IAS/PARK CITY MATHEMATICS INSTITUTE
Report for 1998
Representation Theory of Lie Groups
Organizers: David Vogan, Massachusetts Institute of Technology Jeffrey Adams, University of Maryland
Faculty for the Graduate Summer School:
Anthony Knapp, SUNY at Stony Brook
Roger Zierau, Oklahoma State University
Jian-Shu Li, University of Maryland
Kari Vilonen, Brandeis University
David Vogan, Massachusetts Institute of Technology
Faculty for the Undergraduate
Roger Howe, Yale University
William Barker, Bowdoin College
Faculty for the High School
Naomi Fisher, University of Illinois at Chicago
Cynthia Hays, McCallum High School, Austin, Texas
James King, University of Washington
Robert Stingley, Rice University
The annual three-week Summer Session of the IAS/Park City Mathematics Institute was held in Park City, Utah, July 12-August 1, 1998. The PCMI receives major support from the National Science Foundation.
In 1998, over 225 participants attended the PCMI summer session, attending six separate yet overlapping programs for mathematics researchers, high school teachers, undergraduate faculty, mathematics education researchers, and undergraduate and graduate students.
The research topic for the Graduate Summer School and Research Program was Representation Theory of Lie Groups, organized by David Vogan of the Massachusetts Institute of Technology and Jeffrey Adams of the University of Maryland. The Undergraduate Program, designed to enhance students interest in mathematics in general and understanding of representation theory in particular, was organized by Robert Bryant of Duke University. The Undergraduate Faculty Program was organized by Daniel Goroff, Harvard University, and the new Mathematics Education Research Program was organized by Timothy Kelly of Hamilton College, with assistance from Joan Ferrini Mundy. Both of these groups were involved in their own seminars as well as in those of the other PCMI programs. 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 site directors also participated in the High School Teacher Program. In addition to the lectures and courses developed specifically for each group, there were Cross Program Activities and integrated small-group discussions on topics of general interest, including the Third International Mathematics and Science Study, the National Council of Teachers of Mathematics (NCTM) Standards, and high school geometry curriculum content. 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 six regional, university-based sites where participating high school teachers work in collaboration with the site directors and other faculty. The 1997-99 sites include Purdue University, Rider University, Rhode Island College, the University of Cincinnati, the University of Louisville, and the University of Michigan. 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 to the larger community around them.
Individual sections of this report may be quickly accessed by clicking on the following links.
Mathematics Education Research Program
High School Teacher Program
Undergraduate Faculty Program
Graduate Summer School
Cross Program Activities
Mentoring Program for Women in Mathematics
Mathematics Education Research Program
In 1998 the PCMI unveiled its newest program, the Mathematics Education Research Program, organized by Timothy Kelly of Hamilton College with assistance from Joan Ferrini-Mundy. Six participants, including two lead researchers, worked intensively on research in pedagogical issues in modern mathematics education, collaborating closely with participants in both the High School Teacher Program and the Undergraduate Faculty Program. The topic for the Mathematics Education Research Program was Technology and Teaching of Geometry. It is apparent that interaction at PCMI has been greatly facilitated by the participants of its two newest programs, the undergraduate faculty and the mathematics education researchers, who move easily among the already established programs.
High School Teacher Program
The schedule of classes for the High School Teacher Program was changed in 1998 so that all coursework was covered in the mornings. Afternoons were devoted to presentations by the site groups, who are in the second year of their two-year cycle with PCMI, or to guest presentations. Courses were: 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: Symmetry, Robert Stingley, Rice University. Guest speakers included Gail Burrill, University of Wisconsin and past-president of the National Council of Teachers of Mathematics; Allen Knutson, Massachusetts Institute of Technology; and John Polking, Rice University.
Undergraduate Faculty Program
There were 13 participants in the Undergraduate Faculty Program, now in its second year. This years focus was Linear Algebra, and a variety of activities took place under the guidance of Daniel Goroff, Harvard University. Small working groups met to discuss curriculum and the publishing of textbooks, and to experiment with technology. Guest presentations were made by Guershon Harel, Purdue University and PCMI Site Director; Wilfried Schmid, Harvard University and PCMI Research Program participant; Daniel Goroff, Harvard University; John Polking, Rice University; Roger Howe, Yale University; William Barker, Bowdoin College; and Joan Ferrini-Mundy, National Research Council. A web site of the materials generated can be found HERE.
