Zhu collaborated with Cao Huaidong of Lehigh University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team is one of three teams formed for this purpose. The other teams were the Tian–Morgan team (Gang Tian of Princeton University and John Morgan of Columbia University) and the Kleiner–Lott team (Bruce Kleiner of Yale University and John Lott of University of Michigan). Zhu and Cao published a paper in the June 2006 issue of the Asian Journal of Mathematics with an exposition of the complete proof of the Poincaré and geometrization conjectures. They initially implied the proof was their own achievement based on the "Hamilton-Perelman theory", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest "exposition of Hamilton–Perelman's proof". They also published an erratum disclosing that they had forgotten to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the AJM editorial board issued an apology for what it called "incautions" in the Cao–Zhu paper.
In December 2004, Zhu won the Morningside Medal of Mathematics at the Third International Congress of Chinese Mathematicians (ICCM), a triennial congress hosted by institutions in Mainland China, Taiwan, and Hong Kong on a rotating basis. According to ICCM, "Awardees (of the Morningside Medal) are selected by a panel of international renowned mathematicians with the aim to encourage outstanding mathematicians of Chinese descent in their pursuit of mathematical truth."