MATHEMATICS AND STATISTICS DEPARTMENT

The Department of Mathematics and Statistics continues to be an active and vibrant place where creativity and innovation are abound. The excellence of the faculty continues to be recognized nationally. This year Tom Garrity was awarded the Mathematical Association of America’s Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics. He becomes the fourth member in our department to be given such an honor. The previous winners are Frank Morgan, Colin Adams, and Ed Burger. Tom was also named “Professor of the Year” by College Council here at Williams.

We are very pleased to welcome our new statistician to the faculty. Joining us is Assistant Professor Bernhard Klingenberg who comes from The University of Florida where he received his Ph.D. Also joining us for the year will be Professor Perry Susskind, visiting from Connecticut College, and our own Charlie Stevenson ’93 who will be teaching Precalculus in the fall. We welcome Susan Loepp back from her sabbatical leave and bid a fond farewell to Janine Wittwer who has accepted a position at Westminster College in Salt Lake City. Going on leave this year are Colin Adams, Ollie Beaver (for the fall term), and Tom Garrity. All three plan to pursue their work here in Williamstown.

Due to the efforts of Cesar Silva and Frank Morgan, the National Science Foundation has awarded our department a new five-year grant to fund our renowned “SMALL” summer undergraduate research program. There will be a special session this year at the Mathematical Association of America’s Mathfest summer national meeting in Providence, RI, “On SMALL Mathematics,” organized by Frank Morgan. SMALL alums will reunite and give presentations on their current work.

We are very proud of the accomplishments of our graduating seniors. The Rosenberg Prize for outstanding mathematics senior was awarded to David Jensen ’04 and Aaron Magid ’04; Matt Hoffman ’04 received the Goldberg Prize for best colloquium. Zach Yeskel ’04 was awarded the Morgan Prize for Teaching and/or Applied Math, while Kai Chen ’04 won the Witte Problem Solving Prize. Eric Engler ’04 was applauded for the highest colloquium attendance. Two rising junior majors were honored with the Benedict Prize: Vojislav Sesum ’06 and Ya Xu ’06.

The department is particularly appreciative of the dedication and hard work of the members of the student advisory board, SMASAB (Student Mathematics and Statistics Advisory Board), each of whom were heavily involved in the faculty hiring process, in addition to organizing the department’s advising Ice Cream Socials and Math Snacks. The members of SMASAB were Eric Engler ’04, Matt Hoffman ’04, Neil Hoffman ’04, Kari Lock ’04, Aaron Magid ’04, Charles Baschnagel ’04, Jordan Rodu ’05, and Aaron Pinsky ’06.

Without a doubt, all days spent with mathematics are exciting, but one day this year stood out as especially entertaining: Thursday, March 4, when we celebrated Skewe’s Day. Skewe’s Number, which is approximately equal to 10^10^1034, was the first upper bound for a certain number theoretic quantity involving primes. For some time it was considered to be the largest number used in a piece of serious mathematics (we celebrated it on the date 3/4 in honor of the last exponent 34). The climax of the event was a math and stat version of the $10,000 Pyramid game show involving faculty and students. After a tense runoff, The Dick De Veaux /Neil Hoffman ’04 team were named the winners.

All of the members of the faculty had a busy and productive year. Their individual highlights and achievements are given below.

At the beginning of the summer of 2003, Colin Adams ran a weeklong workshop on knot theory at Wake Forest University for the Mathematical Association of America. He also directed six students in the SMALL summer research program at Williams College. He continued as a member of the Council of the American Mathematics Society, chair of the subcommittee on undergraduate research for the Mathematical Association of America and as humor columnist for the Mathematical Intelligencer. His new book “Why Knot?” was published by Key Curriculum Press in the spring. It presents knot theory in a comic book format, with an attached toy for doing the knot exercises. Over the year, he advised Aaron Magid ’04 on his thesis work on hyperbolic knots in Euclidean 3-manifolds. In the spring, he taught a senior seminar on hyperbolic 3-manifolds, the first time he has done so since coming to Williams. In late spring, he again taught a knot theory workshop for the Ohio section of the MAA.

