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MATHEMATICS DEPARTMENT


Another eventful year has passed for faculty and students in the Math Department. We are most happy to welcome the newest members of the mathematics faculty: Professor Mikhail Chkhenkeli and Professor Susan Loepp. Chkhenkeli is a topologist who comes to Williams from the University of Pennsylvania; Loepp is an algebraist arriving from the University of Nebraska. The year-long search to fill the two positions produced many excellent candidates out of the 1,000 or so applicants, and we are very excited to have found such superb additions in Chkhenkeli and Loepp.

We welcome back Professor Deb Bergstrand, not only from a year's sabbatical leave at Stanford University, but from a long sojourn in the Deans' Office before that. Although she was a superb Dean, it is wonderful to have her back again full-time in the department.

In the fall we were very fortunate to have Professor Philip Maini as Visiting Professor from Oxford University, England. Maini's research is in mathematical biology, and thanks to his interesting and informative talks, we now know not only how wounds heal, but also how zebra stripes and leopard spots are formed.

A sad farewell was bid to Professor Robert Mizner who left Williams for Jackson National Life Insurance Company in Lansing, Michigan, where he will be starting his new actuarial career. Mizner came to the math department in 1989 as a regular faculty member and as the Director of the Quantitative Skills Program. The Program flourished under the guidance, skill and attention of the "Gentle Giant." We are grateful for his excellent efforts and wish him good fortune in his future.

Williams mathematics majors had another outstanding year. The National Science Foundation awarded Fellowships to Math-Computer Science double majors A.J. Burnheim `96 and Chris Umans `96 for study in Computer Science and to Shawn McDougall `95 for study in Cognitive Science; honorable mention went to Jonathan Todd `96 and Jamie Kerman `93. Jasper Rosenberg `96 was class valedictorian; Tarun Ramadorai `96 was awarded the Herchel Smith Fellowship; and Jason Schweinsberg `97 won a prize for his talk "Prediction Intervals for Neural Networks," presented at a spring Statistics Conference at RPI. At the annual spring math majors dinner, the Rosenberg prize for outstanding senior went to Mike Touloumtzis `96; the Morgan prize for accomplishment and promise in applied mathematics went to Andrew Matuch `96; the Goldberg prize for the best senior colloquium went to Jonathan Todd `96. Additionally, the Benedict prize for outstanding sophomore went to Kariane Calta `98, and the first annual Witte Problem Solving Prize went to Jason Schweinsberg `97.

We are grateful to this year's SMERSH members for their diligence and care in meeting with job applicants, helping the department gather student opinions, running the fall and spring Ice Cream Socials and generally being available for giving advice and help. Many thanks go to Daniel Ebert `96, Teon Edwards `96, Barbara Shreve `96, Joanna Barnes `97, Deborah Greilsheimer `97, Kimberly Kehrberger `97, and Kariane Calta `98.

The math faculty had a busy and productive year. Colin Adams was named the Northeastern Section of the Mathematical Association of America Distinguished Teacher of the Year for 1996. He received a grant from the National Science Foundation to support his research in hyperbolic 3-manifolds for the next two years.

He wrote a grant to support the SMALL Summer Undergraduate Research project that was funded for $40,000 per year for three years. It will support eight students per summer and some faculty participants. Over the summer of 1995, he worked with three students Bevin Brennan `97, Deborah Greilsheimer `97, and Alexander Woo `97, and the paper that resulted, entitled Stick Numbers and Composition of Knots and Links, has been submitted for publication. His paper "Unknotting Tunnels in Two-Bridge Link Complements," co-authored with Alan Reid was accepted for publication in Commentarii Mathematici Helvetici. His paper "Hyperbolic 3-Manifolds with Two Generators," will appear in Communications in Analysis and Geometry. The paper "Knotted Tilings," will appear in the Proceedings of the NATO Conference on Long Range Aperiodic Order, ed. by R. Moody and J. Patera. "Splitting Versus Unlinking," will appear in the Journal of Knot Theory and its Ramifications. An article about his work, written by I. Stewart, appeared in the November issue of Scientific American, p.100-101.

He gave the talk "Unknotting Tunnels for Two-Bridge Knots and Links," Low-Dimensional Topology Conference, University of Montreal, Montreal, Canada, June 12-16, 1995. He was the Mathematical Association of America 25 year Banquet Speaker at the AMS/MAA National Meeting, Burlington, VT, Aug. 10, 1995, where he spoke on "Bus Tours of the Universe and Beyond." At the NATO workshop on Long Range Aperiodic Order at the Fields Institute, Waterloo, Canada, Aug. 29, he spoke on "Knotted Tilings."

