Apparently all events in the math department occur in twos (two new faculty members, two tenures, two new Full Professors, two honors). So, not surprisingly, we welcome back two professors from sabbatical leaves: Colin Adams and Tom Garrity. Professor Adams comes back as the new Chair of the math department, following Professor Ollie Beaver. While there are three faculty members going on sabbatical leave 1997-98, Ollie Beaver, Dick De Veaux and Frank Morgan, the last on the list (Frank Morgan) will be away for two years.
Mathematics majors had an excellent year. This year's sophomore pre-registration numbers came in with forty incoming math majors! We welcome them all. Continuing and graduating majors have distinguished themselves with honors and awards. Senior Jason Schweinsberg was awarded a National Science Foundation Fellowship for his graduate studies in Statistics at the University of California, Berkeley. Jason was also valedictorian of the Class of 1997, and he received the Rosenberg prize for outstanding senior at the annual spring math majors dinner. At the same time, the Morgan prize for accomplishment and promise in teaching was awarded to Eric Watson and the Goldberg prize for the best senior colloquium went to Sandeep Bhatt. In addition, the Benedict prize for outstanding sophomore was given to Alex Wolfe, and the recipient of the David Witte Problem Solving Prize was Alex Woo.
This year SMERSH (Students of Mathematics Enjoying Research and Scientific Happiness), so-monikered student advisory board for over ten years, chose to rename itself. The new, more serious acronym (as fits the organization) is SMAB, referring to Student Mathematics Advisory Board. Times and passions change. SMAB was spared the "fortunes" of helping interview job applicants this year, but kept busy apprising the department of student opinions, running the very successful Fall and Spring Ice Cream Socials and generally being available for giving advice and help. Thanks go to Dan Anello, Eric Furstenberg, Linden Minnick, Eric Watson, Aaron Weinberg and Francisca Winston.
Of course, very little could have been accomplished this year without the glue that holds the department together, Marissa Barschdorf. As the departmental secretary, Marissa is simply super! Patiently and cheerfully she gets us through deadlines, crises and the daily big or little details that crop up. Faculty and student alike, we are all very glad and grateful that Marissa is in the department.
All of the faculty had a busy and productive year. Highlights of the year's faculty activities are:
Colin Adams was on sabbatical for the 1996-97 academic year. He spent the fall as a member of the Mathematical Sciences Research Institute in Berkeley, California, which was having a special year devoted to low dimensional topology. He spent the spring working on his research at Williams College.
In the summer of 1996, Colin Adams directed the SMALL Undergraduate Mathematics Research Project. He also directed a group of three students working in the project on research in knot theory. At the end of the summer, he and Prof. Edward Burger presented a mathematical play at the American Mathematical Society Mathematical Association of America Joint Meeting in Seattle on August 10-12. They were asked to repeat the performance for the 1997 Summer Meeting in Atlanta, GA. In addition, they also gave a presentation entitled "Math as Performance Art: How NOT to Excite Students about Mathematics," followed by a series of six workshops on Improv Teaching.
Over the last academic year, Prof. Adams gave a variety of talks, including "Bus Tours of the Universe and Beyond" for the Mathematical Association of America Public Lectures, in Kansas City, "Mel Slugbate's Real Estate in Hyperbolic Space," for the University of California-Berkeley MSRI/Evans Hall Lecture, "Introduction to Hyperbolic 3-Manifolds" and "Waist Size for Knots," at the University of Southern California, "Unknotting Tunnels in Hyperbolic 3-Manifolds," at the University of California-Berkeley, "Mel Slugbate's Real Estate in Hyperbolic Space" and "Waist Size for Knots," at the University of California-Santa Barbara, "Waist Size for Knots," at the Mathematical Sciences Research Institute, Berkeley, CA, "Mel Slugbate's Real Estate in Hyperbolic Space," at California State University-Fresno, "Mel Slugbate's Real Estate in Hyperbolic Space," for the Mathematical Sciences Research Institute Teachers' Conversations Series, "Teaching as Performance Art: How NOT to Excite Students," with Edward Burger, at the LaCEPT Teachers Conference, Louisiana State University, Baton Rouge, LA, "Mel Slugbate's Real Estate in Hyperbolic Space" and "Waist Size for Knots," at the University of Texas, Austin, TX, "Waist Size for Knots," at Columbia University, New York, NY, "Mel Slugbate's Real Estate in Hyperbolic Space," at Fairfield University, Fairfield, CT, "Mel Slugbate's Real Estate in Hyperbolic Space," at Hudson Valley Community College, Troy, NY, " Short Geodesics in Hyperbolic Knot and Link Complements," at the Hudson River Undergraduate Mathematics Conference, Williams College, "Mel Slugbate's Real Estate in Hyperbolic Space," at Colby College, Waterville, ME, "Waist Size for Knots and Hyperbolic 3-Manifolds," at the University of Maine, Orono, ME, , "Mel Slugbate's Real Estate in Hyperbolic Space," at Morehead State University, Morehead, KY, "Systoles of Surfaces and Hyperbolic Knot and Link Complements," at the Catskills Taconic Topology Day, Vassar College, Poughkeepsie, NY and "Waist Sizes for Knots and Links" at the Wisconsin Ph.D. Centennial Conference, Madison, WI.