Professor Ollie Beaver gave invited panel talks on Williams’ Summer Science Program and the Quantitative Skills Program; in September ’03 at the Haverford Conference on Summer Bridge Programs, in October ’03 at the Amherst Workshop on Quantitative Skills Programs, and in April ’04 at the Quantitative Literacy Group conference at Williams. Her paper with Professor Tom Garrity, “A Two-Dimensional Minkowski ?(x) Function,” has been accepted and appears in the Journal of Number Theory. Beaver attended the Joint Mathematics meetings in January at Phoenix. At Williams College, she was the chair of the Winter Study Committee. She continues to teach in the mathematics component of the Summer Science Program.

Professor Edward Burger returned from a yearlong sabbatical at the University of Colorado at Boulder and began his term a Chair of the department. In the fall, he offered a new tutorial version of Math 251, Introduction to Mathematical Proof and Argumentation. He is now writing a text based on this course.

This year Burger advised Kari F. Lock’s number theory research that led to her Senior Honor’s Thesis entitled “Identifying Best Rational Approximations through Sharp Diophantine Inequalities.” His paper, “On Newton’s Method and Rational Approximations to Quadratic Irrationals, dedicated to the memory of Williams Mathematics Professor H. William Oliver, appeared in the Bulletin of the Canadian Mathematical Society. He published Precalculus: A Multimedia Course and Calculus: A Multimedia Course both with Holt, Rinehart and Winston. Together with Michael Starbird from The University of Texas at Austin, he produced the DVD video course, The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, with The Teaching Company. He continues to serve as an Associate Editor of the American Mathematical Monthly.

Professor Burger gave numerous lectures including a Plenary Address at the Conference on Analytic Number Theory and Surrounding Areas, held at the Research Institute of Mathematical Sciences in Kyoto, Japan. He was also an Artist-in-Residence at the Hayground School in New York.

In his second year at Williams, Professor Satyan Devadoss continued to have a blast. In summer 2003, he supervised his team of SMALL summer students in research on coxeter groups. Eric Engler ’04 extended this work as a thesis student under Devadoss.

Devadoss continued research in two areas: his work on particle collisions and real moduli spaces appears in the Notices of the American Mathematical Society. He is also working on computational cartography and mathematical origami, where he will supervise new students in the SMALL program in summer 2004.

Devadoss also taught a new course in spring 2004, funded by NSF/DARPA, on computational geometry. Along with giving talks at Williams, he was invited to speak at conferences in Princeton, Brandeis, and UMass, Amherst.

Professor Richard De Veaux with co-authors Paul Velleman of Cornell and Dave Bock published their introductory Statistics textbook, Stats: Data and Models. This book, published by Addison-Wesley, is intended for a higher introductory level than their Intro Stats book, published last year by the same company. It joins their high school text Stats: Modeling the World, as the third in the series.

De Veaux continued his research and presentations throughout the US and the world on Data Mining, including keynote speeches at the ECAS conferences in San Marco, Italy at the McMaster Advanced Control Consortium at McMaster University, Canada and at the Sixth Annual Winter Workshop: Data Mining, Statistical Learning, and Bioinformatics at the University of Florida. He participated in the DIMACS Working Group on Privacy / Confidentiality of Health Data at Rutgers and was named to the National Academy of Sciences Committee on Assessing Behavioral and Social Science Research on Aging.

De Veaux continued his service on the Data Mining Forum of GlaxoSmithKline and as Associate Editor of Environmental and Ecological Statistics, and Environmetrics.

