Academic Report of School of Mathematical Sciences [2026] No. 091

(Series Report for High-Level University Construction No. 1350)

Title：A Five-Dimensional Counterexample to the Hessian Conjecture, with a Discussion of the Low-Dimensional Picture of the Jacobian Conjecture and the Hessian Conjecture

Speaker：Professor Meng Guowu (The Hong Kong University of Science and Technology)

Time：15:00-16:00, Sept. 18, 2026

Location：Room 1420, Huiwen Building, Yuehai Campus

Abstract: The Hessian Conjecture states that for a polynomial with a nonzero constant Hessian determinant, its Legendre transform is also a polynomial. It implies the Jacobian Conjecture and is known to hold for n ≤ 3. This report presents collaborative work with Yang Liang: Starting from Alpöge’s ternary Jacobian counterexample, we constructed an explicit quinary counterexample of degree 14 using the doubling construction and linear univariate Schur descent. Consequently, the Hessian Conjecture fails for n ≥ 5, remaining open precisely at n = 4; moreover, HC4 ⇒ JC2.

Speaker Profile：Meng Guowu is currently a professor in the Department of Mathematics at the Hong Kong University of Science and Technology. He received his Ph.D. in mathematics from Brown University in 1993, under the supervision of Professor Thomas Goodwillie. In 1986, he traveled to the United States from the University of Science and Technology of China as a student under the CUSPEA (China-U.S. Joint Graduate Program in Physics) and switched to mathematics research in 1989. He joined the Hong Kong University of Science and Technology in the fall of 1993, serving as a lecturer (1993–1998) and associate professor (since 1999), before being promoted to professor. In 2011, he was named a Qiushi Foundation Scholar at the Institute for Advanced Study in Princeton. He has long been engaged in physics-inspired mathematical research, having proposed the renowned Meng-Taubes formula in three- and four-dimensional topology. In recent years, his work has focused primarily on the Kepler problem and its profound connections to relativistic symmetric structures.

Faculty and students are welcome to attend!

Invited by: Xiao Wei

School of Mathematical Sciences

September 14, 2026