A mathematician invited to speak at the International Congress of Mathematics
Every four years, mathematicians from around the world come together for the International Conference on Mathematics. Hosted by the International Mathematical Union, the conference is the largest of its kind. Prestigious accolades such as the Fields Medals, Nevanlinna Prize, Gauss Prize and Chern Medal are awarded at the opening ceremony.
Associate professor Xin Zhou, surveyor at UC Santa Barbara, is among the researchers invited to speak at the union’s 2022 event in St. Petersburg, Russia.
“Being invited to speak at the ICM can be a source of pride for the rest of your career,” said Stephen Bigelow, chair of the mathematics department. “We are happy and proud that Xin has achieved this great recognition.”
Zhou will talk about minimal surfaces, surfaces with constant mean curvature and the relationship between the two. He plans to focus in particular on how the two are related by his proof of the multiplicity one conjecture.
Minimum surfaces are those which satisfy certain equations while maintaining the minimum possible surface under local disturbances. These surfaces are mathematical models for soap films and soap bubbles, because the surface tension of the liquid produces shapes with as little surface area as possible under the constraints of the enclosed volume. Minimal surfaces are fascinating phenomena in themselves and also have applications in materials science and general relativity.
“Since these equations appear naturally in science, mathematicians are interested in solving them,” Zhou explained. However, unlike a line, parabola, or many other familiar shapes, there is no explicit formula that describes these surfaces. They must be implicitly derived in a more abstract way.
A soap film forms a minimal surface covering a helical frame.
In the 1930s, mathematicians discovered solutions for surfaces with a simple closed boundary where the curvature was zero. These local solutions worked for flat, crimped sheets with edging, like a soap film covering a ring of yarn bubbles. In the 1980s, researchers had proven the existence of at least one closed solution, or a minimal surface without border.
Zhou sought to generalize this theory to produce closed surfaces with constant average curvature, like soap bubbles. It is much more difficult because there are less constraints on the solutions. And in math, more freedoms often lead to more headaches.
Mathematicians have found success after many years of work, finding a strategy to prove the existence of infinite solutions to this equation. Still, they encountered a problem. Many seemingly separate solutions were simply multiple copies of a previous solution. This was tantamount to saying that, say, the movie “Forrest Gump” watched back to back is a different movie than the one watched one time.
In recent years, two of Zhou’s colleagues have formulated a hypothesis: the Multiplicity One Conjecture. He proposed that there was a way to produce solutions to the equation that always included a single copy of the minimal surface. In January 2019, Zhou finally produced evidence, describing a method that could generate these true solutions. The following month, he received a scholarship from the Alfred P. Sloan Foundation in recognition of his promising early career accomplishments. He added it to his NSF career award.
“I think people like my job not only because I solved the guesswork, but because it wasn’t an isolated discovery,” Zhou said. “The solution forms a series with my work on minimal surfaces with constant positive curvature. “
Zhou appreciates all the support he has received at UC Santa Barbara. “Most of my important research was done as an assistant professor in the mathematics department at UCSB,” he said. “I would like to express my gratitude to my colleagues, especially Professor Guofang Wei, for their support. “
Zhou is delighted to showcase his achievements primarily as a way to encourage young mathematicians. “I want to inspire some of the students in the department,” he said. He wants to show them that the work done at UC Santa Barbara attracts the attention of other members of the community. “If I was a student, seeing someone’s success in the department would make me excited and more motivated to get into work in my own field.
The 2022 International Mathematics Conference will be held in July 2022.