Fun times with finite group invariants

Jude Quintero ’20 (left) and mathematics professor Michael Penn work on an equation

One of the things Jude Quintero ’20 enjoys most about his Randolph experience is getting to know and work with his professors. This summer, he and mathematics professor Michael Penn are bonding over some advanced algebraic equations.

“We’ve had a lot of fun,” Penn said. “If you understand this level of math and the language, it’s pretty much like doing puzzles all day.”

Quintero and Penn are conducting a Summer Research project called Finite Group Invariants of the Rank 2 Heisenberg Vertex Algebra. Penn explained that invariants are a common theme in mathematics, and they are specifically researching invariants in the reflections and rotations of a triangle. Penn said these algebraic structures are called vertex operator algebras, and examples are currently hard to find.

“Generally in math you want to look for a classification of all types of things, but the classification for these is way in the future,” Penn said. “So, having examples is really helpful and finding these invariant subspaces produces examples for others in the field.”

Quintero, an engineering physics major, is just a rising sophomore at Randolph but has already logged lots of time in the College’s research labs. He grew up five minutes down the road from Randolph and began taking physics classes at the College as a high school junior.

“Summer Research seemed like a really good opportunity, and I was interested in the project because it ties in with string theory,” Quintero said. “It also ties in with a lot of high level physics, and I think this will give me more background in the field. Plus, I like working with Dr. Penn, so that was a plus.”

Quintero and Penn have completed all of their equations for rotations and reflections and the invariants they produce and are now writing a paper with examples. For Penn, that might be the hardest part.

“To paraphrase what a famous mathematician once said, writing up a math paper is the punishment for all the joy you had when you were doing all of the problem-solving and exploration,” Penn laughed.