CITY COLLEGE OF NEW YORK% COLUMBIA UNIVERSITY% DUKE UNIVERSITY% IDAHO STATE UNIVERSITY % UNIVERSITY OF ILLINOIS AT CHICAGO % RICE UNIVERSITY % UNIVERSITY OF TEXAS AT AUSTIN % UNIVERSITY OF UTAH % UNIVERSITY OF WASHINGTON
The Summer Geometry Institute is a multi-university endeavor to integrate the subject of geometry, from high school education to advanced research. To this end, the four week institute will bring together geometers from all sectors of geometry education and research, in a unique and congenial setting. The program is designed to foster communication and deeper insight into geometry at all levels. The conference will consist of four programs operating concurrently: a program for high school teachers of geometry, a program for undergraduate students, a graduate summer school, and a research program. The High School Teacher Program is a two-year program to begin in 1993; it continues through the academic year 93-94, and includes another summer session in 1994 and academic year 94-95 (pending funding). Undergraduate Program students will attend regular class and problem sessions, including a computer lab. The Graduate Summer School is centered around lectures by prominent geometers. The Research Program consists of workshops and informal lectures, the program itself being determined by the participants. Each year a particular topic in geometry is the focus for the Graduate Summer School and the Research Program. Special programs and evening events of general interest are optional, and movement between courses and lectures in different programs is encouraged (see descriptions on page 12). Opportunities for informal and social interaction abound, both within and between groups, making for a gratifying sense of community.
------------------------------------------------------------ PROGRAM FOR HIGH SCHOOL TEACHERS
The Summer Institute for Teachers is designed to promote fundamental changes in the content and teaching of high school geometry.
The 1993 Summer Institute for Teachers is part of a two year program to develop ongoing classroom reforms leading to fundamental changes in the content and teaching of high school geometry. The program is built around a corps of some 35-40 high school geometry teachers from sites at participating universities. The university sites for the 1993-1995 RGI High School Teacher program are: Columbia University, City College of New York, Duke University, Idaho State University, the University of Illinois at Chicago, the University of Texas at Austin, the University of Utah, and the University of Washington. The full group of teachers meets at the Summer Institutes in 1993 and in 1994; and teachers at each site meet throughout the school year following the summer institutes.
The program regards the teacher as the primary agent for promoting and implementing classroom reform. It is designed to enable teachers to make informed decisions about curricular and instructional reforms and to confidently implement change. The summer experience is a time of intensive professional education. By deepening their knowledge of classical geometry and becoming acquainted with geometry as it is investigated and used today, teachers can undertake the transformation of high school geometry. During the school year, teachers focus on making ideas work in the classroom and communicating efforts in reform with other teachers. Following the first summer institute, relatively modest changes are introduced to prepare for deeper changes that will be made in the school year following the second summer institute. University faculty, serving as consultants to the teachers, organize site meetings for teachers to share their experiences in implementing changes. The RGI Teacher Newsletter, written and edited by the RGI teachers and circulated in the fall, winter and spring, helps teachers stay in touch with one another during the school year.
The instituteUs approach to the study of geometry begins with the basic ideas characterizing Euclidean geometry: rigid motions and congruency, flat space, perpendicularity, and parallelism, size-changing transformations and similarity.
It then investigates the evolution of these ideas in non- Euclidean geometry and their relevance to the world around us. Persuasive reasoning and rigorous thinking build on intuitive and experimental activities.
To turn this vision of high school geometry into reality, the participants coordinate a study of higher geometry with curriculum development and classroom implementation. Strong collegial bonds among the individuals in each site group and among the different groups are essential for this cooperative effort.
THE SUMMER INSTITUTES At the first summer institute, teachers immerse themselves in the world of geometry, developing insights from their own learning experiences for presenting geometry to their students. During the second summer institute, teachers have the opportunity to plan projects suggested by their teaching experiences from the previous school year. Teachers give presentations discussing the changes they have made and the materials they have developed, and give demonstration lessons of classroom activities they have used.
The formal program for the summer institute is organized around three courses (listed on the right), and special programs by guest speakers and faculty of the summer institute. Most important, however, are the many opportunities for discussion among the teachers themselves and with participants from the other institute programs.
The Summer Institute Program for Teachers, as well as the programs for undergraduates, graduate students, and researchers in geometry, will stress informal interactions and a sense of community among the participants of the various programs.
COURSES AND LABORATORIES Geometry for the Classroom, taught by Naomi Fisher, University of Illinois at Chicago. This course probes both the content and teaching practices of high school geometry. Using geometric ideas from the standard curriculum and other sources, teaching strategies to promote active student engagement are considered. The course uses Clemens and Clemens, Geometry for the Classroom (Springer-Verlag, 1991) as a resource.
Computers and Geometry, taught by James King, University of Washington. This course explores geometry topics, familiar and less familiar, using software that allows participants to dynamically manipulate geometric objects for visualization, investigation, and problem- solving. After gaining hands-on experience with the software, teachers develop ideas for using computers in their own classrooms. Macintoshes and Windows PCs are used; no prior experience is assumed. Software includes GeometerUs Sketchpad, CABRI Geometre, and Logo.
