broadcast.html for more information. |David Kraines, DMN Faculty Sponsor

Problem Corner Editor Nikifor Bliznashki [email protected] Faculty Sponsor David Kraines...... [email protected] Production Manager Sunny Oakley ...... [email protected] Department of Mathematics Box 90320 Durham, NC 27708-0320 http://www.math.duke.edu/math news/