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Participating students will receive a stipend of $3,200. Participants will be responsible for the cost of travel, food and housing (dormitory housing with full board is available at a cost of about $1,600 for the eight-week program). Descriptions of the programs and application guidelines are below. There are funds for travel to conferences where participants present papers. There are also some funds for books and supplies.
Geometric group theory lies at the crossroads of geometry, topology, and group theory, yet many of its questions can be attacked by undergraduates. In the first two weeks of the REU, we will explore what it means for a group to have a geometry. We'll start with a hands-on introduction to non-Euclidean geometries, tilings, how to draw graph pictures of groups, and how groups can act on spaces. We will also look at an assortment of groups arising from topology and geometry, such as braid groups, reflection groups, and fundamental groups of surfaces. We will discuss algorithmic questions, such as how to tell when two strings of group generators actually determine the same element in the group.
In the remaining five weeks, students will work on research projects, either individually or in groups of 2-3.
The REU will include an introduction to some of the tools of the trade, including library and online resources, Latex (a program to typeset mathematics), and how to get research published.
Students are encouraged to present papers at the Pi Mu Epsilon section of MathFest 2005 in Albuquerque, August 4-6, or at other conferences.
UIUC has a strong geometric group theory community. There will be guest lectures by local faculty, as well as the opportunity to visit working seminars.
Inverse problems, and the algorithms that can solve them, are of great importance in mathematics, engineering, and science. Mathematical inverse problems are at the core of the technology in tomography and CAT scans, MRI, spectroscopy, ultrasound, radar, and many other technologies. One famous inverse problem is the problem of phase retrieval which arises in the experimental uses of diffraction to determine internal structure. In this situation, the diffracted wave form contains complete information about the object that causes the diffraction, but most methods of measuring this wave form yields only the intensity of the signal, and not the phase information. In many experimental situations one is just measuring the intensity of the Fraunhofer diffraction, which is approximately a Fourier transform of the original distribution that caused the diffraction. So the phase retrieval problem becomes the mathematical problem of determining a distribution from the modulus of its Fourier transform. All types of interesting problems and questions in algebra and analysis are involved in solving this particular inverse problem. Phase retrieval, and to a lesser degree other inverse problems, will be the focus of this REU. More information about this REU is available at www.math.uiuc.edu/~jrsnbltt/reu_2005.html.
Students will work on research projects, either individually or in small groups. Students will have the opportunity to use computer labs and learn to write mathematics in TeX. Students are encouraged to present papers at the Pi Mu Epsilon section of MathFest 2005 in Albuquerque, August 4-6, or at other conferences.
To apply to the program, please send:
Questions about this application should be directed to Randy McCarthy, randy@math.uiuc.edu
Last modified February 18, 2005