Professor Thomas Garrity has continued his research in geometry and number theory. His paper “On Relations of Invariants for Vector-valued Forms,” with Z. Grossman (’99) was published in the Electronic Journal of Linear Algebra. His paper “A two-dimensional Minkowski ?(x) function”, with O. Beaver, was published in the Journal of Number Theory. He spoke at SUNY at Potsdam in November and at Sacred Heart University in Fairfield CT in March. In January at the joint AMS-MAA meeting in Phoenix, he received one of this year’s Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics, where he also gave the talk “Functions for the World.” He learned quite a bit from his two theses students, Andrew Marder and Chris Calfee. Finally, at the end of the academic year in May, he was pleased and surprised to find out from College Council that he was chosen this year the “Professor of the Year” at Williams.

Victor E. Hill IV, Thomas T. Read Professor of Mathematics, continued his work on the mathematical theory of change-ringing and revised his lecture-recital “Mathematical Aspects of the Music of Bach,” which he presented for the American Pianists Association in Indianapolis in December and for the Williams Spring Family Weekend in April. He completed his twenty-second year as Archivist of the Association of Anglican Musicians, for which he also serves on the Editorial Board of the Journal and as sole reviewer of recordings.

Professor Stewart Johnson continues his research in dynamical systems, modeling, and optimal control with a focus on optimal periodic switching strategies. He is interested in small rapidly switching cycles that approximate probabilistic behaviors. Professor Johnson is working on establishing the generic existence of such cycles in high dimensions, smooth dependence of performance measures on the size of such cycles, and the types of behavior possible when these cycles degenerate.

Professor Johnson continues as a statistical consultant for research conducted at Neurological Consultants of Bennington by Dr. Keith Edwards, ’69. This research is an ongoing effort to establish the safety and efficacy of Galantamine as treatment for dementia with Lewy bodies.

Professor Johnson remains active in the college wide Quantitative Studies program. The new Quantitative Studies requirement has brought many students into the program who have little interest in science. Professor Johnson is responding by offering a new course to address their quantitative needs in economics, psychological statistics, and life after Williams. The course will focus on real world problems, financial calculations, spreadsheets, and descriptive statistics.

Professor Susan Loepp was on leave during the 2003-2004 academic year and spent that time focusing on her research in commutative algebra. She had a paper published in the Journal of Algebra and she co-authored a paper that appeared in the Rocky Mountain Journal of Mathematics. In addition, her paper co-authored with former thesis student Philippa Charters ’03 was accepted in the Journal of Algebra. Loepp attended the annual Joint Mathematics Meetings in Phoenix Arizona in January. In March, she gave a presentation on Error Correction as part of the Williams College Faculty Lecture Series and she gave an invited talk in April at SUNY Brockport.

Loepp continued advising the research of undergraduates. During the 2003-2004 academic year, she advised the senior honor thesis of David Jensen ’04 and continued advising the research of Kai Chen ’04. In addition, the paper “Constructing Almost Excellent Unique Factorization Domains,” written by Loepp’s SMALL 2001 commutative algebra group, John Bryk, Sonja Mapes, Charles Samuels and Grace Wang, was accepted for publication in Communications in Algebra.

Professor Frank Morgan published two papers on minimal surfaces and foams and has ten others in the works. He is writing two texts for use in Williams courses on Real Analysis and Applied Real Analysis. He appeared in New Scientist and as a CarTalk Puzzler winner.

He advised two honors thesis students. Neil Hoffman worked on double bubbles in spherical, hyperbolic, and Gaussian spaces. Jonathan Lovett, continuing work from Morgan’s senior seminar on Riemannian geometry and general relativity, studied rotations in nonEuclidean norms.

Morgan has given some thirty talks this year, including special graduate schools in Padova, Italy, and in Paris, France.

He is directing the “SMALL” undergraduate research project and organizing a SMALL reunion at the MathFest in Providence this summer. His SMALL Geometry Group, including Vojislav Sesum ’06 and Ya Xu ’06, is studying bubbles in Gaussian space. In July, the group travels to Paris, where Morgan is teaching a graduate school on geometric measure theory and isoperimetric problems.