He spoke on "Tunnel Number for Hyperbolic 3-Manifolds," at the American Mathematical Society Meetings at Northeastern University in Boston, Oct. 8. On Oct. 21, he gave an Alumni Seminar at Williams entitled "Why Knot?". He gave "Mel Slugbate's Real Estate in Hyperbolic Space," at the Seaway Sectional Meeting of the Mathematical Association of America at Skidmore College, Saratoga Springs, NY, Nov. 4. The two talks "How to Cheat your Way to the Knot Merit Badge" and "Knot Theory Workshop" were given at the Northeastern Sectional Meeting of the MAA, Salem State College, Salem, MA, Nov.17-18. He gave "Mel Slugbate's Real Estate in Hyperbolic Space" at Gettysburg College and at Franklin and Marshall College in Pennsylvania on Feb. 15 and 16, 1996. At Calvin College, in Grand Rapids, he gave the talk "Why Knot"? on March 21. At the Sectional Meeting of the American Mathematical Society in Iowa City, March 23, he spoke on "Minimal Cusp Lengths in Hyperbolic 3-Manifolds: Waist Sizes for Knots." He was the keynote speaker for the Tennessee Mathematics Teacher's Association Convention in Memphis, April 26-27, where he gave the two talks "Mel Slugbate's Real Estate in Hyperbolic Space" and "Why Knot"?. He gave "How to Cheat your Way to the Knot Merit Badge" and "Minimal Cusp Lengths in Hyperbolic 3-Manifolds: Waist Sizes for Knots" at the University of California, Davis at the end of May.

He was a co-organizer of the Hudson River Undergraduate Mathematics Conference, which took place on April 20, at Skidmore College in Saratoga Springs, NY, and brought together 330 mathematics students and faculty from New England for a day of mathematics. He also chaired the ad hoc committee on grading at the college and was hiring chair for the Mathematics Department.

Professor Ollie Beaver continued as Chair of the Mathematics Department, as the Director of the Summer Science Program (SSP) for the tenth year, and as the Coordinator of the Mellon Minority Undergraduate Fellowship Program (MMUFP). She also served on the Advisory Board of the Williams College Multicultural Center.

In January, Beaver attended the national meeting of the AMS/MAA in Orlando and the annual MMUFP Coordinators' meeting in New York City. Beaver has also been an invited member of the Science/Engineering Academic Support Network for students of color, sponsored by the New England Board of Higher Education. She participated in the annual Network Conference for minority high school and college undergraduates in Boston in the Fall, and the annual Spring conference of the Compact program for minority graduate students in science and mathematics.

Professor Deborah Bergstrand spent a year on leave as a visiting scholar at Stanford University. She explored issues in math education, including experiences of adolescent girls. She was an invited participant in a working conference on Women in Science sponsored by the NSF in Washington, D.C., in December, and attended the national meetings of the AMS/MAA in Orlando in January.

Professor Edward Burger returned from a sabbatical leave where he was a visiting assistant professor at the University of Texas at Austin and at the University of Colorado at Boulder. In the summer of 1995 he advised the number theory research of Jonathan Todd `96 that evolved into Todd's senior honor's Thesis. Together they wrote a paper entitled "On a Diophantine Approximation Property Involving the Lagrange Constant for Quadratic Irrationals." In October he was honored by being the first ever faculty guest announcer for the Williams College Mucho Macho Moo Cow Marching Band's half-time show at the Williams-Bates football game.

Professor Burger gave numerous invited lectures throughout the year. In May 1995 he addressed the Williams Alumni Association of Colorado. In July he was a speaker at Bennington College's July Program. In October he addressed the Williams Alumni Association of Jacksonville and also gave a lecture at the University of North Florida. In November he spoke to the Williams Alumni Association of Fairfield and Westchester Counties. He then gave a lecture entitled "Uniformly Approximable Numbers and the Uniform Approximation Spectrum" at a special session in number theory at the American Mathematical Society conference held at the University of North Carolina at Greensboro. Also in November he was a guest on WAMC-FM Albany National Public Radio affiliate. In December he was the keynote speaker at the annual conference of the Association of Teachers of Mathematics of New England, held in Portland, Maine.

