To meet increased interest in the many facets of mathematics, we have instituted a new unified major, including special recommendations to students interested in applied mathematics or other sciences, engineering, graduate school in mathematics, statistics and actuarial science, and teaching. In some very good news for our broadening program, we have hired a statistician, Richard De Veaux of Princeton University. The Daily Princetonian ran a major story on the loss of one of their best teachers to Williams. In addition, visiting next year will be Faan Tone Liu of Colorado (analysis) and Robert Manning of Cornell (applied mathematics).
We greeted the new Williams president, Harry Payne, and his wife Deborah with a dinner with Mathematics faculty and students at Professor Frank Morgan's house January 5, during Payne's first week in office. In April Harry and Deborah Payne returned the favor with a dinner for faculty and seniors in the President's House. We are additionally pleased to have professional contact with Deborah Payne who has taken charge as Director of the Science and Math Resource Center.
Our one-man PR committee, Professor Edward Burger, gave two more sensational parents/family weekend talks, which transfixed audiences of hundreds which packed Bronfman auditorium: "Predicting the Future in an Uncertain World: Do We Really Need that Crystal Ball" and "Why I Hate Mathematics But Love Museums."
During this past year Professor Thomas Garrity received tenure, Professor Colin Adams published his Knot Book, and Professor Frank Morgan, Chair, was elected to the Council of the American Mathematical Society. Adams also served as Chair of the Faculty Steering Committee. Next year Professors Cesar Silva and Ollie Beaver will serve as acting chair in the fall and in the spring.
We have enjoyed the presence of three lively research associates: Jerry Bope of Williamstown, Frank Jones of Rice University, and Steven Rudich of Carnegie Mellon.
Marissa Barschdorf continues her amazing service as department secretary.
This year prizes and awards were announced at our spring majors dinner at the Faculty House. The Rosenberg prize for outstanding senior went to Margaret Donavan. The Morgan prize for accomplishment and promise in applied mathematics went to John Alberg and the Morgan prize for accomplishment and promise in mathematics teaching went to William O'Brien. The Goldberg prize for the best senior colloquium went to Derek Bruneau. The Benedict Prize for outstanding sophomore went to Jonathan Todd. Two seniors received honors: John Alberg and Margaret Donavan.
Our student advisory committee, SMERSH, has been instrumental in all our activities. Warm thanks go to Jenny Cotner '94, Teon Edwards '96, Bill O'Brien '94, Emily Shahan '95, Chris Welch '94 and Ted Welsh '95.
Our SMALL student summer research project is taking a sabbatical in this, its seventh year. SMALL plans to return in full force next year.
Last summer (1993) our SMALL student summer research project, under the direction of Frank Morgan, and Colin Adams, included a NSF-funded physics component led by Stuart Crampton and William Wootters.
Altogether, eight faculty supervised twenty-three students in research in Dynamics, Geometry, Knot Theory, Mathematical Modeling, Minimal Surfaces, Number Theory, Quantum Physics and Symmetry. Student research groups, with student leaders, discussed problems, proved new theorems, gave talks at Williams and elsewhere, and submitted research papers for publication. (See the 1992- 1993 Report of Science at Williams for a listing of participants.)
Professor Colin Adams directed the SMALL project at Williams in 1993, in addition to directing a group of six students working on knot theory as part of the program. He was a co- organizer of the Regional Geometry Institute at Smith College in July and gave talks there entitled "Bus Tours of the Universe and Beyond", and "The High School Teacher's Knot Workshop".
At the end of August, he gave a Knot Theory Workshop for high school math teachers in Springfield, MA. He also spoke on "Tunnels in Hyperbolic 3-Manifolds" at the University of Quebec at Montreal and at the Group Theory and Topology Conference at SUNY-Albany, both in October. He gave a talk entitled "Mel Slugbate's Real Estate in Hyperbolic Space" at Siena College in November and at Vassar College in February.
He wrote a paper entitled "Tilings of Space by Knotted Tiles" which shows that space can be filled with identical tiles, all of which are knotted in the shape of any given knot. It has been accepted for publication in the Mathematical Intelligencer. He submitted an article entitled "Tunnels in Hyperbolic 3-Manifolds" for publication. His first book, entitled The Knot Book was published by W. H. Freeman and Co., in March of 1994. It gives an elementary introduction to the mathematical theory of knots.
