**Institute for
Advanced Study/Park City Mathematics Institute (PCMI)**

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the following sections:**

Cross Program Activities

Graduate Summer School

High School Teacher Program

Introduction

Mathematics Education Research Program

Mentoring Program for Women in Mathematics

Oversight Board

Publication Series

Research Program

Steering Committee

Undergraduate Faculty Program

Undergraduate Program

IntroductionThe IAS/Park City Mathematics Institute (PCMI) is an integrated mathematics program that has been sponsored by the Institute for Advanced Study since 1993-94. Participants of PCMI include research mathematicians, graduate students, undergraduate students, mathematics education researchers, undergraduate faculty, and high school teachers. PCMI is a unique program designed to strengthen mathematics education at all levels, in turn benefiting the mathematics community as a whole. By fostering interaction between education and research, PCMI charts new territory in mathematics education reform and mathematics research.

Throughout the year, programs include the year-long High School Teacher Program, the Continuing Outreach Program, the Mentoring Program for Women in Mathematics, and the Publication Series. A major activity of PCMI is the annual three-week Summer Session held in Park City, Utah.

The Summer Session of 1999 marked the 9

^{th}year for PCMI, with over 215 participants in attendance. The program was held at the Inn at Prospector Square in Park City, Utah, from June 20-July 10. Six separate yet overlapping programs for researchers, high school teachers, undergraduate faculty, mathematics education researchers, and undergraduate and graduate students were held during this timeThe research topic for the Graduate Summer School and Research Program was Arithmetic Algebraic Geometry, organized by Karl Rubin of Stanford University and Brian Conrad of Harvard University . The Undergraduate Program, designed to enhance students’ interest in mathematics in general and understanding of arithmetic algebraic geometry in particular, was organized by Robert Bryant of Duke University

.The high school teachers worked with researchers and educators to deepen their knowledge of mathematics and explore new methods of teaching. Teachers-in-residence, selected from alumni sites, and current site directors from the High School Teacher Program also participated. In addition to the lectures and courses developed specifically for each group, there were Cross Program Activities, presentations and small-group discussions on the National Council of Teachers of Mathematics (NCTM) Standards 2000, and lively debates on the role of technology in mathematics education. A complete listing of courses, lectures and activities for each program follows.The interaction which is so integral to PCMI continues during the academic year in selected regional, university-based sites where participating high school teachers work in collaboration with the site directors and other faculty. The Summer Session of 1999 marks the beginning of a new two-year cycle with new teachers drawn from the following sites: California State University of San Bernardino, Rider University, Rhode Island College, the University of Cincinnati, and the University of Michigan at Dearborn. At these sites, the high school teacher participants work closely with university faculty in order to bring about curricular and pedagogical reform, first in their home districts, and then in the larger community around them.

A new feature of the High School Teacher Program is to allow alumni sites to send "second-generation" teachers to PCMI’s Summer Session; these are teachers who join site groups after the sites’ formal participation with PCMI is at an end. (For instance, teachers joined the University of Washington Site group as participants of a PCMI-like organization developed by the Site in 1995 with an Eisenhower grant.) The aim of this new opportunity for second-generation teachers is to serve as a boost to the excitement and activities of the alumni sites, thus assisting them in continuing their work in mathematics education reform. The alumni-site teachers who attended the 1999 Summer Session came from the University of Washington site and the Rice University site, and will attend the 2000 Summer Session as well.

The High School Teacher Program courses at the 1999 Summer Session were:

Building Mathematics in the Classroom, Cynthia Hays, McCallum High School, Austin, Texas;Teaching Mathematics with Technology, James King, University of Washington; andAdvanced Mathematics: Constructible Numbers,Susan Addington, California State University at San Bernardino. Specific topics covered included:Constructing angles and regular polygons; The Golden Ratio; Tetrahedral kites and an origami regular pentagon; Writing in the mathematics classroom; Spherical geometry using the Lenart Sphere; The Golden Mean and paper folding; Impossible constructions; Arcs on the sphere; Theory of origami; The Archimedean solids with polydrons; Rectifying areas; Reflections and other plane symmetries; Tesselations; Pop-up polyhedra.Guest speakers and presentations to the High School Teacher Program included a discussion with Gail Burrill, Mathematical Sciences Education Board, and past-president of the National Council of Teachers of Mathematics;