Advanced Classical Geometry, taught by James Carlson, University of Utah, and John Polking, Rice University. The development of geometry from Euclid to modern times is studied, highlighting the pivotal areas that have led to the explosive development of advanced geometry during this century. The notions of curved space, relativity, transformation groups, and special curves and surfaces will be studied. Specially prepared class notes are provided by the institute.
In a research environment with a specific scientific goal, undergraduate students will gain insight into advanced geometric concepts.
The Summer Institute will provide opportunities for talented undergraduate students to enhance their interest in mathematics in general and geometry in particular. We are seeking undergraduate students at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, we will divide them into two groups, introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be ample time for study groups and individual projects guided by institute advisors, as well as other activities.
The Summer Institute Program for Undergraduates, as well as the programs for high school teachers, graduate students, and researchers in geometry, will stress informal interaction and a sense of community between the participants of the various programs.
A special program has been developed by the Summer Geometry Institute in conjunction with MSRI. Women who are accepted for the graduate or undergraduate program of the Summer Geometry Institute will automatically be invited to attend a special two- week program at MSRI in May, 1993. You need not apply separately for the women's program. See page last pages for details.
The Computer Projects All of the undergraduate students, regardless of grouping, will participate in the Computer Projects Activity. After a brief introduction to the computer facilities, the students will be presented with several problems (or project descriptions) involving the concepts of algebraic geometry from which they will chose one or more to develop into projects. The students will work teams, learning how to use computers and programs like Maple, Mathematica, Macaulay, or GeometerUs Sketchpad to explore questions in algebraic geometry and related subjects. The students will also be encouraged to develop their own projects in consultation with the Computer Projects supervisors. No prior experience with computers is absolutely necessary, but a beginning familiarity with a symbolic manipulation program such as Maple or Mathematica is strongly encouraged.
The Geometry Discussion Group This activity will be mainly oriented to the introductory undergraduate group and the high school teachers. It will consist of four discussion-seminars, each consisting of two meetings which will be held at the beginning and end of a week of the institute. Each such discussion-seminar will be led by an institute advisor and its focus will be on some topic of elementary or classical geometry which is mentioned in introductory geometry courses, but whose development and history are not usually discussed until late in undergraduate or early graduate mathematics training. For example, one such topic will center on the question RWhat do we really mean when we say that you canUt trisect an angle with ruler and compass?UU The discussion will be intended to highlight not only the history of the problem, but its place in the development of geometry as a subject.
An Introduction to Algebraic Geometry This activity is intended for the more advanced students and will consist principally of Miles ReidUs introductory course on Algebraic Geometry (see pages on Graduate Summer School). Some of the more advanced computer projects will be designed to illustrate the concepts encountered in these lectures, and serve as springboards for further investigation. To participate in this program, students should have had some background in abstract algebra, being familiar with the fundamental notions of groups, fields, rings, and ideals. A first course in complex variables might also be helpful, but is not absolutely necessary.
The Problem Book This activity will be intended mainly for the advanced students. Its purpose is to develop additions to the RGI Undergraduate Problem Book, a resource book in geometry which the RGI is assembling. The students will collect and study problems in geometry and algebraic geometry which are related to the material they are encountering in the computer projects and course work. They will produce write-ups of these problems and their resolution which will then be added to the Problem Book. The eventual goal is to produce a good source of problems for students who want to further their understanding of geometry by individual study.
The Graduate Summer School offers an intense introduction to problems and techniques in an active field of research.
The Graduate Summer School is an intensive introduction to a current aspect of research in geometry. The topic for 1993 will be Higher Dimensional Complex Geometry.
The Graduate Summer School bridges the gap between a general graduate education in geometry and the specific preparation necessary to do research on problems of current interest. Graduate students should have completed basic graduate courses in algebra and geometry. They will usually be in their second or third year of graduate school. While most of the participants will be graduate students, we also anticipate that some recent postdoctorates and researchers will attend.
The main activity will be a set of intensive short courses offered by leaders in the field. These lecture series will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The tradition of informal study groups, begun in previous summers by the students themselves, will again be encouraged in 1993. The institute will help to facilitate the formation of such groups.
Graduate Summer School participants can also take advantage of opportunities offered by the other programs. We expect that students will audit some of the sessions in the Research Program, and some graduate students will be actively involved in the undergraduate programs. Graduate students will also be asked to donate some time to various volunteer activities related to running the Summer Institute.
The Graduate Summer School, as well as the programs for high school teachers, undergraduates, and researchers in geometry, will stress informal interaction and a sense of community between the participants of the various programs.
SUMMER SCHOOL LECTURE SERIES
There will be six lecture series, of 5-10 lectures each, offered during the four-week institute. Short course descriptions are listed below. (Schedule is subject to change.)
