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Topic: Utah Summer Geometry Institute
Replies: 1   Last Post: Feb 2, 1993 10:03 AM

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Gene Klotz

Posts: 30
Registered: 12/3/04
Utah Summer Geometry Institute
Posted: Feb 2, 1993 9:56 AM
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N a t i o n a l S c i e n c e F o u n d a t i o n *
(Funding has been requested)


June 20, 1993-July 17, 193
Park City, Utah

H i g h S c h o o l T e a c h e r s
U n d e r g r a d u a t e / G r a d u a t e S t u d e n t s
R e s e a r c h e r s i n G e o m e t r y




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.


The Summer Institute for Teachers is designed to promote
fundamental changes in the content and teaching of high school

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.

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

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.

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

Computers and Geometry, taught by James King, University of
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

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

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.


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

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.




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,, (312) 413-3749.


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

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