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Summer Research Experience: illiMath2002

10 June - 2 August
Program Director: George Francis
email: gfrancis@math.uiuc.edu

In this program students with strong interest and some experience in mathematical graphics participate in research programs of their associated mentor (AM). The program provides the facilities (and any necessary training in their use) for the student to assist their AM in creating visual materials illustrating research in a variety of media, including the production of custom soft-ware, web-pages, animations, print quality images and virtual environments (CAVE and CUBE). The program also provides the student with tutorials in the mathematical background and lectures by specialists in the subjects of their projects. Similar REU programs, illiMath2001 and Audible Sketchpad for the CAVE, were conducted for the past three years.

Please check the Math Department pages for details on applying and getting paid.

Tentative projects for illiMath2002 will be posted in this location over the next several months.

"The Illustrated Analyst"; PI: Brad Henry, AM: Dr. Karen Shuman, VIGRE Postdoc UIUC Math Dept.

The galaxy of mathematics known as Analysis does not lend itself easily to graphical illustration. This project will prove matters otherwise. We will investigate how to hear Fourier transforms, to see minimization algorithms in nonlinear programming, and how to animate certain problems in signal processing witht he tools of computer graphics.

"Train Tracks"; AM: Dr. Peter Brinkman, Doob Postdoc UIUC Math Dept.

Low dimensional topology and geometry, and the associated combinatorial group theory is a rich source of experimental mathematics that lends itself to graphical expression and VR visualization. Generally, illustration lag behind discovery by several years. Not so with these three projects on surface homeomorphism and their mapping tori.

"Alice on the Eightfold-Way"; AM: Dr. Ben Schaeffer, ISL, Beckman Institute, UIUC; CM: Dr. Jeff Weeks, Canton, NY and Prof. John Sullivan, UIUC

The Integrated Systems Lab (ISL) of the Beckman Institute has built the CUBE, which is the first rigid walled 6-sided CAVE in the US. We plan to continue with our long-range project to implement Thurston's Geometries in the CUBE. A collaboration between illiMath2001 students, ISL scientists, and extramural associated mentor, Dr. Jeff Weeks, achieved a proof-of-concept implementations of five of Thurston's eight geometries. These are programmed in syzygy, the evolving CUBE library for distributed graphics computing on Linux, and NT clusters. For the purpose of learning to program the CUBE, we have a small, dual-boot PC cluster running in syzygy in our lab space.

"Phillips and DeWitt Eversions"; PI: Wendy Hubbard, AM: George Francis.

There remains several sphere eversions that have not yet been made computational. Two (technically related) of these are the Tony Phillips eversion (Scientific American, 1966) which is the first published one, and Bryce DeWitt (maybe non-) eversion, which was proposed (but never "proved") in 1967 at the same Batelle conference which spawned the Froissart-Morin eversion. Both eversions are based on horizontal slices (plane curves) of a surface undergoing a regular homotopy.

"Bishop Coaster"; SPI: Ben Farmer and Abdul Hamide, AM: Prof. Paul McCreary, Xavier University, New Orleans;

A remarkable feature of DNA is its ability, during reproductive cycle, to unpack and repack its one-meter length of genetic code into the tiny package (chromosome) that fits inside a cell nucleus. One current model for the shapes DNA can take in this process considers the framing of the core curve of the double helix. Because of its tendency to greatly twist its integral curve in response to small perturbations of its parameters, the Frenet-Serret frame may seem to be the natural choice. Again, it may be the more stable, less sensitive Bishop frame which holds the greater promise. This project is to develop graphical software tools for biologists. With it, scientists can build and refine the shape of space curves, and create camera paths through DNA models.