In 1901, Hilbert proved there is no smooth isometric immersion of the hyperbolic plane H^2 into Euclidean 3-space. (https://en.wikipedia.org/wiki/Hilbert%27s_theorem_(differential_geometry)))

John Douglas Moore, in his 1969 PhD thesis under Shoshichi Kobayashi, Some Geometric Applications of Exteriorly Orthogonal Quadratic Forms, conjectured the natural generalization: that there is no isometric immersion of H^n into E^(2n-1). He also worked toward the conjecture in his 1972 paper Isometric Immersions of Space Forms in Space Forms. (https://msp.org/pjm/1972/40-1/pjm-v40-n1-p17-s.pdf)

Recently I noticed Moore posted a preprint paper on arXiv last November, Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem. (https://arxiv.org/abs/2512.00973) It would appear he has proved his conjecture after 56 years!

He writes on his research interests page that the proof is "based on the generalized Gauss-Bonnet formula developed by Allendoerfer, Weil and Chern." (https://web.math.ucsb.edu/~moore/researchinterests.html)

I find this incredibly cool but don't have much sense for how big of a deal it is; if anyone out there does, let us know! On a personal note, as an undergrad I took ODEs and vector calc under Moore, and later did a summer research project with him on minimal surfaces. He was so warmly encouraging and a huge positive influence on my math trajectory, easily one of my favorite professors ever.

If this result holds up, it's a beautiful capstone on a long and accomplished career.

The CV was last updated this year as it contains a 2026 honor.