Algebra in Pictures
Today, let’s do something amazing to see how ordinary formal calculations can be replaced with pictures, or diagrams.
Read MoreStaff Spotlight: Dr. Vardan Tepoyan
Dr. Vardan Tepoyan is a key member of the High School Projects team in RSM’s Curriculum Department.
Throughout his time at RSM, Dr. Tepoyan has played a critical role in further developing RSM’s high school, Calculus, Pre-Calculus, SAT, and Olympiad level courses.
This year, Dr. Tepoyan has also begun creating EdTech products to support RSM’s unique methodology. These products include interactive mathematical online models designed to improve student understanding and allow hands-on exploration of advanced topics in Geometry, Pre-Calculus, and Calculus. Additionally, the program will be able to generate new problems “on the fly,” supporting an adaptive approach to each student's skill mastery. Online assessments will allow teachers to differentiate within a class and focus on topics that need further improvement and suggest challenging problems for the stronger students.
Prior to working at RSM, Dr. Tepoyan studied mathematical structures at the Institute of Mathematics NAS in Armenia, working with isometric representations of semigroups, C*-algebras, and operator theory. His publications have been cited over 50 times and explore topics like inverse representations, semigroups, and extensions of the Toeplitz algebra. This research is highly beneficial in developing advanced mathematics curricula, particularly in areas like algebra, functional analysis, and problem-solving, for students aiming to deepen their understanding of abstract mathematical concepts.
This article, penned by Dr. Tepoyan for the Journal of Contemporary Mathematical Analysis, focuses on studying mathematical structures called “semigroups,” which are sets where elements can be combined in a certain way, but don't always have to be reversible like numbers in addition or multiplication. In this case, the semigroup is Z+∖{1}, which means the set of positive whole numbers except for 1.
His paper explores how these semigroups can be represented using “isometric operators,” which are mathematical tools used in spaces with distances, like vectors. These operators help map the elements of the semigroup onto a space in a way that preserves certain properties, like lengths.
