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Name: James Long
Honors in: Mathematics
Hometown: West Chester, PA
Major(s): Mathematics and Computer Science
Title of project: "The Collatz Conjecture: A Conjugacy Approach"
Abstract or brief description: The Collatz Conjecture has perplexed mathematicians since its initial proposal over 70 years ago. Many different approaches have been suggested and formulated, but so far, all have fallen short of proving or disproving the conjecture. One of the more promising strategies involves topological conjugacy, which was initially applied to the conjecture by Bernstein and others as early as the 1990s. In this paper, we continue the work pioneered by Bernstein to make statements concerning the Non-trivial Cycles Conjecture. We also derive a necessary condition for an endomorphism of the shift map to induce a conjugacy, as well as classify the dynamics of a particularly interesting conjugacy that was initially discovered by Monks.
How did you get interested in your topic? Dr. Fraboni (who would later become my honors advisor) introduced me to the Collatz Conjecture during my first semester at Moravian College, and invited me and two other students to help him and another professor with a SOAR project that following summer.
Do you intend to research your topic further, if so, how? As it turns out, I have already continued my research on the Collatz Conjecture as a "side project" while I wait to begin my doctoral studies at Lehigh in the fall. Also, since most of my research background is in the same subfield of mathematics as the Collatz Conjecture, there is a very real possibility that I will be able to continue my research as part of my graduate studies.
How did you benefit academically by conducting research/participating in honors? Honors affirmed my desire to attend graduate school and did wonders in preparing me for the eventual process of writing my dissertation.
How has the department (or faculty advisor) in which you studied prepared you for the future? In addition to Honors, I have participated in two SOAR projects, presented at several conferences, and published four papers in academic journals and conference proceedings; I credit all of these things to my interactions with the faculty in the Mathematics and Computer Science Department.
What advice do you have for other students interested in honors? As with choosing a job or graduate school, I feel it is important to choose a project topic that you love. Honors was undoubtedly a trial, but it was one that I was able to endure and enjoy due to my strong attachment to my topic.