Research Seminars are the most important portion of the SUMSRI program.  Professional mathematicians and statisticians contribute problems and lead seminars. At least one of the seminar leaders is African American; the other two are Miami faculty from either the Department of Mathematics and Statistics. Before arriving in Oxford, the students are mailed a card listing each seminar area and prerequisites. They are asked to rank the topics and return the card. They are then assigned an area based on their choice. We hope to give each person his or her first choice, but this is not always possible. During the first four weeks of the program, each seminar director presents a series of lectures to the seminar participants in their area of expertise and assign research problems. These problems are challenging and at the same time easy enough for a very good undergraduate student to get partial results. Each student chooses a problem to work on and consults the appropriate professional. The seminar leaders are asked to meet with their students every day except Friday during the first four weeks and at least twice a week during the last three weeks. We strongly encourage students to work in groups. At the end of the program, the students give an oral presentation on their results and write a paper. These papers are  included in an online Journal published by the Institute.  (All mathematics between dollar signs is written in LaTeX mathematical typesetting language.)

Uncharted Territory of Zero Divisor Graphs and Their Complements Let $\Gamma(\mathbb Z_n)$ be a zero divisor graph whose vertices are nonzero zero divisors of $\mathbb Z_n$ and whose edges connect two vertices whose product is zero modulo n.  Then $\overline \Gamma(\mathbb Z_n)$ represents the complement of $\Gamma(\mathbb Z_n)$.  The authors explore the center of $\Gamma(\mathbb Z_n)$ and $\overline \Gamma(\mathbb Z_n)$.  Further study is done on planarity, independent sets and cliques vertices of minimum degree, and connectivity of $\overline \Gamma(\mathbb Z_n)$.

Kevin Tolliver, Amanda Phillips, Julie Rogers, Frannie Worek, Leigh Cobbs (Graduate Assistant) and Dr. Reza Akhtar.

Jon Middleton, Nikia Thomas, Karen Lostritto, Jared Cunningham, Nancy Ho, Lakeshia Legette (Graduate Assistant) and Dr. Edray Goins (seated)

On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell It is well-known that cubic Thue equations have finitely many integer points, and once one associates these equations with elliptic curves, then there exist algorithms to determine whether they have infinitely many rational points.  In the case of infinitely many rational solutions, we explain how to explicitly find "large" rational points of a cubic Thue equation.  The paper proceeds as follows.  First, we exhibit a map from the cubic Thue equation C having a rational point of inflection to an elliptic curve of the form $E:y^2=x^3-D$, then prove that a "large" rational point on C maps to a rational point of "approximate" order 3 on E.  Second, following an idea of Zagier, we compute rational points of "approximate" order 3 using continued fractions of elliptic logarithms.  Third, we investigate how to modify the algorithm by considering homogeneous spaces when a rational point of inflection does not exist.

A Multivariate Statistical Analysis of Crime Rate in US Cities Kendall and Ralph classified a city as safe or unsafe by using multivariate methods of Principal Components, Factor Analysis, and Discriminant Analysis.  In addition, they discover which variables have salience in the identification of a city being safe or dangerous.  The aforementioned analytical techniques can assist governments in finding out what variables they need to change to improve their city and make it a better place to live.

Kendall Williams, James Lawrence (Graduate Assistant), Ralph Gedeon, and Dr. Vasant Waikar

Nick Imholte and Sara Blight

Educating the States: A Multivariate Statistical Analysis of Education Educating the population is important in every state.  To measure the quality of education in a state, Nick and Sara examined average Scholastic Aptitude Test scores.  They created a model to predict future scores based on variables that affect education.  First, they used the multivariate statistical methods of Principal Component Analysis and Factor Analysis to reduce the number of variables.  Second, they used both of these methods in conjunction with Discriminant Analysis to create a model that predicts future scores.  Finally, they used the results of Discriminant Analysis to conjecture how to improve the quality of education. "Learning is not attained by chance,it must be sought for with ardor and attended to with diligence."--Abigail Adams