At Williams, he chaired the Steering Committee. He was a councilor of the Mathematics Section of the Council on Undergraduate Research. He served on the both the Allendorfer Committee and on the Lecturer's Board of the Mathematical Association of America.
Professor Ollie Beaver finished her third and last year as Williams' Gaudino Scholar. In this capacity she has this year guided and mentored several "experiential learning" Williams student projects including: an internship in a hospital for mentally disturbed adolescents; one in a Washington D public defender's office by mathematics major Heather Morton '94; an ecological expedition to the Bulgarian Black Sea; a video project, "Images of Pluralism; an introspective exploration of Community in Ghana"; an investigation by mathematics major Tarun Ramadorai '96 of how wildlife sanctuaries in India displace village communities; and Roosevelt High School WSP teaching internships for several students including mathematics majors Shawn McDougal '95 and Krystal Williams '96.
Continuing as coordinator of the Mellon Minority Undergraduate Fellows Program, Professor Beaver guided Mellon Fellows in projects mentored by Williams faculty. In the spring four new Mellon Fellows were added. Current Mellon Fellows include mathematics majors Shawn McDougal '95 and Krystal Williams '96 who have been mentored by Professors Colin Adams, Bill Lenhart and Frank Morgan. Under the auspices of a grant to Williams College from the Andrew W. Mellon Foundation, the Mellon Fellowships support research projects by Williams students who are bound for doctoral studies in the humanities, anthropology, astronomy, ecology, geology, mathematics or physics.
Professor Beaver continues as Director of the Summer Science Program, an intensive five-week program for a select group of incoming minority Williams pre-freshmen who have expressed an interest in the sciences. (See Williams College Summer Science Program elsewhere in this Report).
This January Beaver attended the Meetings of the American Statistical Association in Atlanta, Georgia, both to satisfy a budding interest in statistics as well as to meet with candidates for the statistics position. The Meetings were of intrinsic interest, but were rendered even more so, statistically speaking, when Beaver, the only mathematician attending, won the $20 door prize awarded at the gala evening gathering.
Professor Deborah Bergstrand continued her work in the Dean's Office as Dean of first-year students. While the department has greatly missed her, we take pride in knowing of the excellent and dedicated job that she does in Hopkins Hall.
Professor Edward Burger gave the following lectures at Williams College: at the Mathematics Faculty Seminar in September he spoke on "Odd Measures of Irrationality That are Even Effective". In October he gave a Special Freshman Parents' Weekend Mathematics Lecture to about 300 parents and students: "Predicting the Future in an Uncertain World: Do We Really Need That Crystal Ball??" In March he was invited to give a Philosophy Club Lecture and gave a talk entitled "Do Mathematical Truths Exist Independently of Human Cognition?" Finally in April he gave a Special Spring Family Weekend Mathematics Lecture to an audience of over 330 people: "Why I Hate Mathematics But Love Museums".
This past summer, Burger participated in the SMALL Undergraduate Research Program where he advised the Number Theory Group. The group investigated the structure of the continued fraction expansions of certain classes of quadratic irrationals and generalized some results occurring in the literature. His group is currently writing up a paper for publication. Also during the summer, Burger offered a special Elderhostel course at Williams entitled "Discovering Beauty and Truth: A Creative Journey Through Mathematics". Burger then developed a new senior seminar course: MATH 417: Algebraic Number Theory which was offered this past fall. Burger also debuted a new Winter Study course in January: MATH 015: The Comedy Writing Workshop. Here students were creative and inventive in a slightly less traditional academic pursuit. Students wrote and edited original comedy material including monologues, short stories and skits. The class produced the very successful special comedy show: "The Williams Winter Comedy Hour" in late January to a crowd of over 200. The course also spawned a new comedy troupe on campus named Blue Tortilla.
Professor Burger gave a number of lectures outside of Williams College. In June he was invited to deliver a lecture in the Séminaire de Théorie des Nombres at the University of Caen in France. There he gave a lecture entitled "Transcendence and Irrationality Measures". He then attended the Colloquium on Elementary and Analytic Number Theory held by the János Bolyai Mathematical Society in Hungary. There he gave a talk entitled "On a theorem of Erdös regarding sums of Liouville numbers" and then was invited to speak at the University of Debrecen in Hungary where he lectured "On Siegel's Lemma and a Question of Schinzel". When he returned to the U.S. in July, he was a guest on the News at Noon on WNYT-TV (NBC) in Albany where he discussed recent progress on Fermat's Last Theorem. In October he was interviewed about mathematics on WMRA- FM, Virginia National Public Radio affiliate.
