Alexei Kolesnikov

Offering undergraduate students many opportunities to engage in meaningful research experiences is an important part of my teaching. Here are some of the undergraduate research projects I directed at Towson.

### AGAT student activity group

AGAT stands for "Algebraic Geometry and Approximation Theory". This project, generously supported by the Fisher College of Science and Mathematics, offers students of all levels an opportunity to work on open-ended problems that are of current interest to mathematicians. The students working on the project gave a number of presentations at student conferences (at least 10 and counting); it is likely that some of the work will result in papers.

### A factorial power version of Fermat's equation

Matthew Green worked on a version of the famous equation $$x^n+ y^n=z^n$$, where the usual power $$n$$ is replaced by factorial power: $$x^\underline{n}:= x(x-1)(x-2)\cdots (x-n+1)$$. After completely describing integer solutions for the factorial power 2, Matt found that there are also infinitely many non-trivial solutions for the factorial power 3 and that the set of factorial powers for which there exist non-trivial solutions is unbounded. Matt gave talks at two student conferences about this project and he published the results in Rose-Hulman Undergraduate Mathematics Journal. He is now a graduate student in mathematics at the University of South Florida.

### Transportation Security project

The project was sponsored by the Chemical Security Analysis Center. The students' goal was to design a mathematical model to assess and manage the risk to the population involved in transporting toxic chemicals. The student team adapted a minimum cost network flow model with randomized cost coefficients. The results have been presented by the students in talks at professional meetings in addition to formal presentations to the sponsors of the project. The study was published in the Journal of Transportation Security.

### Senior Seminar

Two of my students in this capstone course for mathematics majors at Towson elected to do a research project (typically, the students are asked to do an exposition of an existing result). One project addressed the question of which rational numbers can be written as the sum of squares of two rational numbers; the other was dealing with reflections in curved mirrors (the mathematics behind anamorphic art). The results of both projects were presented at an Undergraduate Mathematics Research Conference at Towson.

One of the students, Kimberly Rausch, continued the project on mathematics of anamorphic art. She was also one of 12 students representing Towson at the 2012 CAA Undergraduate Research Conference. Her paper was accepted (via peer-review process) for Bridges Conference; she gave a talk at that conference in July 2012.