Students from his previous Clay Research Institute graduate school published a paper on double bubbles in a three-dimensional torus, with his all-time favorite illustration. The students commissioned for him a stained glass window of the illustration, in which every pane portrays the appearance of a different kind of double bubble.

Professor Allison Pacelli came to Williams in the fall of 2003. She taught a new 300-level course in the spring, called Polynomial Arithmetic, and received a grant from the interdisciplinary program at the college to develop a new course for non-majors, Mathematics and Politics: Voting, Power, and Conflict. Her paper on “Abelian Subgroups of Any Order in Class Groups of Global Function Fields” appeared in the Journal of Number Theory; in addition, her paper “Subgroups of the Class Groups of Global Function Fields: The Inert Case,” a joint work with Yoonjin Lee at Smith College, was accepted for publication in the Proceedings of the American Mathematical Society. As well as giving various talks at Williams, she spoke at the Five College Number Theory Seminar, Queens University, the College of the Holy Cross, the Canadian Number Theory Conference VIII, and the Number Theory Conference in Honor of Harold Stark.

Professor Pacelli co-organized the Green Chicken Contest at Middlebury College this year with Professor Tapp. Twenty Williams students participated, and succeeding in bringing home the Chicken. She also brought students to the Hudson River Undergraduate Mathematics Conference, at Mt. Holyoke College, and will organize HRUMC at Williams next year.

Pacelli was selected as a fellow for Project NExT (New Experiences in Teaching), a professional development program run by the Mathematical Association of American for recent Ph.D.s. She attended Mathfest 2003 in Boulder, CO, and she organized panel discussions for other NExT fellows at the Joint Meetings in Phoenix and Mathfest 2004 in Providence on the topics of writing textbooks and doing research with undergraduates.

During the summers of 2003 and 2004, Professor Pacelli taught her intensive three-week course for high school students, Number Theory and the Art of Mathematical Proof, at Brown University. She is currently acting as the mentor for the Intel Science Talent Search for one of her students in that program. They are doing research on polygonal numbers.

Professor Cesar Silva taught a new tutorial in spring ’04. The tutorial was addressed to first year and sophomore students and is an introduction to mathematical reasoning using linear algebra as an example.

Silva continued his research in ergodic theory and published three papers, one of them with his former thesis student Darren Creutz ’03. He also revised a paper with John Bryk ’02 that is to appear in the American Mathematical Monthly.

Before the fall started, he wrote a National Science Foundation grant proposal with Frank Morgan to fund our summer undergraduate research program. The grant was awarded in the spring and we expect it will support our program for five years.

He hosted his colleague Sasha Danilenko, from the Institute of Low Temperature Physics in Kharkov, Ukraine, for six weeks in the fall. This visit was made possible by Collaboration in Basic Science and Engineering (COBASE) grant, from the National Academies. They have already published one paper on their research and have several others in progress.

In spring ’04, he was awarded a grant from the Mellon Foundation, together with Robert Benedeto at Amherst and Mike Keane at Wesleyan, to organize a conference on p-adic dynamics. He also was a reviewer for Mathematical Reviews and a referee for several journals.

Professor Kristopher Tapp attended an August conference in Muenster, Germany on “Curvature and Global Shape,” where he spoke about his research on positively curved manifolds, and remained an extra week to work with a collaborator. After returning to Williams, he initiated collaborations with two other colleagues by inviting them to speak in the math seminar, funding their trips through a newly received NSF grant. This summer, he will collaborate with four undergraduates, together investigating questions about the curvature of Lie groups, and will finish writing an undergraduate textbook about Lie groups, based on an advanced course he taught in the fall. He spoke at a Geometry conference in Manhattan, and attended several other conferences. His most recent paper, “Quasi-positive curvature on homogeneous bundles” was accepted for publication in the Journal of Differential Geometry.