In January 1996, he spoke at Williamstown Elementary School and was a speaker at the 1996 Williams in the Northwest Alumni Seminar in Alderbrook, Washington. He was invited to give a lecture at the University of Washington in Seattle and he also spoke at the Washington Middle School and Charles Wright Academy in Tacoma. In February he was the banquet speaker for the Atlanta Williams Alumni Association. In March he was the keynote speaker at the 27th Annual Mathematics Day at Montclair State University. In April he gave a talk at the Third Annual Hudson River Undergraduate Mathematics Conference held at Skidmore College. He also gave invited lectures at Johnson and Wales University and Middlebury College. In May he was invited to speak at Connecticut College and Union College.

Here at Williams, Professor Burger gave a Mathematics Faculty Seminar in September, 1995; represented Division III at the Presidential Colloquium in October; gave a Mathematics Colloquium in February, 1996 and also produced a musical concert with Brian Wecht `97. In April, he gave a Mathematics Colloquium with Professor Morgan and was the keynote speaker at the Parents' Fund Dinner. He gave a Mathematics Faculty Seminar in May.

Professor Richard D. De Veaux received grants from ICI Limited and First USA Bank for his work on artificial neural networks. The funding will enable him to support several undergraduates in the summers to help in the assessment of these networks. It also enabled him to establish a computing laboratory consisting of four Silicon Graphics work stations with state of the art computing, graphics and video capabilities. These machines will be used both by De Veaux's research students and by students in his advanced classes in statistics. Colleagues have been stopping by to make sure De Veaux is okay as he seems to be talking to his computer more and more. On closer examination, they find that he is actually talking to a video image of a student on the computer, usually discussing a homework problem. "Virtual" office hours are becoming a reality.

De Veaux gave several invited talks this year, including "A Guided Tour of Modern Regression Techniques" at the Fall Technical Conference of the American Society of Quality Control in St. Louis and at Dupont's Experimental Design Conference in Wilmington, Delaware; "Neural Networks Untangled" at the meeting of the Boston Chapter of the American Statistical Association held at Smith College; and "Prediction Intervals for Artificial Neural Networks" at the Spring Research Conference of the ASA held at NIST in Gaithersburg, Maryland. He was also an invited panelist at the RPI conference in the panel entitled "What's Hot - What's Next" held in Troy, New York. Dick serves 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 has been invited to be a discussant at this year's Gordon Research Conference in Oxford, England in August, and at the International Statistical Meetings to be held in Sydney, Australia in July. His discussion, "Neural Networks in Practice", (with L.H. Ungar) will appears in Technometrics this August, and "A Guided Tour of Modern Regression Techniques" appears in The 1995 Proceedings of the Section on Statistics in the Physical and Engineering Sciences.

Professor Thomas Garrity has continued his research in higher codimensional CR structures and classical algebraic invariant theory. He has had two papers accepted for publication this year, both written with Professor Robert Mizner. "The Equivalence Problem for Higher Codimensional CR Structures" to appear in the Pacific Journal of Mathematics and "Vector-Valued Forms and CR Geometry" will appear in a collection CR Geometry and Overdetermined Systems, Advanced Studies in Pure Mathematics, published by the Mathematical Society of Japan. He gave a talk "Triangle, Beats and Math" during the fall's First Year's Family Weekend, gave a talk "Now, Why Is Math (Life) So Hard? (Using Math to Understand Complexity)" during the spring's Family Weekend, spoke on earlier work of his on "Factoring Polynomials" at a colloquium at Bennington College in the fall and gave a few talks at the Faculty seminar at Williams. He had three senior thesis students this year: Dan Ebert, whose thesis was "Probabilistic Enumerative Geometry: How Many Reflection Points are Real", Daniel Kim, whose thesis was "Examining Continuity Along Tubular Neighborhoods" and Alex Meadows, whose thesis was "The Construction of the Moduli Space of Quadrilaterals up to Similarity".

Victor E. Hill IV, Thomas T. Read Professor of Mathematics, continued his research in history of mathematics and in the relationship between mathematics and music. He gave his lecture-recital "Mathematical Aspects of the Music of Bach" and his illustrated lecture "Comprehending Beauty in Mathematics and in Music" at Georgia Tech, Armstrong State College (Georgia), the University of Richmond, St. Olaf College, and Carleton College. He also continued his career as a concert harpsichordist and organist, both on campus and on tour. In March he played the premieres of three organ works that he had commissioned for the occasion from young American composers. He has been appointed to his 15th term as Archivist of the (international) Association of Anglican Musicians, and also serves on the Board of Trustees of the Berkshire Unit of Recording for the Blind and Dyslexic (for which he has completed 25 years as a volunteer reader) and on the Executive Board of the Berkshire Chapter of the American Guild of Organists.

