Research
McKay Sullivan, Ph.D.
435-652-7647 • SNOW 139
mckay.sullivan@utahtech.edu
McKay Sullivan, Ph.D.
435-652-7647 • SNOW 139
mckay.sullivan@utahtech.edu
Creativity and tenacity are important characteristics of students who succeed in research. Research in mathematics is challenging and often frustrating. However, for those who stick with it, it can also be exhilarating and rewarding.
The real-world applications of mathematics that interest me require the use of advanced mathematics such as linear algebra, number theory, abstract algebra, and analysis. Students wanting to work with me in one of my main project areas will at least need to have taken linear algebra and preferably at least one of the other courses mentioned above.
If you think you might be interested in conducting undergraduate research with me, you are always welcome to stop by my office (SNOW 142) and talk with me.
Oscillator representations of Lie algebras are used to model particle interactions in quantum physics. These oscillator representations are built using skew-supersymmetric bilinear forms. We classified such forms that are inhomogeneous and have dimension less than or equal to 3 over the real numbers and explored the oscillator Lie algebras that result from them.
Dixie PREP is a summer program that gives interested middle school students a boost in their STEM education. This program includes a course entitled Algebraic Structures. Shanna picked several important abstract algebra topics that were not fully covered in the existing curriculum, and she developed excellent lesson plans and course materials for these topics. Shanna then had the opportunity to teach the Algebraic Structures course during Summer 2020 using the lessons she created.
The security of public-key cryptosystems relies on math problems that are believed to be very difficult. For example, RSA, a widely used public-key cryptosystem, relies on the difficulty of factoring the product of two large prime numbers. It has been shown that large enough quantum computers will be able to complete this factorization and break RSA. This has led to the development and testing of candidate cryptosystems that might be resistant to quantum computer attacks. Crystal took on the challenging task of encoding a complete working model of one such cryptosystem called NTRU. The NTRU cryptosystem employs advanced mathematics, including Linear Algebra, Number Theory, and Ring Theory. She also fully encoded a working model of LLL, one of the fastest know attack algorithms on NTRU. Using this model, we tested several ideas for modifications to the LLL attack algorithm.
Anthony approached me with the request to work with him on a research project in game theory. I asked him his interests, and one of those was green initiatives. Together we worked on an idea for a relay system in a non-cooperative wireless peer-to-peer network. Using a simplified model we built in SageMath, we found a theoretical limit to the amount of energy savings obtainable using a single message relay in a cooperative peer-to-peer wireless network and showed that an auction system from game theory could be implemented to get close to the same amount of energy savings in a non-cooperative network.