Pick’s Theorem,Phyllis Chinn, Humboldt State University;Miras; Advanced technology techniques; Introduction to Java Sketchpad and other Sketchpad topics,all by Annie Fetter of the Geometry Forum at Swarthmore College;Solving cubic and quartic polynomial equations, John Polking, Rice University;Introduction to arithmetic algebraic geometry, Karl Rubin, Stanford University; andTeaching using the history of mathematics,Robert Stein, California State University at San Bernadino. In addition, there was a special session with the participants of the Mathematics Education Research Program.The courses of the High School Teacher Program were designed this year to provide a more cohesive curriculum for the participants. Thus, the topic of the Advanced Mathematics course,

Constructible Numbers,provided the framework for activities used inBuilding Mathematics in the ClassroomandTeaching Mathematics with Technology. Also included in the curriculum was a daily problem session for the teachers.The year-round High School Teacher Program has also been a great success. In the past year, the University of Cincinnati teachers organized and taught a 6-month course for high-school geometry teachers. The course, which was held at the University, was completely subscribed with 30 participants, all of whom received graduate level credits.

The University of Louisville sponsored a high profile visit from Dr. William Schmidt of Michigan State University. Dr. Schmidt, who is a dynamic speaker, is the U.S. National Director of the Third International Mathematics and Science Study (TIMSS). The Site Directors and teacher-participants of the Louisville Site group worked hard at organizing and publicizing this event.

All of the PCMI teachers continue to be active in their sites, either with group activities or with individual presentations on in-service days or at regional, state, and local chapters of the National Council of Teachers of Mathematics (NCTM). Three PCMI teachers made a presentation to their colleagues at the Annual Convention of the NCTM in April, and two of these teachers have been invited to make a presentation in Portugal in the Fall of 1999.

Mathematics Education Research ProgramOrganized by Timothy Kelly of Hamilton College with assistance from Joan Ferrini-Mundy, the Mathematics Education Research Program had 10 participants, including three lead researchers. The topic was

Mathematical Reasoning and Proof, and seminars were organized by the lead researchers. The Mathematics Education Research Program met twice a day for formal sessions. The work of leading researchers in the field, as well as the participants’ own work was explored.Seminar titles included

Sociomathematical norms in calculus and differential equations, Karen King, San Diego State University;Dynamic software in pre-service teacher preparation, Helen Gerretson, University of Northern Colorado;Student proof schemes, Guershon Harel, Purdue University;The ethnography of argumentation¸Erna Yackel, Purdue University Calumet;Social and socio-mathematical norms in a differential equation classroom, E. Yackel;Realistic mathematics education (Freudenthal),E. Yackel;Discussion of Dreyfuss: why Johnny can’t read,E. Yackel;Discussion on White Paper on reasoning and proof for Standards 2000 (Hanna and Yackel),E. Yackel.There were 11 participants in the Undergraduate Faculty Program. This year’s focus was

The Role of Proof,and a variety of activities took place under the guidance of Guershon Harel, Purdue University. The Undergraduate Faculty Program met for two formal sessions each day. A series of readings was assigned at the beginning of the course, with participants discussing both the pre-assigned readings and their own work. Seminar titles includedTransition course;Application of the necessity principle;Proof, all by Guershon Harel, Purdue University. In addition, each participant was expected to make a formal presentation to the group. Undergraduate Faculty Program participants also attended the Undergraduate Program, and the Graduate Summer School, and there was heavy interaction with the Mathematics Education Research Program.The course titles and lecturers for the Undergraduate Program were:

Introduction to elliptic curves,taught by David Pollack of Ohio State University, andIntroduction to Zeta-functions and L-functions, taught by Keith Conrad of Ohio State University. In addition to the regular courses, the 24 undergraduate participants met once each day for problem sessions. They also were encouraged to attend the introductory-level courses of the Graduate Summer School, and many did so.Graduate Summer School and Research Program

The Graduate Summer School met for three formal lectures and one problem session each day. The lectures were very well attended, drawing research mathematicians, graduate students and undergraduate students. This year’s lecturers and course titles were:

