Yiming Zhao will explore new variations of two of the oldest problems in geometry—the isoperimetric problem and the Minkowski problem—with a $434,697 CAREER grant, the National Science Foundation’s most competitive award for early-career faculty who may serve as academic role models in research and education.

Isoperimetric problems go back to the ancient Greeks.

“They wanted to know how to enclose as much area as possible with a thread of fixed length,” says Zhao, assistant professor of mathematics. “The answer is you make the thread into a circle. In the second type of problem, the Minkowski, you find how to reconstruct a geometric shape when you have only partial information.”

The problems are connected. “In classical cases, they can be two sides of the same coin,” says Zhao. “If you know the answer to one, you usually know the answer to the other.”

But not always. Zhao will explore isoperimetric problems or Minkowski problems in various settings when answers to one exist while answers to the other remain elusive.

“In new mathematical variations over the last few decades, sometimes we only know the answer to one,” he says. “I want to use our existing knowledge of one answer to a problem to find the answer to the other.”

Applications of solving these problems extend beyond mathematics into engineering and design.

The CAREER award calls on faculty members to integrate their research into instruction. Zhao will organize a series of events for K-12 students, high school teachers and the public about mathematics at a local science museum, high schools and community centers. These events will expose the fun and exploratory side of Zhao’s research to young students, raise society’s awareness and interest in mathematics and promote mathematics among historically underrepresented populations.

Zhao will encourage youngsters to think about mathematics differently, conducting an educational session for K-12 students at the Milton J. Rubenstein Museum of Science and Technology (MOST), a science and technology museum in downtown Syracuse.

“Math is about discovery, not just about people applying a set of formulas on an exam,” Zhao says. “I plan to get kids involved in an old problem, a toy version of the Minkowski problem I’m working with, which I could easily explain to them.”

Zhao’s project will involve graduate and undergraduate students in research and educational activities. Graduate students will help plan programs for K-12 students and the public, gaining crucial training opportunities to explain research to different audiences.

Zhao's CAREER award brings the total to 10 A&S researchers since 2022. Read about the 2022 and 2023 CAREER grant winners.