Weeks One and Two
A First Introduction to Higher Dimensional Geometry (10 lectures) Miles Reid, University of Warwick, U. K. This is the introductory level course of the Graduate Summer School, designed to be accessible, in the main, to beginning graduate students. The aim is to study examples of curves, surfaces and threefolds, and later in the course, to use these to illustrate the basic ideas of Mori theory. Background reading: M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
Introduction to Intersection Cohomology and Hodge Theory (10 lectures) Eduard Looijenga, University of Utrecht, the Netherlands This course will cover the basics of intersection cohomology and Hodge theory on (possibly) singular varieties, paralleling the classical theory of Lefschetz in the smooth case. The course is designed to be accessible to students with a first course in algebraic topology and a first course in complex manifolds.
Linear Series on Varieties (5 lectures) Robert Lazarsfeld, UCLA This course will aim to provide an introduction to linear series on algebraic varieties. As in the case of curves, important geometric properties are sometimes most clearly revealed by studying the projective embeddings of a given variety. To the extent possible, we will focus on both classical and modern approaches to the subject.
Weeks Three and Four
Higher Dimensional Complex Geometry (10 lectures) J nos Koll r, University of Utah This course will cover the fundamentals of the Mori program to produce minimal models of surfaces and threefolds and to classify them through the use of extremal rays. We will also consider applications of these ideas to other questions in algebraic geometry. Higher Dimensional Complex Geometry, Asterisque vol. 166 (Math. Soc. France, 1988), will be the fundamental reference for material in this course.
Introduction to Mirror Manifolds (5 lectures) David Morrison, Duke University One of the most spectacular developments in threefolds in the last few years concerns a totally unexpected symmetry among families of Ricci-flat threefolds. Discovered by theoretical physicists studying Rsupersymmetry,S this still conjectural theory would, for example, allow the reduction of RimpossibleS computations concerning curves on threefolds to relatively well- known computations about variations of Hodge structure. This course will introduce mirror symmetry and present some of the mounting evidence for the correctness of the theory.
Birational Transformations with Small Centers: Flips and Flops (6-8 lectures) Shigefumi Mori, R.I.M.S., Japan At the heart of the theory of minimal models and the resulting classification of threefolds is the ability to control and organize birational transformations for which the set of adherence values at any point of the domain or range space has dimension less than or equal to one. These birational transformations are the so-called flips and flops, which will be the subject matter of this course.
The Research Program offers researchers a stimulating environment for discussion, collaboration, and individual work.
The research topic for the 1993 Summer Institute is Higher Dimensional Complex Geometry.
Complementing the highly structured Graduate Summer School, which is directed at younger mathematicians, this program addresses the needs of many geometers already carrying out research. The program is designed to introduce active areas of research around specific topics. The informal format generates a lively exchange of views and information, with established and newer researchers working together spontaneously. We especially encourage new and recent Ph.D.s to apply to the Research Program if they are working in the field of Higher Dimensional Complex Geometry.
For 1993, there will be at least one formal session per day. Topics for additional workshops and working groups will be chosen at the beginning of the institute by the participants themselves.
Research Program participants may wish to sample some of the Graduate Summer School lectures. It is also expected that many researchers will be interested in other aspects of the institute as well, occasionally giving talks or seminars for the undergraduates or high school teachers. The rest of the time will be free for work and informal discussion.
The Summer Institute Research Program, as well as the programs for undergraduates, graduate students, and high school teachers, will stress informal interaction and a sense of community between the participants of the various programs.
As funding in each area is limited, some guidelines for application are provided here, so that interested postdoctorates can apply to the correct program. Some mobility between programs is expected (and encouraged), but postdoctorates should carefully read all of the information, and then apply to the one program most appropriate for them.
Graduate Summer School: Funding will go to graduate students, and to those postdoctorates who are not mainly interested in the research program, or are not working in the field of Higher Dimensional Complex Geometry. Postdoctorates should also be within four years of the date of their Ph.D.
Research Program: Funding will go to established researchers, and to postdoctorates in the field of Higher Dimensional Complex Geometry. (This program is NOT for graduate students.) Recent Ph.D.s with limited expertise in the main topic should apply to the Graduate Summer School instead.
JOINT SEMINAR: ISSUES IN THE TEACHING OF MATHEMATICS
Participants in the Research and High School Teacher Programs will be invited to participate in a seminar on Issues in the Teaching of Mathematics. This seminar will meet three hours a week throughout the summer institute. Each week's program will feature an outside guest speaker who will give a presentation in the seminar and participate in the seminar discussions. Additional information: Naomi Fisher, University of Illinois at Chicago, U37158@uicvm.uic.edu, (312) 413-3749.
JOINT WOMEN'S PROGRAM WITH MSRI
Women who apply to and are accepted for the graduate or undergraduate program of the Summer Geometry Institute will automatically be invited to attend a special two-week program from May 17-28, 1992 at the Mathematical Sciences Research Institute in Berkeley, California. Women students do not need to apply separately for this program. Funding for this special program has been requested. This program will introduce participants to career opportunities in research mathematics in general, and in algebraic geometry in particular. Small problem- solving groups will work with MSRI mentors on "mini-research" projects at a level appropriate for the individual participant. Some of these mentors will continue their work with participants of the program during the Summer Geometry Institute. Additional