Figure 1. An illustration* of “Determinant Values on Lattices,” a new result by mathematicians Wooyeon Kim and Hee Oh. Credit: Hee Oh, Google Gemini.

By Siobhan Roberts

 

In real-world applied settings, a matrix is a digital machine—a powerful mathematical tool that transforms data. Matrices rotate 3D objects in computer graphics and transmit data through layers of a neural network; they encrypt messages for cybersecurity and solve large engineering equations; and matrices are the foundational language of quantum computing.

 

In mathematics, researchers investigate the theory and properties of matrices—and, inextricably, lattices. In this domain, two mathematicians made key advances during the Spring 2026 program on Geometry and Dynamics for Discrete Subgroups of Higher Rank Lie Groups at the Simons Laufer Mathematical Sciences Institute in Berkeley, with support from the U.S. National Science Foundation.

 

Wooyeon Kim, of the Korea Institute for Advanced Study, and Hee Oh, of Yale University, a co-lead program organizer, announced their results during a workshop at SLMath in April, and recently posted the paper (138 pages), “Determinant Values on Lattices.”  

 

A determinant is a single number calculated from a square matrix that reveals some of its core geometric and structural properties. The new research focuses on how determinant values are distributed across highly structured, discrete families of matrices.

“In many familiar settings, matrices describe transformations of space: they can stretch, rotate, or shear a grid,” Dr. Oh explained. “In our problem, the grid is not a grid in ordinary space. It is a grid inside the space of all (d)-by-(d) matrices. Thus each point of the grid is itself a matrix. We ask how the determinant values of those matrix-points are distributed as we look farther and farther out in the grid.”

 

“Our main result says that, unless there is a specific arithmetic reason preventing it, the nonzero determinant values follow a remarkably even and predictable pattern,” Dr. Oh said.

 

Matrices with determinant zero, called singular matrices, are special. One might expect them to be negligible, but for some lattices they appear in numbers large enough to contribute an additional term of the same order as the main term. Dr. Kim and Dr. Oh identify when this extra contribution occurs.

 

Dr. Oh is supported by National Science Foundation grant DMS-2450703. She is among the prestigious roster of plenary speakers at the International Congress of Mathematicians in Philadelphia in July 2026. She is an inaugural Fellow of the American Mathematical Society and was AMS vice president from 2021 to 2024.

 

Inspiration for this research dates to the 1987 proof of the Oppenheim conjecture by Fields Medalist and Abel Prize laureate Grigory Margulis. Dr. Margulis was Dr. Oh’s Ph.D. advisor. The Oppenheim conjecture, which concerns the values of quadratic expressions at integer points, asserts that if an indefinite quadratic form in at least three variables is not proportional to an integral quadratic form, then its values on integral vectors are dense in the real line. In particular, such a form takes values arbitrarily close to zero on integral vectors. For a 2-by-2 matrix, the determinant is a quadratic expression, thus connecting the determinant problem to the quantitative work of Dr. Margulis and his collaborators.

 

Dr. Oh first became interested in the determinant problem in about 2004. Over the next two decades, she returned to the problem every few years. “I had worked out part of the expected picture, but I could not find a way through the main difficulties,” she said. “There was not just one obstacle, but several layers of them.”

 

The turning point, she recalled, was June 28, 2024, during a visit to the Korea Institute for Advanced Study in Seoul. Wooyeon Kim, then a fourth-year graduate student at ETH Zürich, gave a job talk titled “Moments of Margulis Functions and the Quantitative Oppenheim Conjecture.”

“I was extremely impressed, and I immediately thought that he might be the right person with whom to revisit the determinant problem,” Dr. Oh said. “I carried the question for almost two decades, but the decisive progress came through a new collaboration with Wooyeon and through ideas he had developed as a graduate student. A long-standing problem met a fresh viewpoint at exactly the right moment.”

 

Two concentrated periods of working together in the same place were also crucial for progress: first a summer 2025 visit at KIAS, and then the Spring 2026 program at SLMath, where the researchers zeroed in on the final result.

At SLMath in March 2026, however, they found an error. “We discovered a serious gap in one of our key estimates,” Dr. Oh said. “We had treated certain maps as one-to-one when they were not, and this affected an essential part of the proof.”

 

“The special semester at SLMath allowed us to work side by side during the most difficult stage,” Dr. Oh said. “I do not think we would have overcome the crisis in quite the same way without that concentrated period together.”

 

Dr. Oh said, “Determinants are fundamental throughout mathematics and its applications. A nonzero determinant tells us that a square system of linear equations has a unique solution. The absolute value of the determinant measures how a linear transformation changes volume. A zero determinant means that a system has collapsed into a lower-dimensional space. A very small determinant often indicates that a system is close to singular and may be unstable.

 

“These ideas appear in numerical computation, statistics, optimization, data analysis, and physical modeling. Our theorem asks how frequently different determinant values occur within highly structured, discrete families of matrices. One way to view it is as a description of how often such matrices are regular, singular, or close to singular as we look farther and farther out in the lattice.

 

“The methods may ultimately be more transferable than the final counting formula. We developed tools for estimating how often structured families of matrices approach singular configurations, for treating families of higher-degree polynomials in a uniform way, and for counting matrices of bounded rank.

 

“This line of research may be useful in future problems involving integer or algebraic matrices, low-rank structures, or other polynomial quantities associated with matrices and tensors. For example, I believe that a related approach may be possible for determinants of symmetric matrices.

 

“In short, the ideas developed in the proof may later be useful in other settings, including some that are closer to applications.”

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* Dr. Oh generated the illustration with Google Gemini: She uploaded a draft of the research paper and asked for a suitable image. Commenting on the resulting allusion to a black hole (upper left), Dr. Oh said: “In the image, the black hole suggests a mysterious source producing many Diophantine lattices whose determinant values we are trying to understand. Our theorem applies to Diophantine lattices; lattices with algebraic entries provide concrete examples. In general, however, we do not know how to describe the source of all Diophantine lattices.”