Summer Institute 1996

Project Descriptions

Project Afton

This is a model of the flow of very viscous fluids on horizontal and inclined rigid surfaces. Such flows are of interest in a large variety of engineering problems such as uniform coating of magnetic media, and also becuase of fundamental issues in the underlying physics. At present, gravity dominated flows into a dry region which is surrounded by fluid are being explored. Codes are being developed for the numerical solution of the partial differential equation which groverns the flow, and for for visualization of the output. A demonstration video of a simple case which duplicates the results of some experimental work numerically has been produced. Now the codes are being adapted to allow the investigation of more general situations which have not been studied either experimentally or theoretically. This project will involve the investigation and implementation of methods for analyzing and visualizing the numerical output to provide an understanding of what the simulations are describing.

Essential qualifications: C and Fortran programming and some familiarity with partial differential equations (undergrad methods of applied mathematics cour level, Boyce and DiPrima textbook level).

Combinatorial Models of Algebraic Surfaces

The goal of this project will be to develop and implement algorithms and supporting code for the construction and visualization of combinatorial models of smooth surfaces in dimension three. These models will be obtained by pasting triangles together following a combinatorial scheme and represent the real solutions of a homogeneous polynomial in three variables. This tool will be used to explore open questions about the number of component of the surfaces with respect to the degree of the polynomial. The ultimate goal is to integrate such a program to the PISCES platform.

Essential qualifications: C programming and some knowledge of multivariable calculus.

The DS Tool Project

Dynamical systems theory is devoted to studying the equations which model systems that evolve in time. This includes such diverse areas as the motion of pendulums and spacecraft, the flexing of beams and shells, the spread of disease, and the bursting of neurons. At the Geometry Center, we are developing software to aid in the study of such systems of equations. Students will work with us to put together demonstration software in hypertext, the language of World-Wide Web pages, to show off the capabilities of our computational and visualization software for dynamical systems.

Essential qualifications: C programming and some background in differential equations.

A more detailed discussion of the DS Tool Project is available.

Software for Euclidean and non-Euclidean transformations

Homogeneous transformation matrices are ubiquitous in computer graphics, a significant part of the Geometry Center's work. We have various disparate programs for constructing such matrices -- representing Euclidean similarities, or hyperbolic or spherical isometries -- and converting among different forms, but we'd like a general-purpose software tool, that among other things is able to factor a matrix into primitives such as rotations and translations. For Euclidean transformations, there is a published factoring algorithm; for non-Euclidean cases, we'll have to invent one.

Essential qualifications: C or a related language, knowledge of linear algebra, and an interest in learning non-Euclidean geometry.

The Java Math Browser

Java is a programming language, developed by Sun Microsystems as a language for implementing executuable web pages. The goal of this project is to use Java to extend current web browser technology to communicate mathematics more effectively. Since Java is still under development, the exact specification of this project may evolve between now and next summer. However, the project will certainly focus on Java programs for manipulating and adding behaviors (like the ability to link to other web pages) to 3D geometric objects defined in OOGL, an "Object Oriented Graphics Language" developed at the Geometry Center.

Essential qualifications: familiarity with some object oriented programming language.

The K-12 Curriculum Project

The 1996 Summer Institute will provide a unique opportunity for students who are interested in Mathematics Education. The Kaleidotile software and the Shape of Space video have both been the focus of previous summer projects. Preliminary versions exist of curriculum materials that support the use of those tools at the secondary and/or undergraduate levels. Students this summer will work under the supervision of an experienced editor on a subset of those materials, to finish getting them into a publishable format.

Essential qualifications: mathematical background in group theory and/or topology and programming skills (e.g., C, Perl) or visual design skills (e.g., graphics, videos, hypertext).

The Oriented Matroid Manipulator Project

Convex polyhedra and directed graphs are two of the many seemingly unrelated mathematical objects that can be described inside the theory of oriented matroids. What these two structures have in common are notions of incidence (e.g points contained in planes, edges contained in trees of the graph) and linear independence (e.g points that do not lie on a line, or edges in a graph not in a loop). The immediate goal of the project is to implement algorithms to compute oriented matroids and exchange between any two of their representations. The ultimate goal is to provide software for investigating problems such as the subdivisions of high dimensional cubes. For example, what is the smallest number of tetrahedra needed to subdivide a three dimensional cube?

Essential qualifications: C programming and use of methods in linear algebra and combinatorics.

The PISCES Team

The equation x² + y² + z² = 1 defines the sphere implicitly as the set of points that satisfy a certain equation. In general, implicitly-defined surfaces are difficult to compute and visualize, especially if the surface contains singularities such as cusps or curves of "self intersection." This summer project will attempt to develop and implement algorithms and supporting code for the computation of implicitly-defined surfaces with singularities.

Simple examples of the kind of surfaces we will be working with include the intersecting planes defined by y² + z² = 0 and the cone defined by x² + y² - z² = 0. The code we develop will be incorporated into the Pisces Software, which is used by mathematicians, scientists, and engineers worldwide. Therefore the code written will be heavily documented by the participants and will be frequently reviewed by the project leader.

Essential qualifications: C programming, data structures, multivariable calculus, linear algebra.

Exhibits for the Topological Zoo

The Topological Zoo is part of the Geometry Center web site, and consists of multimedia exhibits illustrating a variety of mathematical topics. A typical Zoo exhibit contains pages explaining the underlying mathematical theory, images, animations, and even embedded interactive applications. The purpose of the Zoo is to provide mathematics students and teachers with a graphical reference library. The goal of this project is to create two new "exhibits" for the zoo. One will focus on non-Eucliean geometry, and the other on the theory of curves and curve evolution.

Essential qualifications: a strong math background including knowledge of multivariable calculus.

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Created: Oct 12 1995 --- Last modified: Jan 29 1996
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