Burger was named a Visiting Scholar at James Madison University and in October gave a University Lecture entitled "Art, Beauty and Mathematics: Finding the Sexiest Rectangle". He also gave a colloquium lecture for the mathematics department at JMU. In November he was the invited breakfast speaker at the Conference for the Association of Teachers of Mathematics in New England held in Springfield. There he presented "Magic with Mathematics: Is the Equation Faster than the Eye?" In March he gave two lectures at Michigan Technological University: "How to Divide and Conquer the Real Numbers" and then "Does S 1/n! Really Converge? An introduction to p-adic analysis". In April at the first Hudson River Undergraduate Mathematics Conference at Siena College, he gave a lecture entitled "Continued Fractions: Uncovering the Hidden Structure of Numbers". Also in April he was the main speaker at the M.A.A. Connecticut Regional Dinner Meeting held at Connecticut College where he gave his lecture on "How to Divide and Conquer the Real Numbers". While at Connecticut he also gave a mathematics colloquium entitled "A Rational Approach to Transcendental Numbers".
His article, "On Mahler's Compound Bodies" appeared in the Journal of the Australian Mathematical Society . His article, "On Frequency Distributions of Partial Quotients of U- Numbers", co-authored with T. Struppeck appeared in Mathematika. His paper "Sur les Quotients Partiels de U- Nombres Dans Un Corps de Séries Formelles" co-authored with E. Dubois was recently accepted for publication in Comptes Rendus de l'Acádémie des Sciences Paris.
Next year Professor Burger will be on a sabbatical leave and will hold Visiting Assistant Professorship positions at the University of Texas at Austin and the University of Colorado at Boulder. He will also be a Visiting Member of the Center for Number Theory Research in Sydney, Australia.
David Dailey from Williams' Center for Computing joined the department this year as a part time visiting instructor to teach the introductory level statistics course, MATH 140. During this time he continued his research in graph theory and combinatorics, publishing an article "On the Graphical Containment of Discrete Metric Spaces" to appear in the journal, Discrete Mathematics. He has also continued to explore algorithmic approaches and statistical applications of this class of graph theoretic problems, and has resumed some research on topics of non associative binary operations.
Professor Thomas Garrity continued his collaboration with Professor Robert Mizner on the interplay between algebraic geometry, differential geometry and partial differential equations, primarily concentrating on examples from CR structures. They wrote the paper "The Equivalence Problem for CR Structures of Codimension 3 or Greater". Their earlier paper "Invariants of Vector- Valued Bilinear and Sesquilinear Forms" has been accepted for publication in Linear Algebra and Its Applications.
Victor E. Hill IV, Thomas T. Read Professor of Mathematics, continued his research in history of mathematics, giving talks ranging from the high school level to faculty research seminars on "story problems" and on the development of geometrical reasoning. He also continued his career as a concert organist and harpsichordist, both on campus and on tour. In November he played the recital rededicating the newly-rebuilt Roosevelt organ at the First Congregational Church in Great Barrington; on this program he played the premiere of "Fanfare on the Old Hundredth" by Paul Hamill, composed for the occasion and dedicated to Professor Hill. For 1994-95 he has been appointed to a 13th term on the Board of the Association of Anglican Musicians.
Professor Stewart Johnson continued his research in applied dynamical systems. His main work has been in the area of Hybrid Systems, an emerging subfield of control theory. His recent paper "Basic Hybrid Systems" will appear in the International Journal of Bifurcations and Chaos. He is completing a joint project with John Guckenheimer that lays a theoretical foundation and creates basic tools for analyzing and working with hybrid systems. He is looking forward to a sabbatical leave in the 1994-95 academic year and plans to start the year by attending the special Control Theory Conference of the Society of Industrial and Applied Mathematicians and being involved in the Hybrid Systems conference at the Mathematical Sciences Institute at Cornell. His recent popular paper in mathematical biology "Sex Ratio and Population Stability" has appeared in OIKOS.
