"Supervising this Hackman Scholars project was an opportunity for me to reacquaint myself with my research on binary quadratic forms," says Prof. Wendell Ressler, advisor for Nart Shalqini's summer research on class numbers binary quadratic forms over Z[λ]. "Part of the benefit was to have someone to talk to about the ideas of the field. Nart also helped me by working out details of some proofs, and by finding several mistakes in my work. My main goal this summer was to better understand generalized binary quadratic forms, and we succeeded at that."
The research relates to Algebraic Number Theory which was crucial for solving the celebrated Fermat’s last theorem. There are no direct applications to the real world, the student researcher explains. "One can only appreciate the beauty of pure mathematics. The goal of this research is to provide a formula that gives the number of equivalence classes of binary quadratic forms given a discriminant d," Nart adds.
Nart plans on pursuing mathematics in graduate school and becoming a mathematician. "Doing undergraduate independent research will prepare me for both grad school and my future mathematical career," comments the researcher.
"When challenged, developing a tolerance for obstacles is very important in order to be motivated to continue working on problems."
"Graduate school involves a lot of work and independent study," Nart continues. "This research taught me how to be patient with hard problems. When challenged, developing a tolerance for obstacles is very important in order to be motivated to continue working on problems. This research also taught me how to recognize the time I should try taking different paths when a particular path is not leading to the solution; in other words, learning when and how to change perspectives. Besides that, it has also helped me improve my communication skills in math. I learned how to better put my thinking process in words.
"Research is very different from class work because you do not have a specific set of tasks you need to complete. Instead, you are digging information from different sources and trying to find the right path for the solution. Therefore, it taught me how to give structure to my own work and manage my time accordingly."
Prof. Ressler adds, "Doing a Hackman in math is new and challenging for most students. This Hackman was the first time Nart worked on a problem without a known answer; he even had to think about how to correctly frame the question. He also had to search the literature more thoroughly than any class had required. This Hackman was also the first time Nart did math full time; he had to budget his time while working independently. For example, that required him to develop the discipline to step away from a proof when stuck and then devise fruitful ways to reengage."
"Working one-on-one with a professor is a valuable experience. First of all, one receives individualized attention. This makes students and professors get to know each other better. Therefore, both the student and the professor can recognize a student’s strength and weaknesses in a much quicker time," explains Nart. "Besides that, working with a professor one-on-one makes the student better understand the thought process involved in solving a problem. It also accelerates the learning time of a student, since a student can ask more questions (without hesitation) and receive answers immediately."
And the math major adds, "All these reasons also boost a student’s confidence in himself."