In previous years the participants of the Undergraduate Program have been split into introductory and advanced sections, with classes running concurrently each day. At the suggestion of the undergraduate lecturers, the 1998 schedule was re-structured so that all of the participants could attend both of the daily lectures if they wished. As a result, nearly 100% of the undergraduates attended both of the offered courses each day. There were a significant number of visitors from the other programs observing these lecture as well. The course titles and lecturers were: Continuous Symmetry, William Barker, Bowdoin College; Introduction to the Representations of Lie Groups, Roger Howe, Yale University.
Graduate Summer School
The Graduate Summer School met for three formal lectures and one problem session each day. This years lecturers and their course titles were: Representations of Semisimple Lie Groups, Anthony Knapp, SUNY at Stony Brook (due to illness, Professor Knapps lectures were delivered by Peter Trapa, Massachusetts Institute of Technology); Representation Theory and Dolbeault Cohomology, Roger Zierau, Oklahoma State University; Theta Correspondences and Dual Pairs, Jian-Shu Li, University of Maryland; The Method of Coadjoint Orbits, David Vogan, Massachusetts Institute of Technology; and Topological Methods in Representation Theory, Kari Vilonen, Brandeis University.
The Research Program included one or two organized seminars each day. Researchers also attended Graduate Summer School lectures and Undergraduate Program advanced lectures, participated in working groups with the Undergraduate Faculty, and participated in the small group discussions on geometry curriculum. Titles for 1998 research seminars were:
The Theory of Types for P-adic Groups: Plancherel Theorem and Conductors, Philip Kutzko, University of Iowa
Characters of Covers of SL(n), Jeffrey Adams, University of Maryland
The Schwartz Space for the Hypergeometric Fourier Transform, Patrick Delorme, Institut de Mathematiques
Minimal Orbits and Associated Representations, Pierre Torasso, Université de Poitiers
Poles of Intertwining Operators Via L-functions and Endoscopy, Freydoon Shahidi, Purdue University
Types and Hecke Algebras: construction, Julee Kim, Institute for Advanced Study
Duality and Hecke Algebras, Anne-Marie Aubert, École Normale Supérieure
Double Fibration Transforms, Joseph Wolf, University of California at Berkeley
On Classifications of Square-Integrable Representations, Gordon Savin, University of Utah
Analytic Structure of Compactifications of Symmetric and Locally Symmetric Spaces, Lizhen Ji, University of Michigan
Geometry and Computation in Coxeter Groups, William Casselman, University of British Columbia
Bernstein Degree and Characteristic Cycles, Kyo Nishiyama, Kyoto University
The Barbasch-Vogan Conjecture, Wilfried Schmid, Harvard University
Tensor Products of Singular Representations, Alexander Dvorsky, Rutgers University
A Comparison of Zeros of L-functions, Ravi Raghunathan, California Institute of Technology
Orbital Varieties, Weyl Group Representations, and Robinson-Schensted Algorithms, Peter Trapa, Massachusetts Institute of Technology
Positivity of Littlewood-Richardson Coefficients; or, the Discovery of Tensor Products Among the Bees, Allen Knutson, Massachusetts Institute of Technology
Construction of Minimal Representations for Complex Classical Groups, Alexander Astashkevich, University of California at Davis
Generalized Spherical Functions on Symmetric Spaces, Jing-Song Huang, Hong Kong University of Science and Technology
Gelfand Pairs Associated with Heisenberg Groups, Gail Ratcliff, University of Missouri at St. Louis
First Occurrence for the Dual Pairs U(p,q), U(r,s), Annegret Paul, University of California at Berkeley
Group Actions on Central Simple Algebras, Daniel Sage, University of Utah;
Langlands Parameters and Derived Functor Modules, Paul Friedman, Aarhus University;
Equivariant D-modules on a Semisimple Lie Algebra and a Homomorphism of Harish-Chandra; Markus Hunziker, Brandeis University;
The (Symplectic) Geometry of Mackey Theory, Francois Ziegler, Pennsylvania State University;
The Differential Operator for the Double Fibration Transform: An Example, Jodie Novak, University of Northern Colorado;
Classification Theorems for Unitary Representations, Susana Salamanca-Riba, New Mexico State University.