Professor Stewart Johnson continues his research in applied dynamical systems and modeling. He is working on finding common chaotic attractors in different hydrodynamic models as part of a project at Cornell University on Constrained Buoyancy-Driven Convection funded by the American Chemical Society/Petroleum Research Fund. He also continues his work on control theory and hybrid systems, and is on the Program Committee of the Sixth Annual Hybrid Systems Conference, to be held this year at Cornell.

Johnson supervised an honors thesis this year conducted by Takeshi Yokoo on using wavelets to analyze dense-target radar data.

Professor Frank Morgan has continued his study of optimal shapes and structures, including work on the long-standing open question of whether the bees' hexagonal honeycomb is really the most efficient way to get infinitely many unit areas in the plane. He has just received a new research grant from the National Science Foundation.

Morgan published four papers and wrote a report for the December 1995, issue of the Mathematical Association of America FOCUS on the recent breakthrough on the Double Bubble Conjecture. The new result says that the familiar double soap bubble is the most efficient way to enclose and separate two equal volumes of air. This result, announced by Joel Hass at Morgan's special session on "Soap Bubble Geometry" at the Burlington Mathfest, followed earlier work by undergraduates in Morgan's Geometry Group, part of the Williams SMALL undergraduate research project. This summer of 1996 his Geometry Group will study nets of geodesics on polyhedra and optimal shapes for double droplets of immiscible fluids.

Morgan also published Calculus Lite, which covers all of single-variable calculus in 281 pages.

During the year Morgan has given some 45 talks, including lectures at an IMA regional graduate course last summer, an assembly at his old high school on "New Geometry For A New Century", and the Bullitt Lecture at the University of Louisville. With Ed Burger he gave a special April Fools Day colloquium at Williams on incorrect proofs of the 1800's, including Fermat's Last Theorem and the Four Color Conjecture, and then proved by request that "I am Bill Clinton".

Morgan has launched a live call-in Math Chat on local cable TV, which is already the talk of the town. The production team includes Eric Watson `97, Aaron Dupuis `99, Charmaine Mangram `99, Liz Claflin `99, Ted Melnick `98, Ramona Nicholson `98, and Alexei Erchak (SMALL `96). Math Chat is featured in Peterson's MathLand on the MAA web page at http://www.maa.org/.

Morgan chairs the steering committee for the Hudson River Undergraduate Mathematics Conference, which also includes Professors Adams and Burger and Eric Watson `97. At the 1996 conference at Skidmore College, 330 faculty and students, with the largest delegation from Williams, presented about 100 talks. The 1997 conference will be at Williams on Saturday, April 12. Dr. Benoit Mandelbrot will give the invited address on fractals.

Morgan serves on the innovative program committee for the Seattle Mathfest for the summer of 1996. William Perry, Secretary of Defense and a Ph.D. mathematician, has been invited to give an address. Professors Colin Adams and Ed Burger of Williams will present a proof of a new theorem in the form of a play. Morgan will moderate a panel discussion and analysis of an invited talk by Joel Hass on the double bubble conjecture.

Morgan sits on the Council of the American Mathematical Society and a number of national committees, including the national teaching award committee and the Morgan undergraduate research prize committee. He served on the new Riemannian geometry panel at the National Science Foundation. In Williamstown, he works with local schools as a member of the Leadership Team for PALMS (Partnerships Advancing the Learning of Mathematics and Science). At Williams College, he serves as chair of the Calendar and Schedule Committee. This fall he introduced a national mailing to prospective Williams students with promise in mathematics.

In August 1996, Morgan received an honorary Sc.D. degree from Cedar Crest College in Allentown, Pennsylvania.

Professor Cesar Silva was on sabbatical leave during the Fall semester at the State University of New York at Albany, where he gave several talks in their ergodic theory seminar. He also visited the University of Toronto for two weeks.

He began the summer with a visit to his collaborator P. Thieullen at the University of Paris, and then at Williams he participated in the SMALL summer research program. His group consisted of Balakrishna Narasimhan `97 and Michael P. Touloumtzis `96, who completed research in ergodic theory and a manuscript that was submitted for publication. Michael Touloumtzis went on to write a thesis under him. Silva and Touloumtzis spoke on this work at the Hudson River Undergraduate Mathematics Conference, in April 1996.

