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The Institute for Advanced Study/Park City Mathematics
Institute (PCMI) was begun in 1991 at the University of Utah as one of the
National Science Foundation’s Regional Geometry Institutes. In 1993
the Institute for Advanced Study was asked to assume institutional
sponsorship, and the name was changed in 1994 in order to reflect the new
affiliation. The first Summer Session under this new sponsorship was
held in 1994.
1994 Summer Session
Gauge Theory and the Topology of Four-Manifolds
Organizers: John Morgan, Columbia University Robert
Friedman, Columbia University
The 1994 Summer Session was held in Park City, Utah, from July 10 to 30
and was attended by 80 graduate students, 56 researchers, 20 undergraduates
and 34 high school teachers. The following four programs were held:
the Graduate Summer School, the Research Program, the Undergraduate Program
and the High School Teacher Program. The Research Topic was Gauge
Theory and the Topology of Four-Manifolds.
Courses and Lecture Series:
Graduate Summer School: Introduction to Gauge Theory, John Morgan,
Columbia University; Introduction to Complex Surfaces, Coherent
Sheaves, and Algebro-Geometric Computation of Donaldson Polynomials, Robert
Friedman, Columbia University; Hermitian Differential Geometry and
Holomorphic Vector Bundles, Jun Li, Stanford University; ASD
Connections on Cylinders and the L2 Moduli Space, Cliff Taubes, Harvard
University; Decompositions of Four-Manifolds along Homology
Three-Spheres and Computations of Donaldson Polynomials, Ron Stern,
University of California at Irvine; Spaces of ASD Connections Singular
along a Riemann Surface and Universal Relations Among the Donaldson
Polynomial Invariants, Tom Mrowka, California Institute of Technology;
Introduction to Geometric Invariant Theory, David Gieseker,
University of California at Los Angeles.
Undergraduate Program: Computer Projects, Robert Bryant, Duke
University; Tilings and the Topology of Surfaces, John Harer, Duke
University; An Introduction to Riemann Surfaces and Connections, Karen
Uhlenbeck, University of Texas at Austin.
High School Teacher Program: Mathematics in the Classroom, Naomi
Fisher, University of Illinois at Chicago; Technology for Teaching
Mathematics, James King, University of Washington; Advanced
Classical Mathematics, John Polking, Rice University and John Wood,
University of Illinois at Chicago.
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1995 Summer Session
Nonlinear Wave Phenomena
Organizers: Luis Caffarelli, University of Texas at Austin Weinan
E, Courant Institute
The 1995 Summer Session was held in Park City, Utah, July 9-29.
Participating were 51 graduate students, 33 researchers, 20
undergraduate students, and 38 high school teachers. The
Research Topic for 1995 was Nonlinear Wave Phenomena. There were four
programs at the Summer Session: the Graduate Summer School, the Research
Program, the Undergraduate Program, and the High School Teacher Program.
Courses and Lecture Series:
Graduate Summer School: Nonlinear Schrödiger Equations, Jean
Bourgain, Institute for Advanced Study; Harmonic Analysis, Wavelets and
Applications, Ingrid Daubechies, Princeton University; Nonlinear
Waves: Patterns, Oscillations, Singularities, and Stochasticity, David
W. McLaughlin, Courant Institute and Alan Newell, University of Arkansas; Waves
in Random Media, George Papanicolaou, Stanford University; High
Frequency Nonlinear Hyperbolic Waves, Jeffrey B. Rach, University of
Michigan.
Undergraduate Program: An Introduction to Linear and Nonlinear Waves, Roger
Knobel, University of Texas - Pan American; An Introduction to Solitons, Steve
Cox, Rice University; The Computation of Linear and Nonlinear Waves, Richard
Palais, Brandeis University.
High School Teacher Program: Building Mathematics in the Classroom, Naomi
Fisher, University of Illinois at Chicago and Cynthia Hays, McCallum High
School, Austin, Texas; Technology for Teaching Mathematics, James
King, University of Washington; Mathematical Modeling in the Classroom,
Frank K. Lester Jr., Indiana University; Advanced Mathematics, John
Polking, Rice University.
All-Institute Activities:
Mathematics, Music and Waves, Robert Bryant, Duke University; Nonlinear
Waves in Traffic, Barbara Keyfitz, University of Houston; Shock Waves
and Some Recent Problems, Cathleen Morawitz, Courant Institute; Surface
Water Waves, Kathleen Socha, Oregon State University.
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1996 Summer Session
Probability Theory
Organizers: Srinivasa Varadhan, Courant Institute Elton Pei
Hsu, Northwestern University
In 1996, over two hundred participants attended the PCMI Summer Session,
which was held for the first time at the Institute for Advanced Study in
Princeton, NJ. The dates were June 23 to July 13, and the Research
Topic was Probability, organized by Elton Pei Hsu, Northwestern University,
and Srinivasa Varadhan, Courant Institute.