Cross Program Activities
Held four times each week, this years Cross Program activities were structured around three themes: the Third International Mathematics and Science Study (TIMSS), the NCTM Standards, and geometry curriculum. In addition, one formal presentation each week was on a mathematics theme. Formal presentations were as follows:
Lie Groups and Geometry, Roger Howe, Yale University
Policy Implications from the Third International Mathematics and Science Study (TIMSS), William H. Schmidt, Michigan State University
The Mathematics of Juggling Patterns, Allen Knutson, Massachusetts Institute of Technology, (Research Program participant) and Greg Warrington, Harvard University (Graduate Summer School participant);
Setting the Stage for Standards, Gail Burrill, University of Wisconsin, past-president of National Council of Teachers of Mathematics (NCTM)
Tilings, Quasi-Crystals and Continuous Symmetry, Lorenzo Sadun, University of Texas at Austin (Undergraduate Faculty Program participant)
How to Read Your PCMI T-Shirt, Robert Bryant, Duke University, PCMI Steering Committee
Each week on Thursday there were integrated small group discussions on the topics of TIMSS, NCTM Standards, or Geometry Course Curriculum, and these small groups made presentations to the larger group during the Cross Program Activity on Fridays.
The computer lab, under the direction of James King of the University of Washington, was equipped with a variety of computer hardware and software, providing Windows, Macintosh, and Linux platforms for participants use. The lab was a valuable resource for educational and computational work as well as Internet access, and it was in use around the clock.
This year the computer lab was used to test new technology specifically designed for the use of mathematicians and mathematics educators by the Microsoft Corporation. The project, known as MathWorld, is still in development, however components of it were available during the PCMI Summer Session, specifically V-chat and Netmeeting. Microsoft generously donated all of the development for the technology, as well as the software and the server from which it was run. In addition, they sent several people to PCMI to assist with setting up the lab and provided funding for an on-site technical support person for the entire three weeks.
Participants had numerous opportunities for informal and social interaction throughout the summer session, including pizza parties, field trips to area attractions, barbecue dinners, and daily breakfast and lunch. An outdoor tent served as the dining area and was a popular venue for both social and academic gatherings. All of these resources and activities were designed to encourage casual interaction and promote a sense of community among the participants. The professional relationships and friendships formed in this way, extending throughout the mathematics community, are among PCMI's most important goals.
On Tuesday, July 21, 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 Park City Community Church. The program for the evening included works by Beethoven, Brahms, and Scriabin. On the preceding Sunday evening, Robert Taub gave a pre-concert lecture to the PCMI participants.
Progress continues on the publication of the lecture notes from each years Graduate Summer School in the PCMI Lecture Series. This past year saw the publication of Volume 4, Gauge Theory and Four Manifolds, from the 1994 Graduate Summer School. This volume joins Volume 1, Geometry and Quantum Field Theory, Volume 2, Nonlinear Partial Differential Equations in Differential Geometry, and Volume 3, Complex Algebraic Geometry, from the 1991, 1992, and 1993 programs, respectively. Volume 5 is expected to be published before the end of 1998, and Volumes 6 and 7 will be in production by the end of the year. There are plans to publish material from the High School Teacher and Undergraduate Programs. The PCMI Lecture Series allows material generated during the summer session and academic year site program to be shared with a wider audience.
Many 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 11-21. 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 33 participants included graduate students, undergraduates, postdoctoral scholars, and senior researchers. The Womens Program is funded by the National Science Foundation.
The undergraduate lecture series, Representations of Finite Symmetry Groups, was given by Lisa Mantini, Oklahoma State University. The graduate lecture series, Theory of Unitary Representations, was given by Leticia Barchini, Oklahoma State University, Rebecca Herb, University of Maryland, and Susana Salamanca-Riba, New Mexico State University. Guest lecturers were Mark Goresky, Member, Institute for Advanced Study, and Sigurdur Helgason, Massachusetts Institute of Technology, Member, Institute for Advanced Study.
Karen Uhlenbeck led a Women in Science Seminar, a daily informal discussion group which included biographical readings on such women mathematicians as Emmy Noether and Evelyn Fox Keller. Special guests to the Women in Science Seminar were Bhama Srinivasan, University of Illinois at Chicago, LaVerne Gill, Princeton Theological Seminary, and Joan Feigenbaum and Anna Gilbert, both of AT&T Research.
The IAS/Park City Mathematics Institute and the Mentoring Program for Women in Mathematics mutually support and interact with each other. The Womens Program has enabled the PCMI 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 further during the PCMI Summer Session, thus encouraging these women to stay in the field of mathematics.