Professor Adams published the following papers: "Splitting Versus Unlinking," C. Adams, Journal of Knot Theory and its Ramifications, Vol. 5, No. 3 (1996) 295-299, "Unknotting Tunnels for Two-Bridge Knots and Links," C. Adams and A. Reid, Commentarii Mathematici Helvetici 71 (1996) 617-627, and "Knotted Tilings," C. Adams, in The Mathematics of Long Range Aperiodic Order, ed. by R. Moody, Kluwer Academic Pub. (1997) 1-8.
Professor Adams served as editor for a special issue of the mathematics journal Chaos, Solutions and Fractals that is devoted to knot theory and its applications. He also wrote the following papers, which are being submitted for publication: "Isoperimetric Curves on Hyperbolic Surfaces, C. Adams and F. Morgan, "Maximal Cusps, Collars and Systoles for Hyperbolic Surfaces," C. Adams, "Systoles for Hyperbolic Knot and Link Complements," C. Adams and A. Reid, "Waist Size for Cusps in Hyperbolic 3-Manifolds," C. Adams, and "Why Hyperbolic Space Will Never Go Metric," C. Adams. The expository paper "Exploring Knots," by C. Adams, and students E. Furstenberg (Williams), J. Li (Michigan), and J. Schneider (St. Johns) has been accepted for the Activities section of Mathematics Teacher Magazine. The three students also produced a paper entitled "Stick Knots" that will appear in the special issue of Chaos, Solitons and Fractals.
Professor Ollie Beaver finished two and a half years as Chair of the department. At the same time, she closed another chapter by stepping down from her post of ten years as the Director of the Summer Science Program. Each summer, under Professor Beaver's direction, the SSP has brought to campus 12-14 Williams pre-first years who come from minority groups that have traditionally been under-represented in the sciences. The participants are students who have shown promise in science studies and have declared interest in pursuing a science major at Williams; the purpose of the Program is to nurture that interest. The Program has enjoyed marked success in retention of participants in the sciences, as is evidenced by the numbers of SSP "alumni" who major in the sciences, who write science theses and who enter science professions after graduating from Williams. Professor Beaver will continue to teach in the Program. She also continues on the Advisory Board of the Multicultural Center, and was invited to be in the Window on Williams panels given as part of fall orientation activities. In the mathematics department, she gave a faculty seminar on "Partially Mixing Transformations."
Professor Deborah Bergstrand attended the 28th Southeastern Conference on Combinatorics, Graph Theory, and Computing at Florida Atlantic University, Boca Raton, where she spoke on "Outer Crossing Number: A New Graph Parameter with Results for K(m,n)." In April she spoke on "Drawing Graphs in Special Ways: An Introduction to Outer Crossing Numbers" at the 4th Hudson River Undergraduate Mathematics Conference, held at Williams College. She gave faculty seminars on "Conjectures on Crossings," and "Graph Crossings on the Torus: Not All Black and White," and a Bronfman Bag Lunch talk on "Keeping Your Edges Uncrossed: Lessons in Skiing and Graph Drawing."