Introduction to elliptic curves and modular forms, Joe P. Buhler, Reed College;Isawa theory for elliptic curves,Ralph Greenberg, University of Washington;Deformation of Galois Representations,Fernando Gouvêa, Colby College;Serre’s Conjecture, Kenneth Ribet, University of California at Berkeley;Open Questions,Alice Silverberg, Ohio State University. Two lectures by John Tate of the University of Texas at Austin were inserted into the regular schedule of the Graduate Summer School.A highlight of PCMI this year was the announcement of the proof of the Taniyama-Shimura-Weil Conjecture. This exciting development stunned the audience at Kenneth Ribet’s lectures and caused organizer Brian Conrad of Harvard University to give two supplemental lectures at the graduate level on the results of the new proof:

Shimura’s Construction of weight 2 modular representationsandElaboration on last 15 minutes of Shimura’s proof. The announcement, made from the PCMI Summer Session by organizer Brian Conrad and invited researcher Richard Taylor, was written up in a subsequent issue of the Notices of the American Mathematical Society.The Research Program included one or two organized seminars each day. Researchers also attended Graduate Summer School lectures and Undergraduate Program advanced lectures, joined in informal activities with other participants, and participated in the small group discussions on the NCTM Standards 2000.

Titles for 1999 research seminars were:

p-adic semi-stable Galois represenations I,Jean-Marc Fontaine, University of Paris XI;The local Langlands conjecture, Richard Taylor, University of California at Berkeley;p-adic semi-stable Galois representations II,Jean-Marc Fontaine;Wild ramification and deformation rings, Brian Conrad, Harvard University;L-functions in p-adic etale cohomology, Matthew Emerton, University of Michigan;Slopes of p-adic modular forms;Lawren Smithline, University of California at Berkeley;The local Langlands conjecture II, Richard Taylor;Heights of elliptic curves and black hole entropies, Stephen Miller, Yale University;Wild ramification and deformation rings II, Brian Conrad;L-functions, curves, and roots of unity, Christian Popescu, University of Texas at Austin;Some abelian varieties with visible Tate-Shafarevich groups, William Stein, University of California at Berkeley;Automorphic L-functions over function fields, Winnie Li, Pennsylvania State University;Relation between L-value and period integral for quotients of JAmod Agashe,_{0}(N),University of California at Berkeley;Modularity of Q-curves and a generalized Fermat equation, Jordan Ellenberg, Princeton University;The hidden structure of de Rham cohomology, Robert Coleman, University of California at Berkeley;Deforming Galois representations, Ravi Ramakrishna, Cornell University;Geometrically simple abelian varieties over finite fields, Hui (June) Zhu, MSRI;The rank of abelian varieties in infinite Galois extensions, Siman Yat-Fai Wong, Brown University;Finding similarities of Tate-Shafarevich elements,Catherine O’Neil, Harvard University;Lowering levels of reducible Galois representations, Shuzo Takahashi, Harvard University;Modules of Iwasawa algebras coming from Galois cohomology, Susan Howson, MIT.Held four times each week, the formal Cross Program activities were focused on mathematical topics, on the NCTM Standards 2000, and on the use of technology in mathematics education.

Titles of Cross Program Activities were as follows:

An introduction to arithmetic algebraic geometry, Karl Rubin, Stanford University;Geometer’s Sketchpad – Teaching Geometry with Dynamic Software, James King, University of Washington;Cryptography and Mathematics,János Csirik, University of California at Berkeley;The use of technology in the mathematics classroom: a panel discussion, John Polking of Rice University, William McCallum of the University of Arizona, Franco Salialo of York University, William Stein of the University of California at Berkeley, Jeff Farmer of the University of Northern Colorado, Helen Gerretson of the University of Northern Colorado, and Art Mabbott of the University of Washington Site;Pre-concert lecture, Robert Taub, Institute for Advanced Study;Juggling Permutations, Joe Buhler of Reed College and Phyllis Chinn of Humboldt University;Principals and Standards for School Mathematics (Standards 2000),Gail Burrill, MSEB and past-president of NCTM;Breakout discussions between mathematics researchers, students, and educators about the NCTM standards,organized by Alan McRae of Washington and Lee University;Plenary session reports from the breakout discussions on NCTM Standards 2000,moderated by John Polking of Rice University;Technology Troika: three examples of technology in mathematics education,organized by Alan McRae of Washington and Lee University, moderated by James King of the University of Washington, illustrations presented by William McCallum of the University of Arizona, Ken Collins of the Duke University Site, and John Polking of Rice University;My students are Greek to me,Guershon Harel, Purdue University;Informal discussion on ethno-mathematics, moderated by Herb Clemens, panelists were Laverne Bitsie-Baldwin of Kansas State University and Susan Addington of California State University at San Bernardino;Theory and examples of congruences of modular forms,Matthew Emerton, University of Michigan.The High School Teacher Program sponsored a "Polyhedra Building Party" in which all participants were invited to attend and take part in construction activities. An enthusiastic crowd filled the room throughout the evening, including participants from all programs. The activity, originally scheduled to last from 7:30 p.m. to 9:30 p.m., was well under way by 7:00 p.m. and did not end until 10:30 p.m..

