In the academic off-season, Chapman University welcomes motivated students across all disciplines to OURCA’s Summer Undergraduate Research Fellowship (SURF) program. While many students are tanning at the Southern California beaches, selected students, along with a faculty mentor, spend over 30 hours a week immersed in their proposed research project.
Math majors Sarah Alexander (’18) and Nadiya Upegui (’18) were among the 16 fellows selected to participate in SURF 2017, under the mentorship of Dr. Peter Jipsen. Alexander describes her impetus for applying to SURF simply: “We just wanted to understand partial algebras better, specially effect algebras.”
But what is an effect algebra? When asked to describe this in layman’s terms, Upegui laughed knowingly and explained, “An effect algebra in general is just an algebra where some products are undefined. In division for example, when you divide by zero, we consider that something that is undefined. So, effect algebra is the same way: we just have some things that don’t have values.” At its core, Upegui and Alexander worked toward discovering a new avenue to define these undefined products. But the process by which one defines the undefined is no simple task.
The pair began their summer researching the different partial algebras in existence to draw conclusions and gain insight around how these effect algebras connect to one another. Alexander notes, “We did that through analyzing their axioms…so first we kind of had to wrap our head around what the axioms meant.” Eventually, by comparing these partial algebras, the Alexander and Upegui aim to make these partial algebras into total algebras, or algebras that we commonly know of and understand.
Both fellows discussed their research process, beginning with a lot of reading to understand what exactly it means to be a partial algebra. These readings lead to hours of trial and error; “We’d try to prove something, and then maybe find it’s already been discovered, or this doesn’t really work the way we want it to,” Alexander explains. Upegui adds, “Before we can just jump into a proof, we need to have a sort of understanding of what’s going on…and then we try to prove things.”
The pair identify the ultimate goal of their project is actually related to physics. Upegui aptly describes that effect algebras derive their name from effects in physics, essentially, “just a fancy term for something that happens in nature.” By understanding effect algebras, Alexander and Upegui are working to provide the building blocks for physicists to understand the natural functioning of the universe.
Alexander and Upegui became interested in the study of partial algebras through Chapman’s Abstract Algebra course that focused on group theory. “I really loved that class. I just fell in love with it,” confesses Upegui. The pair approached Professor Jipsen with a desire to learn and discover, asking his advice on research directions and avenues to explore. Under his mentorship, the team found that partial algebras was a burgeoning area of research in mathematics, and applied for SURF with the determination to make real discoveries and contributions to the field.
Upegui and Alexander both agree SURF was transformative for their future goals, teaching them tenacity, patience, and teamwork. Upegui notes, “One of the coolest parts of this program is we actually get to talk to the other [SURF fellows]. There are so many things that people have said and I’ve thought that’d be super cool to research in the future! I learned so much even from just the faculty seminars. It gets you out of your zone!”
Are you interested in applying to SURF for a summer of discovery and personal development? Alexander and Upegui shared their advice for applicants and students new to research. Alexander emphasizes, “Don’t get discouraged. I feel like in any field of research, you’re researching something so you’re trying to find something new. Sometimes it’s a little daunting. I think no matter what you’re going to find something interesting, if you don’t lose hope in yourself!”
OURCA encourages all students interested in becoming a 2018 SURF fellow to attend the program information session on Monday, February 12th at 1 PM in Argyros Forum 209A.
SURF Applications are due March 2nd at 4 PM. Application requirements and qualifications can be found here.
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