The team’s work provides a new result in spectral graph theory — that a bounded degree graph must have sublinear second eigenvalue multiplicity. The proof requires clever insights relating the spectrum of a graph with the spectrum of small pieces of graphs.

How did this great result come about — did the team of five researchers hover in front of a whiteboard for weeks on end, scribbling ideas and crossing them out? Not exactly. This was a summer research project (before COVID-19) where the team mostly interacted remotely using Slack, an instant-messaging app where users can share messages, images, internet links, videos and more. 

Initially, the team had one in-person meeting where they got together and summarized the existing ideas, shared their perspectives of the problem and showed pitfalls they had encountered before.

“I would tell the group, when you read previous literature, you don't see what didn't work, right?” said Jiang. “When we read a math paper, it's always about what works — nobody's going to write an article about ‘these are the things I tried and I failed, sorry guys.’”

After that initial face-to-face meeting, the research team moved their entire discussion online over Slack.

“We still comment nowadays through Slack, and I feel like it's a fantastic way to communicate. Not only can we have a threaded discussion, and also we can just keep everything in one place. It's not like emails that you have to sort through. It keeps an ongoing record, which was great,” said Jiang.

After the problem percolated in the back of Jiang’s mind for a year, and then four other fresh minds were added, new ideas and directions started to flow.

“When you stop working on a problem for a while and then come back to it, definitely you get new ideas, and my explanation is that there are little people working (as he again gestures with his moving fingers behind his head), but I can't verify that,” said Jiang, smiling. “Definitely, when I look at the same problem once, twice, three times, I always get fresh ideas. But also, the other members in the group bring new ideas as well, so it's a combination of these two things.”

Over the summer, the team truly had fun working on the problem. We asked Jiang to discern what his specific contributions were to the solution.

“Let me actually be a bit conservative here, because I think the tradition in mathematics is that we try to push the frontier of human knowledge in a collective way. That's part of the reason why, for example, we don't sort authors by their contribution. I guess that's also why you see in other articles they try not to split the credits between the authors, and I want to keep that tradition, but also to answer your question," he said.

“I did have fun, and I can tell you at the beginning, we were a little bit stuck. We made some partial progress, but I guess by hitting those roadblocks we just learned a lot about what we needed at the end. And that was great experience, because at least for me personally, I feel like doing research is also about the experience. If you can get some results quite easily and straightforwardly, then you view the end result as less rewarding — but this is not like that. It’s quite challenging. When we did have a breakthrough and solved it, it was very rewarding and everyone was super excited.”

At 33 years old, Jiang is an old soul, respectful of the history and traditions of mathematics. 

When he was a graduate student at Carnegie Mellon, he was inspired by the teaching of Professor Po-Shen Loh. Jiang was kind of lost in what he wanted, in which direction he wanted to do research. Loh was teaching something called extremal combinatorics, which is a subfield of combinatorics.

"His passion and enthusiasm just lit me up," said Jiang. "While listening to his teaching, you could feel that he just loves the subject, and it was entirely an inspirational experience."

Jiang believes in paying it forward in his own classroom. He strives to inspire his students because his own experience was like that.

“I feel if you can share your enthusiasm with the younger generation, that will just do a lot of good to the community," said Jiang.

“At the end of day, mathematics is about passing down the knowledge to the future generation, because once math is written in papers but nobody is reading them, it's dead. Working as a group, especially with undergrads or graduate students, has its own merits. In particular, the younger generation is also very creative, and they can just forget about the boundaries and limitations that our minds were trained with. They can break barriers."

Rhonda Olson

Manager of Marketing and Communication, School of Mathematical and Statistical Sciences