10/23/2020 Peter Durantine

Students Explore World of Non-unique Factorization Atoms

For five weeks last summer, Franklin & Marshall mathematics major Mitchell Young spent 40 hours a week calculating and notating patterns in the results from a study that he and a team of students conducted under the tutelage of Assistant Professor of Mathematics Nicholas Baeth. 

The students examined two different settings of non-unique factorization. It is part of a larger body of research in Factorization Theory, a relatively young field (about 70 years old) in algebra where Baeth has focused his scholarly work.

“Whole numbers factor uniquely as a product of prime numbers—12 factors as 2 times 2 times 3, and that’s the only way you can do it,”Baeth said. “But there are many algebraic structures where factorization is not unique. The goal of my research is to take some sort of algebraic structure and measure the degree to which factorization is not unique.” 

  • Mathematics major Mitchell Young, a vocal music performance minor who also played trumpet in F&M’s Symphonic Wind Ensemble, zeroed in on a setting called "Tn" or upper-triangular n-by-n matrices. Mathematics major Mitchell Young, a vocal music performance minor who also played trumpet in F&M’s Symphonic Wind Ensemble, zeroed in on a setting called "Tn" or upper-triangular n-by-n matrices. Image Credit: Deb Grove

Young, a junior, and three other students – sophomores Greg Heilbrunn, Peter Liu and Santure Chen – worked with Baeth to study a particular semigroup and measure just how non-unique factorization is by computing various quantities. How do they do this?

“For whole numbers, primes are your atoms; they’re your basic building blocks. In the more general setting, you don’t necessarily have primes. You have what we call atoms, and these are your basic building blocks that cannot be factored any farther,” Baeth said. “There are many different ways you could measure the degree to which factorization is not unique.”

Young, a vocal music performance minor who also played trumpet in F&M’s Symphonic Wind Ensemble, zeroed in on a setting called "Tn" or upper-triangular n-by-n matrices. 

“Within this setting, I derived how non-unique the factorization is, as well as quantifying the ‘prime-ness’ of all the atoms,” he said. “This is because, unlike our primary numbers, atoms can have varying degrees of ‘prime-ness.’"

Once the Hackman Summer Scholar “had an idea of what was happening in this setting or an answer to a question I wanted to solve, I would run several test examples and write up lemmas, theorems, and proofs to explain my findings.”

Young’s project is part of Baeth’s larger research that he and three other students worked on together. In addition to Young, two other students are also presenting their efforts at the Autumn Research Fair.

“Whether it’s DNA built up in terms of nucleotides or molecules built out of atoms, you want to figure out what your basic building pieces are and how they all fit together,” the professor said.  “In a general sense, that’s what the students are doing. They’re looking at these types of matrices that appear all over in mathematics and physics, and trying to understand these basic pieces and how they fit together.”

Young found the research opportunity exceptional.

“The experience was really enlightening on what math research is truly like,” he said. “Working on a particular question I wanted to answer, or proof for a theorem, for hours or days only made the ‘Eureka!’ moment when I finally thought of a solution even more gratifying and exhilarating.” 

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