Solving mathematical mysteries

Across Junliang Shen’s chalkboard, visitors will see formulas, long lines of equations, and diagrams drawn by graduate students, colleagues, and of course, Shen himself. 

To Shen, the formula for his research is simple: communication and collaboration with others, thorough reviews of literature in the field, and persistence. The process can be slow and arduous, but he's motivated by the challenge.

Shen is Associate Professor of Mathematics in Yale’s Faculty of Arts and Sciences (FAS), where he conducts research on algebraic geometry and related fields. His work in topology, geometry, and mathematical physics earned him the 2024–25 Arthur Greer Memorial Prize for Outstanding Scholarly Publication or Research from the FAS. The award celebrates outstanding research conducted by ladder faculty members in the social or natural sciences, broadly construed, who are untenured at the time that the work is completed or published.

The Greer Memorial prize is just the most recent award on Shen’s extensive list of accolades. Earlier this year, he received the 2024 Sloan Research Fellowship in mathematics, and in 2018 he received the SwissMAP Innovator Prize. 

He described receiving the Greer prize as a “great encouragement,” and credits his colleagues and department for creating a “very friendly and creative environment.” He also emphasized the importance of having fun while pursuing one’s research questions. 

“As a mathematician, getting this recognition is fun and great, but I always think that the most rewarding moment is when I get a new idea for a project I’ve thought about for a long time and was stuck on,” he said.

Solving mathematical mysteries

Shen’s primary field of exploration, algebraic geometry, might be reminiscent to anybody who encountered algebra and geometry in middle and high school. Algebraic geometry is the study of spaces, which can be described by algebraic equations.

Take the equation for the unit circle, x² +y² = 1, Shen said as he sketched the equation on a chalkboard. That algebraic equation gives you a geometric shape, but when the equation changes, the shape becomes more complicated. This place, where algebra and geometry interact, is what fuels Shen’s curiosity.

“Algebraic geometry connects two worlds: one is the algebraic world, which is about equations and numbers,” Shen explained. “The other world is geometry, which is about shapes.” People have explored both worlds for thousands of years, revealing deep connections between them—and the study of both algebra and geometry are “moving forward in exciting ways,” Shen says. 

One of Shen’s related research interests is the algebraic cycle, a persistent unresolved problem in the field and a question Shen has been trying to unravel since graduate school. The algebraic cycles, which are simpler geometric shapes formed by subvarieties, can be combined to understand the overall structure of an algebraic variety—a larger, more complex space.

These mathematical structures, however, can be very hard to discover in certain geometric spaces, according to Shen. “It’s quite mysterious. You know they should exist, but you don't know where to find them,” he said. 

Recent literature has shown that vector bundles, mathematical structures used to analyze the geometric and topological properties of spaces, may serve as a potential solution. With his collaborators, Shen is exploring how vector bundles can be used to discover algebraic cycles. 

“We realized that these vector bundles can be used to produce the algebraic cycles we’ve been looking for for years,” he said. Shen and his collaborators have been considering this question for almost a decade, and someone else’s work on vector bundles was key to pushing his own inquiries forward.

The case of vector bundles and algebraic cycles is a great example of why communication and collaboration are so essential to mathematical exploration, Shen said. “This is important, because I think even for those genius, really smart people, there are things that they cannot think of, but others can.”

Shen naturally gravitated toward mathematics in his early life. During elementary and middle school, he participated in math Olympiads, finding amusement by answering more difficult problems than the work he received in school. This persistence and curiosity followed him to college, and he earned his BA in Mathematics at Peking University in Beijing, China, and his PhD at ETH Zürich in Switzerland.

“I started to learn more modern math and I was attracted by the beauty and the structures involved there,” he said of his path to algebraic geometry. “After taking some courses and talking to senior people, that’s when I started to be interested in this direction.”

Reflecting on the perks of choosing algebraic geometry as his primary field of expertise, Shen highlighted the flexibility and relative ease of doing his research. Any idle moment can be turned into a mathematical opportunity, because modeling problems can be as simple as using his imagination. 

“Every day when I wake up in the morning, I can decide to think about the question that I like the best. I don’t need too many tools to do the work. Most of the calculations are happening in my mind, and that is the great freedom I like about mathematical research.” When the work proves to be more extensive, Shen said, quantitative software like Maple can help him visualize equations and shapes.

Sharing algebraic geometry with up-and-coming mathematicians

This summer, Shen was invited to speak on the topic of enumerative geometry at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. Founded by the late Harvard mathematician Oscar Zariski in 1954, the Summer Research Institute is the largest and most influential conference in algebraic geometry and meets once every decade.

The conference served as a full circle moment, where Shen could both share his findings and also introduce the younger generation to theoretical mathematics, just as he experienced as a graduate student. He is happy to contemplate the long line of mathematicians who have come before him, and who will hopefully come after.

“If there was a time machine, I’d be happy to see what advancements would happen in two thousand years,” he reflected. “Likewise, if [ancient Greek mathematicians] Euclid and Archimedes traveled to present time today, I think they would be very excited about what has happened and the two thousand years of advancements on ideas that they initiated.”

Shen’s foundational love for mathematics and intellectual curiosity are woven throughout his pedagogy and instruction, whether he's working with his three graduate students or conducting a lecture. 

“Teaching is very helpful for my research because it pushes me to explain concepts that I’m still figuring out to a different audience, and to work through my ideas,” he said. “This helps clear up my mind and, in the process, can sometimes help me find things I haven’t thought of before.”

In late June, Shen visited Pisa, Italy, to deliver lectures for the University of Pisa’s Department of Mathematics Summer School Series. In early July, he served as a mentor for a group of ten researchers consisting of graduate students and postdocs in the Graduate Student and Postdoc Bootcamp for the Summer Research Institute at Colorado State University. Rooted in the framework of collaboration, Shen’s unwavering commitment to teaching and learning represents a true vocational calling to build on the work of those who came before him. 

“For thousands of years, people have explored shapes and numbers. These are the building blocks of mathematics, and my research continues the great tradition of exploring something human beings started many years ago,” he said. “I’m excited to see what will happen in the many years and what kind of contribution I can make towards this subject.”

To nominate an FAS faculty member to be featured in this series, please email fas.dean@yale.edu.