In the summer of 1996 Professor Bergstrand supervised three students as part of the SMALL Undergraduate Research Project: Sydney Foster (Swarthmore `97), Francisca Winston `97, and Alexander Woo `97. Alex went on to do an honors thesis with Professor Bergstrand on outer crossing numbers. They are preparing a joint paper for publication.
Professor Edward Burger received a $100,000.00 two year grant from the Educational Advancement Foundation with Michael Starbird from the University of Texas at Austin to support their projects to reform mathematics courses for liberal arts students. A major component of their program is the writing of the text, The Heart of Mathematics: An Invitation to Effective Thinking, to be published by Springer-Verlag in the fall of 1998.
Burger published "Small Solutions to Systems of Linear Congruences Over Number Fields" in a special issue of the Rocky Mountain Journal of Mathematics (vol. 26, 1996); "Uniformly Approximable Numbers and the Uniform Approximation Spectrum" in the Journal of Number Theory (vol. 61, 1996); "On the Structure of Quadratic Irrationals Associated with Generalized Fibonacci and Lucas Numbers," co-authored with Christopher Kollett `95, in the Fibonacci Quarterly (vol. 34, 1996); "Does Summation of 1/n! Really Converge? Infinite Series and p-adic Analysis," co-authored with Thomas Struppeck, in the American Mathematical Monthly (vol. 103, 1996); "On Liouville decompositions in local fields" in the Proceedings of the American Mathematical Society (vol. 124, 1996); and "Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day," co-authored with Frank Morgan, in the American Mathematical Monthly (vol. 104, 1997).
Burger was named Chair of the Program Committee for the fall 1997 Northeastern Mathematical Association of America Meeting to be held in Springfield. He also was a member of the Steering Committee for the 1997 Hudson River Undergraduate Mathematics Conference. Burger reviewed numerous articles for Mathematical Reviews and served as referee for the Journal fur die Reine und Angewandte, the Journal of the London Mathematical Society, the American Mathematical Monthly, and the Fibonacci Quarterly. He also was a reviewer for the National Science Foundation; served as a member of the Ph.D. dissertation committee for Kwok Choi from the University of Texas at Austin, and advised the Masters defense for Michael Silberglitt from the University of Colorado at Boulder.
Professor Burger gave numerous invited lectures throughout the year. In June, 1996, he was an invited speaker at the Discovery Learning in Undergraduate Education Conference, held in Austin, Texas. In August, together with Colin Adams, he gave two Keynote Addresses at the AMS-MAA Summer Meeting in Seattle, Washington. Together they also ran five teaching workshops. In October, he gave the Homecoming Lecture at the University of Richmond, and in November was the Breakfast Speaker at the Association of Teachers of Mathematics of New England Conference, held in Providence, Rhode Island.
In January, 1997, together with Colin Adams, he gave a Keynote Address at the 1997 LaCEPT/LaSIP Louisiana Board of Regents Conference, held at Louisiana State University. In February he gave the 1997 Dan E. Christie Lecture at Bowdoin College, and in March gave an address at MATHCOUNTS, held at the University of Hartford. Also in March he delivered the Keynote Address at the 1997 Association of Teachers of Mathematics of New England Conference, held in Cromwell, Connecticut. In April he gave a talk at the Hudson River Undergraduate Mathematics Conference, held at Williams College, and in April he spoke at the Five College Number Theory Seminar, held at Amherst College. In addition, he gave an invited address at the Contemporary Artists Center and colloquia at Bennington College and Bowdoin College.
Here at Williams, Professor Burger was a featured speaker for the Society of Alumni during Reunion Week in June. In September he gave the First Year Assembly Lecture to the Class of 2000, and October he gave a Mathematics Faculty Seminar. In January, 1997, he addressed the Development Steering Committee, and in February, the Garfield Republican. In March, he gave a Mathematics Faculty Seminar and in May he was a speaker in the Writing Workshop Colloquium. Also in May he gave an address to the Class of 1949 50th Reunion Committee.
In the summer of 1996, Burger directed the number theory research of three students as part of the SMALL Undergraduate Summer Research Program. In the fall he was a Celebrity Guest DJ on WCFM, and the faculty advisor for the Williams College Bowl team. Finally, in the spring he was a Guest Judge for the first annual Williams Electronic Media Awards. He continued his research in Four Dimensional Topology and Gauge Theory.