His paper "A Skew Product Entropy for Nonsingular Transformations" with P. Thieullen appeared in the Journal of the London Mathematical Society . His paper "Rank-One Weak Mixing for Nonsingular Transformations" with T. Adams and N. Friedman was accepted for publication in the Israel Journal of Mathematics. He also was a reviewer for Mathematical Reviews.

Silva presented the invited talks "Continuous Spectrum for Rank-One Infinite Measure Preserving Transformations", at the University of Paris VI, in June, 1995, "Relatively Weak Mixing and Infinite Skew Product Entropy" at the Boston meeting of the American Mathematical Society in October, 1995, and "Modeling Objects with Fractals" and "Undergraduate Research in Dynamical Systems" at The Hotchkiss School in April, 1996.

MATHEMATICS COLLOQUIA

Colin Adams, Williams College
"How Tailors Measure the Waist Size of a Hyperbolic Knot"
"Knotted Tilings of the 3-Sphere"
"Stick Numbers of Knots"
"Real Estate in Hyperbolic Space: Investment Opportunities for the 90's"
Olga R. Beaver, Frank Morgan, and Cesar Silva; Williams College
"Quasi-Measures"
Andrew Beveridge `91, Yale University
"Disorientation and Amnesia: Random Walks on Graphs"
Kenneth Brakke, Susquehanna University & The Geometry Center
"Soap Films and Covering Spaces"
Edward Burger, Williams College
"An Introduction to the Uniform Approximation Spectrum"
"On the Aubry-Thue Theorem and the Geometry of the Adeles"
Edward Burger and Frank Morgan, Williams College
"Math Fools"
Mikhail Chkhenkeli, University of Pennsylvania
"Wonders of Dimension 4"
David Cruz-Uribe, Purdue University
"When Order Matters: Conditionally Convergent Series"
Richard De Veaux, Williams College
"Prediction Intervals for Artificial Neural Networks"
"Bayesian Prediction for Artificial Neural Networks"
Ted Floyd, Williams College
"Mathematics and Biology"
Thomas Garrity, Williams College
"On Relations of Invariants"
"Some Invariants for Polynomial Ideals"
"Now, Why is Math (Life) so Hard? (Using Math to Understand Complexity)"
Victor E. Hill IV, Williams College
"Comprehending Beauty in Mathematics and in Music"
"Further Thoughts on Comprehending Beauty and Elegance"
Amy Jones, Vassar College
"On Two-Generator Knots and Links"
William Lenhart, Williams College
"The Minimum Weight Triangulation Problem"
Susan Loepp, University of Nebraska
"Rational Numbers and Polynomial Rings: The Complete Story"
Philip K. Maini, Williams College
"On The Mathematical Modeling of Wound Healing"
"Red-Gray Squirrel Competition, Skin Organ Pattern Formation, and How to Beat the Breathalyzer"
Robert Manning, University of Maryland, College Park
"Can You Model DNA as a Flexible Rod?"
Helen Moore, Bowdoin College
"The Mathematics Behind Those Amazing Minimal Surfaces"
Frank Morgan, Williams College
"Minimal Surfaces Around the World"
"Energy-Minimizing Harmonic Maps"
"The Soap Bubble Geometry Contest"
"An Isoperimetric Theorem for the Thread Problem and a $100 Challenge"
"Does An Optimal Honeycomb Exist"
"The Hexagonal Honeycomb Conjecture Continues"
James D. Murray, University of Washington, Seattle
"Sex and the Single Alligator and What Wolf Pack Interaction Tells us About Intertribal Warfare"
John Rafter, Vanderbilt University
"Universal Algebra: Abstract Algebra's Abstraction"
Margaret Robinson, Mt. Holyoke College
"Number Theory in p-adic Fields"
Kimberly Ruane, Florida State University
"An Introduction to Geometric Group Theory"
Marc Sanders, University of Minnesota
"Algebraic Topology and Fixed Point Theorems"
Patrick Shanahan, University of California
"Newton Polygons and Knots"
Cesar Silva, Williams College
"Ergodic Theory of Tiling Systems"
Karen Smith, Massachusetts Institute of Technology
"What are `Singularities' in Algebraic Geometry?"
Michael Starbird, University of Texas, Austin
"The Free Surface Problem"
"Untangling the Dread 2-Torus - Topology to the Rescue"
John Sullivan, University of Minnesota, Minneapolis
"Soap-Bubble Singularities as Duals of Deltahedra"
Kendrick Zetie, Oxford University
Juggling and Math