Johnson supervised two honors theses this year conducted by John Alberg '94 and Margaret Donavan '94. He continues his work developing the applied curriculum and is drafting a differential equations textbook based on his approach in MATH 210: Differential Equations and Vector Calculus.
Professor Robert Mizner continued his collaboration with Professor Thomas Garrity on algebraic theory and the geometry of CR structures. Together, they wrote a paper on "The Equivalence Problem for CR Structures of Codimension 3 or Greater". In the fall, Mizner spoke on aspects of this work at the Special Session on Differential Geometry at the meeting of the AMS held in Syracuse. He also spoke to students at Middlebury College on the representation theory of the symmetric group. This spring, Mizner completed an additional paper: "An Elementary Introduction to Some Advanced Topics in Differential Geometry".
Professor Frank Morgan has continued his research on minimal surfaces, including applications to the structure of materials. He has published 4 papers and has 12 other projects in the works, including the second edition of his Geometric Measure Theory book and a new text Calculus in One Semester. He has given a number of talks, including a series of lectures as distinguished foreign lecturer at the First Summer Research Institute of the Mathematical Society of Japan, a series of lectures on geometric measure theory at the International Center for Mathematics Research in Trento, Italy, the Taft lectures at the University of Cincinnati, a talk to the Williams Alumni Club of Boston, and about 20 other talks to groups ranging from research conferences to high school teachers and students. He served on the panel of lecturers for the Council on Undergraduate Research.
Last summer the SMALL undergraduate research Geometry Group, advised by Morgan, mathematically predicted the shape of double salt crystals; later electron microscope photographs resembled their predictions. The work led to a joint paper by Morgan and two of the students, Christopher French '95 and Scott Greenleaf (Bates '94), Morgan's first joint paper with undergraduates. Scott continued the work in a successful honors thesis at Bates, for which Morgan was called in as the outside examiner. Morgan now has two new undergraduate research students: Joseph Masters on a Bronfman grant and Shawn McDougal on a Mellon grant.
During January, Morgan taught a Winter Study course Math for the World, in which students conceived and implemented ways to promote mathematics. They ran a math booth with prizes at Williams and at the local Stop and Shop supermarket, held a successful Math Bowl team competition, and wrote a letter to the Today program (still unanswered).
Morgan helped organize an undergraduate mathematics conference in Albany on April 9, 1994, at which over 300 faculty and students participated as equals. There were some 70 talks in eight concurrent sessions, two-thirds by undergraduates. Professor John Conway of Princeton gave the invited address.
Morgan served a term as chair of the Williams ad hoc faculty committee to advise concerned students on Latino issues, which participated in the very successful hiring of two new US-Latino studies faculty.
Last fall Morgan was elected to the Council of the American Mathematical Society.
This coming year, on sabbatical, Morgan plans to spend time at Bonn, Germany; Vitoria, Brasil; Queens College, New York City; Melbourne, Australia; and Trento, Italy.
Professor Cesar Silva gave the following talks: "On Quotients of Nonsingular Actions and Self-Joinings", at SUNY-Albany on November 5, 1993, and "Nonsingular K-automorphisms", at the Ergodic Theory Seminar at Northeastern University in Boston on May 24, 1994. He was a workshop discussion leader at the Science Engineering Academic Support Network, organized by the New England Board of Education in Boston on October 2, 1993.
Silva continued his research in nonsingular ergodic theory and dynamical systems and had the following paper accepted for publication: "A Skew Product Entropy for Nonsingular Transformations", with P. Thieullen in the Journal of the London Mathematical Society.
Silva also taught a tutorial on Fractals, Chaos and Dynamical Systems and a Winter Study course titled What is a Beautiful Proof?
Professor Dave Witte took advantage of a sabbatical this year to concentrate on his research. He completed three research papers, "Tessellations of Solvmanifolds", "Products of Similar Matrices", and "Superrigidity of Lattices in Solvable Groups". In the fall, he spent a month divided between the Universities of Minnesota and Wisconsin and, in the spring, he spent a month at each of Stanford University and the University of Chicago. He gave talks at the schools he visited and at Williams. He also attended national meetings of the American Mathematical Society in Vancouver and Cincinnati, and a national meeting of the American Association for the Advancement of Science in San Francisco.