Participating were 60 graduate students, 50 researchers, 20 undergraduate
students, and 38 high school teachers in four programs: the Graduate Summer
School, the Research Program, the Undergraduate Program, and the High School
Teacher Program.
Courses and Lecture Series:
Graduate Summer School:
Mathematical Finance: An Introduction, Marco Avellaneda, Courant
Institute; Independent and Dependent Percolation, Jennifer Chayes,
University of California, Los Angeles; Stochastic Spatial Models, Rick
Durrett, Cornell University; Stochastic Analysis on Path and Loop Spaces,
Elton Pei Hsu, Northwestern University; Stochastic Analysis on
Space of Paths on a Riemannian Manifold, Dan Stroock, Massachusetts
Institute of Technology; Scaling Limit of Interacting Particle Systems,
Horng-Tzer Yau, Courant Institute.
The Undergraduate Program was organized by Gregory Lawler, Duke University,
and Emily Puckette, Occidental College and offered the following courses: Contemporary
Probability, Gregory Lawler; Random Walk, Emily Puckette; and Random
Walk Simulations, Lester Coyle, Duke University.
High School Teacher Program:
Building Mathematics in the Classroom, Naomi Fisher, University of
Illinois at Chicago and Cynthia Hays, McCallum High School (Austin, Texas); Technology
for Teaching Mathematics, James King, University of Washington; and Advanced
Mathematics, John Polking, Rice University and Virginia Warfield,
University of Washington.
Cross Program Activities:
Geometry and Elementary Probability, Robert Bryant, Duke University; Education
Seminar: Pedagogical principles and the teaching of mathematics with
particular reference to the teaching of linear algebra, Guershon Harel,
Purdue University; CBL (Calculator Based Laboratory) Demonstration, Dennis
Donovan, Rice University; The Ph.D. Job Market, Moderator: John
Polking, Rice University; Panelists: Gerard Ben Arous, École Normale Supérieure;
Elizabeth Brooks, Duke University; Jennifer Chayes, University of
California, Los Angeles; Mike Cranston, University of Rochester; Carolyn
Cross, University of California, San Diego; Education Seminar:
Students’ Proof Schemes, Guershon Harel, Purdue University; On the
History of the Institute and the School of Mathematics, Armand Borel,
School of Mathematics, IAS; Teaching Seminar: Student Attitudes and
Calculus Reform, Jack Bookman, Duke University; Investigating the
Complexity of Learning and Teaching, Carolyn A. Maher, Rutgers
University; Probability with Infinitesimals, Gregory Lawler, Duke
University; Teaching Seminar: Block Scheduling in High School; Electronic
Publication: An informal discussion, Rene Carmona, Princeton University
and Gregory Lawler, Duke University; Investigating the Complexity of
Learning and Teaching, Part II - Participant Interaction, Carolyn A.
Maher, Rutgers University; Content Characteristics of Eighth Grade
Mathematics Classes: A Three Nation Study, Alfred Manaster, University
of California, San Diego; Poincaré and the Improbable: Historical
Interlude on Randomness and Determinism, Daniel Goroff, Harvard
University; Pictures of Particle Systems, Richard Durrett, Cornell
University; and Rice University School Math Project, Anne
Papakonstantinou, Rice University.
Pilot Program:
In 1996, the Undergraduate Faculty Program was begun on a pilot basis.
This program was designed to enhance the teaching ability of a small group
of undergraduate faculty. Faculty attending gained experience with
recent advances and experimental techniques in their fields and explored
ways to incorporate these into their classroom instruction.
Special Events:
Joan Ferrini-Mundy, a member of the PCMI Steering Committee, organized a
two-day conference for professional mathematics educators to explore the
feasibility of a program for researchers in mathematics education as a
component of the PCMI.
On June 27, one hundred attendees of the Second Conference for African
American Research Mathematicians (AARMS) joined PCMI participants for an
afternoon of research talks and breakout sessions, followed by an evening
banquet.
On July 11, the Institute honored the National Science Foundation and
its director, Neal Lane, with a day-long celebration of the NSF’s
commitment to excellence in mathematics and to the integration of research
and education. Leaders in the public and private sectors, along with
participants in the PCMI and the DIMACS Research and Education Institute,
recognized the NSF’s successful Regional Institutes in the Mathematical
Sciences (RIMS) and acknowledged its long history of supporting developing
mathematicians. Following opening remarks by Phillip A. Griffiths,
Director, Institute for Advanced Study, Jake Garn, United States Senate-Utah
(Retired), and Leo F. Klagholz, New Jersey Commissioner of Education, Neal
Lane gave the Keynote Address, Integration of Research and Education:
Vision, Value and Vulnerability. Then, leaders of two NSF RIMS
programs, John C. Polking, IAS/PCMI, and David P. Dobkin, DREI, described
their activity. The afternoon session began with a panel discussion, A
Cornerstone for the Future: Connecting Mathematics Researchers and
Educators, moderated by Hyman Bass, Adrain Professor, Columbia University.
