Academic Report of the School of Mathematical Sciences No. 062

(Series Report on High-Level University Construction No. 1084)

Lecture Title: Masser’s Conjecture on Equivalence of Integral Quadratic Forms

Speaker: Dr. Han Li (Associate Professor, Wesleyan University)

Date & Time: July 1, 2025, 3:00–4:00 PM

Venue: Huiwen Building, Room 1420

Abstract: In the realm of quadratic forms theory, a fundamental problem lies in determining the equivalence of two given integral quadratic forms. This problem, framed in terms of matrices, seeks to ascertain whether there exists a unimodular integral matrix X that satisfies the equation A=X′BX, where A and B are given symmetric n-by-n integral matrices, and X′ denotes the transpose of X. While a straightforward decision procedure exists for definite forms, the situation is more complex for indefinite forms. Surprisingly, it wasn’t until the early 1970s, with the work of C. L. Siegel, that a solution was found for indefinite forms. In the late 1990s, D. W. Masser conjectured the existence of a polynomial search bound for X in terms of the heights of A and B, for n greater than or equal to 3. The goal of this presentation is to discuss our resolution of this conjecture, achieved jointly with Professor Gregory A. Margulis. Our approach involved translating the problem into one concerning the actions of Lie groups on homogeneous spaces, and subsequently solving it using tools from ergodic theory, harmonic analysis, and representation theory. The talk should be largely accessible to undergraduate students familiar with abstract algebra and functional analysis.

Biography: Dr. Han Li’s research interests span Lie groups, discrete subgroups, dynamics on homogeneous spaces, and their intersections with number theory and geometry. He earned his PhD in Mathematics from Yale University in 2014. Before joining Wesleyan University, he conducted postdoctoral research for one year at the University of Texas at Austin and the Mathematical Sciences Research Institute (MSRI) in Berkeley.

All faculty and students are welcome!

Invited by: Wei Xiao

School of Mathematical Sciences

June 30, 2025