The Womens Program Committee assists the organizers in planning and promoting the program and recruiting lecturers and participants. Members include: Fan Chung, Professor, University of Pennsylvania; Ingrid Daubechies, Professor, Princeton University; Antonella Grassi, Professor, University of Pennsylvania; Sarah Greenwald, Graduate Student, University of Pennsylvania; Nancy Hingston, Professor, The College of New Jersey; Rhonda Hughes, Professor, Bryn Mawr College; Robert MacPherson, Professor, Institute for Advanced Study; Jane Scanlon, Professor, Rutgers University; Diane Souvaine, Professor, Rutgers University; and Lisa Traynor, Professor, Bryn Mawr College,.
Representations of Finite Symmetry Groups, Lisa Mantini, Oklahoma State University
Theory of Unitary Representations, Leticia Barchini, Oklahoma State University; Rebecca Herb, University of Maryland; and Susana Salamanca-Riba, New Mexico State University
Theta Correspondence, Anne-Marie Aubert, Ecole Normale Superieure; Annegret Paul, University of California at Berkeley
Peter-Weyl Theorem with motivation, Sigudur Helgason, Massachusetts Institute of Technology and Institute for Advanced Study;
SL(2,R) - Introduction to infinite dimensional representations, Sigurdur Helgason, Massachusetts Institute of Technology and Institute for Advanced Study
Classification of Lie Groups; Mark Goresky, Institute for Advanced Study;
List of Seminars
Eigenvalues of Invariant Operators Associated with Multiplicity Free Actions, Gail Ratcliff, University of Missouri at St. Louis
Geometric Interpretation of Macdonald Polynomials and N! Conjecture, Carol Chang, University of California at San Diego
On the Plancherel Formula for Hecke Algebras, Anne-Marie Aubert, Ecole Normale Superieure
Howe Duality and Unbounded Realizations of Symmetric Domains, Christina Leslie, University of California at Berkeley
First Occurrence in the Theta Correspondence for Real Dual Pairs, Annegret Paul, University of California at Berkeley
Creating Female-Friendly Classrooms, Gail Ratcliff, University of Missouri at St. Louis
Cells and Representations of Symmetric Groups, Bhama Srinivasan, University of Illinois at Chicago
Representations of Graded Hecke Algebras, Cathy Kriloff, Idaho State University
The Orbit Method and P-adic Lie groups, Monica Nevins, Massachusetts Institute of Technology
Twisted Torsion on Compact Hyperbolic Spaces - a Vanishing Result, Maria Fung, Cornell University
Regular Functions on Nilpotent Orbits, Dana Pascovici, Massachusetts Institute of Technology
Realizing Representations with the Penrose Transform, Jodie Novak, University of Northern Colorado Feminism VS the Working Woman Scientist, Women-in-Science Seminar
College Teaching as a Career; Women in Science Seminar Panel Discussion; moderator: Janet Talvacchia, Institute for Advanced Study; panelists: Nancy Hingston, College of New Jersey; Rhonda Hughes, Bryn Mawr College; Stephanie Frank Singer, Haverford College; Aimee Johnson, Swarthmore College; Cynthia Curtis, Princeton University; Antonella Grassi, University of Pennsylvania
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Phillip A. Griffiths, Director, Institute for Advanced Study
Hyman Bass, Adrain Professor of Mathematics, Columbia University
Ronald L. Graham, Chief Scientist, AT&T Research
Shirley A. Hill, Professor Emeritus, University of Missouri-Kansas City
Leo F. Klagholz, 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.
John C. Polking, Rice University
David Vogan, Massachusetts Institute of Technology
Jeffrey Adams, University of Maryland
Editor, PCMI Lecture Series:
Daniel S. Freed, University of Texas at Austin
Graduate Summer School:
David R. Morrison, Duke University
High School Teacher Program:
Naomi Fisher, Mathematics Education Reform Network, University of Illinois at Chicago
Cynthia Hays, Teacher of Mathematics and Department Chairperson, McCallum High School Austin, Texas
High School Teacher/Computer Program:
James R. King, University of Washington
Mathematics Education Research Program:
Joan Ferrini-Mundy, National Research Council
Timothy Kelly, Hamilton College
Nathaniel Whitaker, University of Massachusetts at Amherst
John Morgan, Columbia University
Research Program/Women's Program:
Karen Uhlenbeck, University of Texas at Austin
Undergraduate Faculty Program:
Daniel Goroff, Harvard University
Robert L. Bryant, Duke University
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please send comments or concerns to C. Jordan