University of Georgia, Summer 2005
The VIGRE Research Experience for Undergraduates in Mathematics at UGA will run for five weeks, May 16 - June 17, 2005. Students will work in teams guided by faculty mentors. Research teams will meet approximately 30 hours a week. The research experience is open to all majors, and it is not restricted to University of Georgia students.
Twelve stipends (either $1500 plus dorm and meals, or $2250) will be awarded to United States citizens and permanent residents. Participants may register to receive course credit. Students are responsible for the cost of tuition and fees. We encourage students to stay in residence halls and to subscribe to the student meal plan. Travel expense money is available for non-UGA students.
A final research report is required of all students.
This program is supported by a VIGRE grant from the National Science Foundation and by the Franklin College of Arts and Sciences of the University of Georgia.
Research topics
Ramsey Theory - Akos Magyar
Ramsey theory is a beautiful area, as its principal results are both natural, easy to formulate and non-trivial. Roughly speaking, it deals with finding regular structures in large but otherwise arbitrary sets. The theorem of Ramsey itself (often called the "party theorem") can be formulated as follows: let k be a natural number. Then for any sufficiently large dinner party you can find a group of k people, who are either (a) mutual acquaintances, or (b) complete strangers to one another. The smallest such dinner party is only known however for k=3 and k=4.
There are three basic approaches to such problems: via combinatorics, ergodic theory and elementary Fourier analysis. A famous example is Szemeredi's theorem, stating that if a set contains a "positive proportion" of the natural numbers, then it must contain arbitrary long arithmetic progressions. This was first proved via combinatorics, and elements of the proof have become influential on the development of the subject. Later a different proof was given using ideas from dynamics going back to Poincare, and more recently by a spectacular application of Fourier analysis (which won the Fields medal in '98).
As deep as such results may be, nevertheless they require little formal knowledge, and our aim will be to understand the basic ideas behind them through exercises, problems and studying articles, with an eye toward some related open problems.
Polynomial Splines, Bezier Curves, Barycentric Coordinates - Tatyana Sorokina
The design of surfaces (airplanes, automobiles, molecules) and image processing (animation, compression, transformation) start with the concept of polynomial splines. Polynomial splines are pieces of polynomials glued together. However, cartesian coordinates are not good enough, and barycentric coordinates are what is actually needed. In the first part of the research project we will learn how to use barycentric coordinates to construct polynomials of one and two variables. The second part of the project will focus on how to join several pieces of polynomials together to create a curve or a surface of a desired shape.
The methods used will be a blend of basic linear algebra, geometry, and calculus of several variables. There will be a minimal amount of formal teaching, and a lot of easy, not so easy, and very difficult problems to solve. Students will be asked to present their solutions to the group. Students interested in computer science will have an excellent opportunity to apply programming skills. Students who like theoretical aspects will not be forced to program.
Program details
Schedule and requirements
The program will run during UGA's May session. Research teams will meet approximately 30 hours a week. Participants may not enroll for other classes or hold other employment during the period of the program. Every student must submit a final report.
Financial support
Twelve stipends will be awarded to students who are United States citizens or permanent residents. Stipend recipients will have a choice of two types of support:
(a) $2250
(b) $1500 plus residence hall and meal plan
Travel expense money is available for non-UGA students. Every qualified applicant will automatically be considered for a stipend award as well as for housing and travel reimbursements.
Other sources of financial support are available through the UGA CURO program and from mathematics faculty research grants.
Housing and meals
Students can arrange to stay in a double occupancy room in a campus residence hall. A Monday through Friday meal plan is available in the summer. If you choose support option (b) we will pay for your residence hall and meal plan. Please contact Jennifer Peterson for assistance in arranging these details.
Course credit
Program participants have the option of receiving 3 semester hours of college credit for the VIGRE Research Experience by registering for MATH 4950, Research in Mathematics (May Session, Summer Semester 2005). Students who register for course credit are responsible for the cost of tuition and fees (UGA Bulletin).
If you are not already a student at UGA, you must apply for transient admission (UGA Undergraduate Admissions) to receive college credit. Transient students for the summer semester should apply for admission by April 1.
Application
Deadline March 1, 2005
Required documents
* Application form (for admission to the program and for financial support)
* Unofficial transcripts
* Personal statement of academic goals (Explain why you want to participate in the program.)
* Two letters of recommendation from faculty
Mailing address
VIGRE Summer Program
Department of Mathematics
University of Georgia
Athens GA 30602-7403
Contact information
Jennifer Peterson
Educational Program Specialist
jennifer@math.uga.edu
(706) 542-2577Dr. Akos Magyar
amagyar2000@yahoo.comDr. Tatyana Sorokina
sorokina@math.uga.edu
This page was created on November 15, 2004. It was last revised on November 30, 2004.