Panelists included Judy Ann Brown, DREI Teacher Program/DIMACS Leadership
Program (K-8), Margaret Cozzens, Division Director, ESIE, National Science
Foundation, Joan Ferrini-Mundy, Director, MSEB, National Academy of
Sciences, Cynthia Hays, PCMI High School Teacher Program, Donald J. Lewis,
Division Director, DMS, National Science Foundation, and Fred S. Roberts,
Director, DIMACS. A concert by Pianist Robert Taub,
Artist-in-Residence, Institute for Advanced Study completed the program.
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1997 Summer Session
Symplectic Geometry
Organizers: Yakov Eliashberg, Stanford University Lisa Traynor,
Bryn Mawr College
In 1997, over 250 participants attended the PCMI summer session held at
the Inn at Prospector Square in Park City, Utah, June 29 to July 19.
Programs offered were: Graduate Summer School, Research Program,
Undergraduate Program, High School Teacher Program, and, for the first time,
the Undergraduate Faculty Program. There were 80 graduate
students, 80 researchers, 20 undergraduate students, 7 undergraduate
faculty, 39 high school teachers, 5 Teachers-in-residence, and many guest
lecturers and visitors.
The research topic for the Graduate Summer School and Research Program
was Symplectic Geometry and Topology, organized by Yakov Eliashberg of
Stanford University and Lisa Traynor of Bryn Mawr College.
Courses and Lectures Series:
Graduate Summer School: Quantum Cohomology, Alexander Givental,
University of California at Berkeley; Holomorphic Curves and Dynamics and
Topology in Dimension Three, Helmut Hofer, Eidgen Technische Hochschule;
Hamiltonian Group Actions and Symplectic Quotients, Lisa Jeffrey,
McGill University; Morse Theory and Singularities, Robert MacPherson,
Institute for Advanced Study; Mechanics: Symmetry and Dynamics, Jerrold
Marsden, California Institute of Technology; Introduction to Symplectic
Topology, Dusa McDuff, SUNY at Stony Brook; Floer Homology, Dietmar
Salamon, University of Warwick; and The Geometry of the Seiberg-Witten
Invariants, Clifford Taubes, Harvard University.
Undergraduate Program: The Geometry of Differential Equations and
Conservation Laws, Lucas Hsu, University of Arizona; Symmetry and
Symplectic Geometry, Robert Bryant, Duke University; and Computer Lab
Session, Richard Palais, Brandeis University.
High School Teacher Program: Building Mathematics in the Classroom, Naomi
Fisher, University of Illinois at Chicago, and Cynthia Hays, McCallum High
School, Austin, Texas; Technology for Teaching Mathematics, James
King, University of Washington; and Advanced Mathematics, James
Carlson, University of Utah.
Cross Program Activities:
What Is Symplectic Geometry?, Robert Bryant, Duke University; Panel
on the Ph.D. Job Market, Judith Arms, University of Washington, Jim
Bryan, MSRI, James Carlson, University of Utah, and Luis O’Shea, Cornell
University; Third International Mathematics and Science Study (TIMSS), Daniel
Goroff, Harvard University; Upper Bounds for the Writhing of Knots and
the Helicity of Vector Fields, Dennis DeTurck, University of
Pennsylvania; The Use of Discovery Method Teaching in Undergraduate
Courses, Virginia Warfield, University of Washington; Pre-Collegiate
Mathematics Education Around the World, Moderator: Naomi Fisher,
University of Illinois at Chicago; Panelists: Veronique Lizan, École
Normale Supérieure de Lyon, Richard Hind, Stanford University, Hansjorg
Geiges, Eidgen Technische Hochschule, Tadashi Tokeida, McGill University,
and Donald Davenport, University of Michigan Site; The Language of
Mathematics and K-16 Mathematics Education, Stephanie Frank Singer,
Haverford College; The Mathematics of the Rainbow, David Fried,
Boston University; Mathematics and Music, Edward Rothstein, The New
York Times, and Robert Taub, Institute for Advanced Study; Marsden’s
Toys, Jerrold Marsden, California Institute of Technology; The
Connection Between High School Mathematics and Collegiate Mathematics, Moderator:
John Polking, Rice University; Panelists: Andrew McInerny, Bronx Community
College (CUNY), Kimya Moyo, School for Creative and Performing Arts,
Cincinnati, Jennifer Slimowitz, SUNY at Stony Brook, Ronald Stern,
University of California at Irvine, Mark Tomforde, Gustavus Adolphus
College, and Christopher Sinclair, University of Arizona.
Cross-program interaction inspired a prize-winning entry in Park City’s
annual July 4 Parade. Participants from all programs joined together
to form the "Human Legendrian Wave Front" and to design the
"Bubble Mobile" (a car covered with balloons of every shape and
color), among other attractions. In addition to being a semi-serious
demonstration of knot theory, the design of the entry grew out of intense
collaborations and mathematical discussions. In recognition of this
effort, parade officials bestowed the "Most Creative"
award upon the PCMI.
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questions and comments to C.
Giesbrecht
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