On June 28, through the generous sponsorship of the Huntsman Foundation, PCMI hosted a concert by Robert Taub, Artist-in-Residence at the Institute for Advanced Study. PCMI participants and community members attended the piano concert at the (new) St. Mary’s Church. Robert Taub gave a pre-concert lecture to the PCMI participants during the Cross Program Activity on the day of the concert.

Rapid progress was made in 1998-99 on the publication of the lecture notes from each year’s Graduate Summer School. This past year saw the publication of Volumes 5, 6, and 7. It is expected that Volume 8 (from 1998) will be published in late 1999 or early 2000. The PCMI lecture series, which is published by the American Mathematical Society (AMS) now includes the following titles: Volume 1,

Geometry and Quantum Field Theory, Volume 2,Nonlinear Partial Differential Equations in Differential Geometry, Volume 3,Complex Algebraic Geometry, Volume 4,Gauge Theory and Four Manifolds;Volume 5,Hyperbolic Equations and Frequency Interactions; Volume 6,Probability Theory and Applications; Volume 7,Symplectic Geometry and Topology. All titles are available from the AMS. There are plans to publish material from the Undergraduate Programs in an AMS series in the future.The 1999 Summer Session was made possible by the generosity of the following foundations and individuals:

The National Science Foundation

W.K. Kellogg Foundation

Michael Bloomberg

Frank and Peggy Taplin

Ladislaus von Hoffmann

James and Elaine Wolfensohn

Albert and Ellen Schwartz Philanthropic FundThe IAS/Park City Mathematics Institute is governed by an Oversight Board which consists of:

Chairperson:

Phillip A. Griffiths, Director, Institute for Advanced StudyBoard Members:

Hyman Bass, Professor, University of Michigan

C. Herbert Clemens, Professor, University of Utah

Ronald L. Graham, Professor, University of California at San Diego

Shirley A. Hill, Professor Emeritus, University of Missouri-Kansas City

David Hespe, New Jersey Commissioner of Education

Robert D. MacPherson, Professor, School of Mathematics, Institute for Advanced Study

William A. Schreyer, Chairman Emeritus, Merrill Lynch & Co., Inc.

Elaine B. Wolfensohn, New York, New York.

Steering Committee 1999Members of the Steering Committee plan and manage the activities of the PCMI as follows:

Convener (outgoing):

John C. Polking, Professor, Rice UniversityConvener (incoming):

C. Herbert Clemens, Professor, University of Utah1999 Organizers:

Karl Rubin, Professor, Stanford University

Brian Conrad, Professor, Harvard University2000 Organizers:

Avi Wigderson, Professor, School of Mathematics, Institute for Advanced Study

Steven Rudich, Professor, Carnegie-Mellon UniversityEditor, PCMI Lecture Series:

Daniel S. Freed, Professor, University of Texas at AustinGraduate Summer School:

David R. Morrison, Professor, Duke UniversityHigh School Teacher Program:

Susan Addington, Professor, California State University at San Bernardino

Cynthia Hays, Teacher of Mathematics, and Department Chairperson, McCallum High School Austin, TexasHigh School Teacher Program/Technology Director:

James R. King, Professor, University of WashingtonMathematics Education Research Program:

Joan Ferrini-Mundy, National Research Council

Timothy Kelly, Professor, Hamilton CollegeRecruitment:

Nathaniel Whitaker, Professor, University of Massachusetts at AmherstResearch Program:

John Morgan, Professor, Columbia UniversityWomen's Program:

Chuu Lian Terng, Professor, Northeastern UniversityUndergraduate Faculty Program:

Daniel Goroff, Harvard UniversityUndergraduate Program:

Robert L. Bryant, Professor, Duke University

MENTORING PROGRAM FOR WOMEN IN MATHEMATICSMany of the women undergraduate and graduate students participating in the IAS/Park City Mathematics Institute Summer Session attended a preliminary workshop at the Institute for Advanced Study from May 17-27. The program, organized by Chuu-Lian Terng of Northeastern University and Karen Uhlenbeck of the University of Texas at Austin, emphasized the content and culture of mathematics and included lectures, seminars, working problem groups, mentoring and networking sessions and the opportunity to meet and interact with leading mathematicians. The 40 participants included graduate students, undergraduates, postdoctoral scholars, and senior researchers. In addition to the registered participants, the lectures and seminars were extremely popular with students and mathematicians from the surrounding area. All lectures were open to the public.

Karen Uhlenbeck led a Women in Science Seminar, a daily informal discussion group geared for the undergraduates and new graduate students, but well attended by all participants. Two highlights of these seminars were an interview with Joan Feigenbaum, a research scientist at AT&T, and panel discussions on working in industry and in academe.

The Women’s Program has enabled the IAS/Park City Mathematics Institute to increase significantly the number of female participants in its Summer Session. It has also provided female students with an opportunity to form professional friendships and collaborations that develop still further during the PCMI Summer Session and beyond. In this way, the Women’s Program provides strong encouragement and real support for women to remain in the field of mathematics. The Women’s Program of 1999 was funded by the National Science Foundation.

Lectures and Seminars:The undergraduate lecture series,

Codes and Curves,was given by Judy Walker, University of Nebraska at Lincoln. The graduate lecture series,The Arithmetic of Elliptic Curves, Modular Forms, and Calabi-Yau Varietieswas given by Wen Ching Winnie Li of Pennsylvania State University, Noriko Yui of Queens University (Ontario, Canada), and Alice Silverberg, Ohio State University. Expository lectures on the research topic were given by two members of the local mathematics community:Why Zeta Functions,by Peter Sarnak, Princeton University;An introduction to Galois Representations,by Chris Skinner, Institute for Advanced Study.The research seminar was organized by Lisa Fastenberg of University of Massachusetts at Amherst. Seminar titles were:

Uniform bounds for rational and integral points on curves,Lisa Fastenberg;The intermediate Jocobian of some rigid Calati-Yau three-folds, Helena Verrill, Max Planck Institut fur Mathematik;The difficulty of pattern classification,Christine Heitsch, University of California at Berkeley;Plane Curve Singularities, Hilbert Schemes, and Jet-Bundles, Heather Russell, Harvard University;Supersingular abelian varieties over finite fields, Hui (June) Zhu, MSRI;Formal patching of wildly ramified covers, Rachel Pries, University of Pennsylvania;Stark’s conjecture in the octohedral case,Karrolyn Fogel, California Lutheran University.A participant Seminar included the following titles:

Sophie Germain and Fermat’s Last Theorem,Karrolyn Fogel, California Lutheran University;Algebraic number fields and their L-functions,Alina Cojocaru, Queen’s University (Canada);Knots and varieties, Heather Dye, University of Texas at Austin;Canonical heights and polynomial dynamics,Susan Goldstine, McMaster University;Braids, Mari Campbell, University of California at San Diego.

Planning CommitteeThe Women’s Program Committee assists the organizers in planning and promoting the program and recruiting lecturers and participants. Members include: Alice Chang, Professor, Princeton University; Ingrid Daubechies, Professor, Princeton University; Joan Feigenbaum, AT&T Research; Antonella Grassi, Professor, University of Pennsylvania; Nancy Hingston, Professor, The College of New Jersey; Rhonda Hughes, Professor, Bryn Mawr College; Robert MacPherson, Professor, Institute for Advanced Study; and Lisa Traynor, Professor, Bryn Mawr College.