MATHEMATICS STUDENT COLLOQUIA

Megan Barber `96
"The Invariants of the Symmetric Group"
Megan Barber `96, Jennifer Tice `97, Brian Wecht `97
"Double Salt Crystals"
Julia Belladonna `96
"Hilbert Basis Theorem"
Alice Bernheim `96
"How to Use Wedging, Bunching and Braiding to Create Wall-Hangings from Knots"
Bevin Brennan `97, Deborah Greilsheimer `97, Alexander Woo `97
"SMALL Undergraduate Research, Summer 1995"
Alain Cunqueiro `96
"Maschke's Theorem on Finite Groups"
Daniel Ebert `96
"Tarski-Godel's Incompleteness Theorem"
Teon Edwards `96
"Playing with Crayons and String: Coloring a Knot"
Teon Edwards `96, Brian Wecht `97
"What It's Like Doing Research"
Gregg Frey `96
"The Construction of Polygons with Compass and Ruler"
Lisa Hauser `96
"Handy-Decorating Lesson 4: How to Tile your Bathroom Aperiodically"
Lisa Howard `96
"Taking a Trip Down Devil's Staircase: The Cantor Set and Function"
Daniel Kim `96
"Spies R Us: The World of Public Key Codes"
David Lee `96
"Game Theory"
"Traffic Congestion"
Andrew Matuch `96
"Logistic and Discriminant Analysis: Categorizing Irises"
"Logistic Regression: Another Approach to Classification"
Alexander Meadows `96
"Brouwer's Fixed Point Theorem, and How to Win a Nobel Prize (Even if you're a Mathematician)"
Balakrishna Narasimhan `97 and Michael Toumoultzis `96
"Weakly Wandering Sets"
Nicholas Raleigh `96
"Percolating Polynomials, Finding the Number of Real Roots of a Polynomial"
Tarun Ramadorai `96
"The Banach-Mazur Game"
Sarah Rispin `96
"Surreal Numbers or How to Create Something Out of Nothing"
Jasper Rosenberg `96
"Plya Theory of Counting"
Patricia Scott `96
"A Chemotactic Model for Biological Pattern Generation"
"A Nice Partial Differential Equation Without (Local) Solutions"
Barbara Shreve `96
"Peaucellier's Inversor: How to Draw a Straight Line Without a Straight Edge"
Jonathan Todd `96
"This Talk's on Random Walks or, will Lassie Come Home?"
Michael Touloumtzis `96
"Fractal Compression"
Christopher Umans `96
"Subresultants and Euclid's Algorithm"
Allison Verderber `96
"The Secret of Nim: Games with Fibonacci Representations"
Jerome Williams `96
"Aproposition to End America as we Know it"
Krystal Williams `96
"Guiga's Conjecture"
Takeshi Yokoo `96
"Obscure Matrix Inversion Technique Gives Insights into Regression Analysis"

POSTGRADUATE PLANS OF MATHEMATICS MAJORS

Megan Barber
Working at Skadden, a law firm in New York City
Alice Bernheim
Interning at Microsoft this summer, then entering the Ph.D. program in Computer Science at the University of Washington
Alain Cunqueiro
Teaching high school math in New Jersey through Teach for America
Daniel Ebert
Entering medical school at Vanderbilt University School of Medicine
Teon Edwards
Working in science education, then attending graduate school
Gregg Frey
Working for an architectural firm in Lancaster, then entering graduate school at Berkeley in architecture
Lisa Hauser
Traveling around the world then settling in Venezuela working for the oil industry
Lisa Howard
Teacher/intern at Woodberry Forest Summer School in Virginia this summer, then teacher/intern at the Casablanca American School in Morocco
Daniel Kim
Consultant at Price Waterhouse, Office of Government Services in Arlington, Virginia
David Lee
Working for a consulting firm as a consultant in Kuala Lumpur, Malaysia
Alexander Meadows
Pursuing a Ph.D. in Mathematics at Stanford University
Nicholas Raleigh
Teaching at St. Marks School in Southborough, Massachusetts then entering the M.Ed. program at Harvard Graduate School of Education
Jasper Rosenberg
Software Developer for PTCG in Boston
Jonathan Todd
Pursuing a Ph.D. in Mathematics at the University of Wisconsin, Madison
Michael Touloumtzis
Working for Computer Associates International, Inc. in Westwood, Massachusetts
Christopher Umans
Pursuing a Ph.D. in Computer Science at the University of California, Berkeley
Takeshi Yokoo
Studying martial arts in Japan for one year, then entering medical school


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