BenU faculty since 2010
Ph.D., Colorado State University (2010)
M.S., Colorado State University (2006)
B.S., University of Evansville (2004)
Calculus with Analytics I, Calculus with Analytics II, Abstract Algebra I, Mathematical Research
Computational Group Theory, Graph Theory, Number Theory and the use of technology in the classroom
- Magic Graphs and Quasigroups: A Quasigroup is a set of elements with a binary operation whose multiplication table forms a Latin square. Latin squares are precisely what you get when you solve a Sudoku game. A Latin square is in a class of special combinatorial objects called magic squares. In fact a Latin square is a semimagic square. In graph theory semimagic squares can be identified with a super magic labeling of the complete bipartite graph on n points. In group theory a graph which describes the multiplication table for a group is called a Cayley Graph. Unfortunately for quasigroups it seems that many different Cayley graphs can be constructed for one quasigroup. In this project I want to study these super magic labeled graphs to see if one can use them to construct a graph similar to the Cayley Graph for a group. The hope is that this new graph would be more useful for answering several questions about quasigroups.
- Cryptography and Quasigroups: A quasigroup is a set of elements with a binary operation whose multiplication table forms a Latin square. Latin squares are precisely what you get when you solve a Sudoku game. These algebraic structures have applications in many areas including the field of Cryptography. Cryptography is the study of secure communication when a third party is present. Recently quasigroups have been used in several cryptographic applications including Message Authentication Codes. In this project we will continue the work of a former student to study properties of this algebraic object to determine why quasigroups are useful in this field.
- Cryptography in Group Theory: Public key cryptosystems have been used for secure communication between two parties. This system is used most often when the two individuals who wish to communicate have not met prior to the communication. It is used often in online transactions. Most of the algorithms currently used rely on modular arithmetic in Zp however the need to ensure security has led to explorations in the field of noncommutative groups. In this project we will study how noncommutative groups are used for developing new approaches and study several of the open questions associated with their use.
- The Algebra of Rewriting: In mathematics, one method of defining a group is by a presentation. Every group has a presentation. A presentation is often the most compact way of describing the structure of the group. However there are also some difficulties that arise when working with groups in this form. One of the problems is called the word problem which is an algorithmic problem of deciding whether two words represent the same element. I want to study the word problem on group extensions. Currently there is a procedure called coset enumeration which can be used to address this problem, however it has difficulties with memory when the groups reach a certain size. In this project we will continue the work of a former student to compute in the group extension using a modified coset enumeration technique. This method is derived using the Cayley graphs for the two smaller groups.
- Ellen Ziliak An algorithm to express words as a product of conjugates of relators, in Computational and Combinatorial Group Theory and Cryptography, Contemporary Mathematics, vol. 582, Amer. Math. Soc., Providence, RI, 2012, pp. 187-199.
- Thomas G Wangler and Ellen M Ziliak, Increasing Student Engagement and Extending the Walls of the Classroom with Emerging Technologies, in Research Perspectives and Best Practices in Educational Technology Integration, IGI Global, Hershey, PA, 2013, pp. 44-60.