The Collegian
Wednesday, June 12, 2024

Senior explores connection between math and physics

Senior physics and math major Jeff Zheng presented his math honors thesis, "Rank One Perturbations of Self-Adjoint Operators and Applications" in a 40-minute PowerPoint presentation and question-and-answer session to the math and physics departments of the University of Richmond last Monday.

Zheng has been studying physics and math since high school in Nanjing, China, he said, and has always done well. Although he describes himself as more of a "physics guy," he likes math, too, and finds it useful and important to physics.

When Zheng told mathematics professor Bill Ross he wanted to do a mathematical physics research project, Ross said he knew just the topic. He suggested a paper by Barry Simon, a mathematical physicist from the California Institute of Technology.

The paper is on Schroedinger operators applied to quantum mechanics, which Ross described as the small-scale study of atoms. Simon wrote the paper as a physicist, Ross said, but it has great mathematics in it.

"Because [Zheng is] a physics major," Ross said, "I think he can appreciate what Barry Simon is doing."

Ross said mathematical physics extracts the mathematics from physics problems. The problem that Zheng was working on was a problem that originated in physics but is now an interesting mathematics problem, he said.

Simon's proof of the main theorem in his paper was correct, Ross said, but he didn't contribute any details of why it was true. Zheng filled in all of the technical details, Ross said.

Zheng used Simon's formula to write a mathematical program that generates formulas. Ross said he called it the cool formula generator. You plug in simple output and it generates a formula that a computer can't do.

"I think that's the piece he enjoyed doing the most," Ross said.

Zheng read a graduate-level book on the operator theory which gave him the basics he needed to know while studying abroad at McGill University in Canada.

He did background reading on linear algebra, real analysis and complex analysis over the summer. Zheng then took a course on an almost-graduate-level with Van Nall this semester on some of the other subjects he needed to learn.

Zheng worked on his thesis between five and ten hours a week for a year, he said. Ross said they had met about two or three times a week, but Zheng had worked a lot on his own, especially with computer programming.

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Toward the end of the semester, they started working on the weekends and had Sunday afternoon meetings to get the thesis done. Mathematics professor Michael Kerckhove read the thesis as the second person required by the program.

"Jeff is a phenomenal mathematics student," Ross said. "I would almost say he's my equal."

Zheng was one of only seven honors students that Ross has had in his 19 years at Richmond. The math honors program consists of two courses, a thesis, read and examined by a second person in the department, and a presentation to the department.

After graduation, Zheng will attend Massachusetts Institute of Technology to get his Ph.D in cosmology. He was recruited by physics professor Max Tegmark.

Contact reporter Sarah Craig at sarah.craig@richmond.edu.

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