Professor Mikhail Chkhenkeli investigated the problem of representing homology classes of 4-manifolds by smoothly embedded 2-spheres. He wrote two papers: "2-spheres in 4-manifolds" and "Characteristic 2-spheres in 4-manifolds." In November, Chkhenkeli was an invited speaker at the MAA Meeting in Boston. In April he gave the talk "The Exotic Four-Dimensional World" at the Hudson River Undergraduate Mathematics Conference. At Williams he gave Mathematics Faculty Seminars in October and May. In November he gave a talk at the Bronfman Bag Lunch. Chkhenkeli advised the research of Eric Soskin `99 in applications of graph theory to the World Wide Web. He organized the Nineteenth Annual Green Chicken Math Contest and monthly mathematics conundrums. In the summer of 1996 he taught an accelerated course in Pre-Calculus at the summer program organized by the Institute for the Academic Advancement of Youth (The Johns Hopkins University). In the summer of 1997, he is an invited by the IAAY to teach an accelerated course in Mathematical Logic and Reasoning.
Professor Richard D. De Veaux gave several invited talks this year, including "Confidence Intervals for Neural Networks," at the Spring Research Conference, American Statistical Association, Washington, D.C., "Bayesian CART," at the Interface Meeting, Sydney, Australia, "Multivariate Control," Joint Statistical Meetings, Chicago, "Prediction Intervals for Neural Networks," Statistics Dept., University of Connecticut, Storrs, CT, and "Neural Networks in Practice," ASA Chapter Meeting on Data Mining, Chicago. He was the organizer and panel leader for an "International Competition of Nonlinear Methods," at the Gordon Research Conference on Statistics in Chemistry and Chemical Engineering, Oxford, England, and gave two one day seminars in "Data Mining" in Belgium and the Netherlands. Professor De Veaux served as the General Methodology Chair for the ASA's joint statistical meetings, to be held in Anaheim in 1997, and continued his work as Associate Editor of Technometrics. He also became an Associate Editor for the journal Environmetrics this year. He received a post-doctoral fellowship to study for the year 1997-1998 at the French National Institute for Agronometric Research (I.N.R.A.) in Montpellier, France, where he will spend his sabbatical leave.
Professor Thomas Garrity has been on leave this past year, continuing his research on higher codimensional CR structures and classical invariant theory and starting to investigate generalizations of continued fraction algorithms. His paper "The Equivalence Problem for Higher Codimensional CR Structures," co-authored with R. Mizner, appeared in the Pacific Journal of Mathematics. In November of 1996, he gave an invited talk "Higher Codimensional Geometry and Linear Algebra" at the fall meeting of the Northeast section of the MAA at the University of Massachusetts at Boston. In April, he gave an invited talk "Higher Codimensional Geometry" at the Valley Geometry Seminar at UMass-Amherst.
Also in April, at the Hudson River Conference, Professor Garrity gave a talk "On Writing Numbers."
Victor E. Hill IV, Thomas T. Read Professor of Mathematics, was the featured Saturday evening speaker at the national summer meeting of the Mathematical Association of America, stepping in for Secretary of Defense William Perry, who had to cancel. Prof. Hill gave his multi-media lecture-recital "Mathematical Aspects of the Music of Bach" to a capacity audience. He also gave this presentation at Union College, Skidmore College, and for the Spring Family Weekend at Williams. At Carleton College he lectured on "Zero and the Null Set: a Mathematical Talk on Nothing."
Professor Hill has been appointed to a 16th year on the Board of the Association of Anglican Musicians, an organization of professionals throughout the Anglican Communion, and to a second term on the Editorial Board of the Association's Journal. He also serves on the Board of Trustees of the Berkshire Unit of Recording for the Blind & Dyslexic and on the Executive Board of the Berkshire Chapter of the American Guild of Organists. He is a reviewer for Mathematics Magazine.
Professor Stewart Johnson continued his research in control theory, working on exploiting the unique properties of hybrid controls (i.e. discrete/digital and continuous/analog combined) to attain optimality without sacrificing stability. Professor Johnson also worked with Paul Steen and Wenjie Hu at Cornell University finding common chaotic attractors in several fluid dynamic models of turbulence.
