BenU faculty since 2010
Ph.D., Colorado State University (2010)
M.S., Colorado State University (2006)
B.S., University of Evansville (2004)

Courses Taught
Computer Programming, Data Structures and Algorithms II, Mathematical Universe, Trigonometry, Calculus I, II, III, Linear Algebra, Abstract Algebra I and II, Discrete Math, and First Year Seminar Course.

Research Area

My scholarly interests lie in three main areas which include the study of symmetric spaces, applications of abstract algebra, and effective pedagogy in my classrooms.

Recent Publications (based on current research)

• 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. (Indexed in ERIC & Thomson Reuters)
• C. Buell, A. Helminck, V. Klima, J. Schaefer, C. Wright and E.Ziliak, On the Structure of Generalized Symmetric Spaces of SL_n(F_q). Communications in Algebra (May 11, 2017 http://www.tandfonline.com/eprint/mMcGQfZRCfaPcRwyM6YU/full )
• C. Buell, A. Helminck, V. Klima, J. Schaefer, C. Wright and E.Ziliak, On the Structure of Generalized Symmetric Spaces of SL₂(F_q) and GL₂(F_q). Note di Matematica. Volume 37, Issue 2 (2017) 1-10.  http://siba-ese.unisalento.it/index.php/notemat/article/view/18692
• C. Buell, A.G. Helminck, V. Klima, J.Schaefer, C. Wright, and E. Ziliak, Orbit decompositions of unipotent elements in the generalized symmetric spaces of SL₂(F_q). In A. Deines, D. Ferrero, E. Graham, M. Seong Im, C. Manore, and C. Price, editors, Advances in the Mathematical Sciences: AWM Research Symposium, Los Angeles, CA, April 2017, pages 69-77. Spring, Cham, https://doi.og/10.1007/978-3-319-98684-5 , 2018.
• E. Ziliak, Calculus 1 Course Comparison: Online/Blended or Flipped?, in Handbook of Research on Blended Learning Pedagogies and Professional Development in Higher Education, IGI Global, Hershey, PA, 2018, pp 224-258. https://www.igi-global.com/book/handbook-research-blended-learning-pedagogies/190385
• C. Buell, A. Helminck, V. Klima, J. Schaefer, C. Wright and E.Ziliak, Orbit decompositions of Generalized Symmetric Spaces for SL₂(F_q). Communications in Algebra (January 20, 2020 https://doi.org/10.1080/00927872.2019.1705471)
• DeLegge, A. and Ziliak, E. “The Math Games Seminar: A Mathematical Learning Community,” Journal of Humanistic Mathematics, Volume 11 Issue 2 (July 2021), pages 148-166. Available at: https://scholarship.claremont.edu/jhm/vol11/iss2/7
• Buell, C., Helminck, A., Klima, V., Schaefer, J., Wright, C., & Ziliak, E. (2023). Fixed-point group conjugacy classes of unipotent elements in low-dimensional symmetric spaces of special linear groups over a finite field. Journal of Algebra and Its Applications. https://doi.org/10.1142/s0219498824501421
• R. Aliakseyeu, N. Oliven, E. Thieme, and E. Ziliak. Counting the generalized and extended symmetric spaces of SO(3,Fq). Accepted in Involve Volume 17

Current Research

• 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, Ph.D.
Professor, Mathematical and Computational Sciences
[email protected]

 

Research

This project involves studying symmetric spaces of the special orthogonal group over a finite field. The study of symmetric spaces comes from a generalization of symmetric matrices. Symmetric matrices are those matrices that are symmetric across the main diagonal meaning the numbers above and below the diagonal are equal. When working with matrices whose entries contain real numbers, the set of symmetric matrices define the symmetric space which was classified by Elie Cartan. These spaces were then generalized to symmetric spaces over other general fields, where the definition of symmetry becomes slightly modified. The work on this project has two main tasks (1) writing algorithms to compute data for larger matrix groups so that we can observe patterns and make a conjecture and (2) proving conjectures from patterns observed already. This work generally involves linear algebra concepts related to eigenvalues and eigenvectors along with modular arithmetic. Those who have taken at least Calculus II or a programming course would be qualified.

Research could be in-person or remote at Benedictine University.