Professor Johnson supervised two senior theses. Sandeep Bhatt `97 worked on control theory and modified classic theories of optimal controls to apply to new cases of hybrid controls. As a particular application, he demonstrates an optimal switching strategy for balancing an inverted pendulum. Laura Christensen `97 worked on mathematical biology and numerically compared several different dynamic models of HIV infection and treatment.
Professor Susan Loepp came to Williams in the fall of 1996 after a two year post doctoral position at the University of Nebraska. During the year, she gave several invited talks including "Excellent Rings with Local Generic Formal Fibers" at Mathfest in Seattle, "On a Question of Matsumura" at an American Mathematical Society meeting in Lawrenceville, New Jersey, and "How I explain My Research to Undergraduates, at a Mathematical Association of America meeting in Boston. In addition, she attended an Algebraic Geometry conference in Albany and the annual Joint Mathematics meeting in San Diego. She continues her research in commutative algebra. Her paper "Constructing Local Generic Formal Fibers" appeared in the January 1997 issue of the Journal of Algebra. Professor Loepp also supervised a mini-thesis written by Deborah Greilsheimer titled "Characterization of Completions of Integral Domains."
Professor Frank Morgan has been named Dennis Meenan `54 Third Century Professor of Mathematics. Also, next year he will hold Princeton University's new 250-anniversary Visiting Professorship for Distinguished Teaching. He was featured in the Williams Parents' Newsletter, spring 1997.
Morgan studies the geometry of soap bubble clusters and other structures in nature and in materials. He has just written a paper with Professor Colin Adams on the least-perimeter way to enclose areas in hyperbolic surfaces. This summer his SMALL undergraduate research Geometry Group will study how the structure of clusters of fluids, such as oil, water, and mercury (which do not mix) may change when additional fluids are added.
During the year Morgan published five papers and has some ten others in the works. He has given some forty talks, ranging from mathematics seminars to his popular Soap Bubble Geometry Contest.
A second edition of Morgan's Calculus Lite appeared this spring, and a second edition of Riemannian Geometry will appear early next year.
Morgan's 1996 April Fools' Day colloquium with Professor Edward Burger on famous wrong 19th century proofs appeared in the March, 1997 American Mathematical Monthly as "Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fool's Day." (The article concludes with a "proof" that "I am Bill Clinton.")
Morgan's live call-in Math Chat TV show on local cable TV has spawned a national Math Chat column every other Friday in The Christian Science Monitor. He hopes to take the show to Princeton next year. In February he was interviewed on a Mutual Radio Network radio program on "Coffee Bubbles." For more information, see his web page via the department page at http://www.williams.edu:803/Mathematics/.
Morgan chaired the steering committee for the 1997 Hudson River Undergraduate Mathematics Conference, held for the first time at Williams. Some 400 students and faculty from schools throughout New York and New England participated in 120 talks in sixteen simultaneous sessions. About 600 attended the keynote address by Professor Benoit Mandelbrot, "Father of Fractals" and the most popular mathematics speaker in the world. The steering committee also included Burger and Eric Watson `97.
Professor Cesar Silva taught a senior seminar in measure theory along with his other courses and continued his research in ergodic theory. He supervised a group in the SMALL summer research program. His group consisted of Erich Muehlegger `97, Andy Raich `98, and Wenhuan Zhao, who completed research in ergodic theory and two manuscripts that were submitted for publication. Erich Muehlegger also wrote a thesis in ergodic theory. Both Andy and Erich presented their work at the Hudson River Undergraduate Mathematics Conference, in April, 1997. Silva submitted for publication to Ergodic Theory and Dynamical Systems his paper "Zd Staircase Actions," with Terrence Adams. He published "Prime III-lambda Automorphisms: An Instance of Coding Techniques Applied to Nonsingular Maps," with A. del Junco, in Algorithms, Fractals and Dynamics, Ed.: Y. Takahashi, Plenum Publishing Company, New York, 1995. He also was a reviewer for Mathematical Reviews. Silva was an invited guest for Math Chat in July 14, 1996, presented a talk at the Hudson River Undergraduate Mathematics Conference in April, 1997 and gave several talks at Williams on his work. He organized a one week conference with Joe Rosenblatt and Karin Reinhold in Ergodic Theory at the State University of New York in Albany in June, 1997.