Daniel Stern
Cornell University, NY·Mathematics & Statistics
Soap films, black-hole boundaries and liquid crystals all settle into shapes that minimize some kind of energy. Mathematicians study these equilibrium shapes as solutions to geometric variational problems.
- Fields
Mathematics & Statistics
Physics
- Topics
differential geometry
minimal surfaces
harmonic maps
- Grants
- 1 known grant, through Jun 2027
What they work on
- Use new constructions of extremal metrics for Laplacian and Steklov eigenvalues, the natural vibration frequencies of a shape, to produce many new minimal surfaces of prescribed topology in low-dimensional spheres and balls.
- Study how minimal surfaces can be constructed in generic ambient spaces using mapping methods related to the eigenvalue-optimization techniques.
- Investigate the correspondence between the abelian Higgs model and codimension-two minimal submanifolds, and explore a similar link between the SU(2) Yang–Mills–Higgs equations and codimension-three minimal varieties.
- Develop existence, compactness and regularity theory for harmonic maps on manifolds of dimension three or higher with bounded Morse index into general targets, combining variational and new analytic methods.
More about the research
Two central examples are minimal surfaces, which model soap films, and harmonic maps, energy-minimizing mappings between curved spaces that appear in computer graphics and in the study of liquid crystals. This project in differential geometry asks when these objects exist and what structure they have. It draws connections to spectral geometry, the study of a shape's natural vibration frequencies, and to equations from particle physics.
Why it matters
This is pure mathematics. It deepens understanding of energy-minimizing shapes and maps and their links to vibration frequencies and particle-physics equations. Because such objects describe equilibrium states, the results could eventually inform the modeling of systems in physics, materials science and engineering, such as soap films, liquid crystals and black-hole boundaries.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Otis Chodosh
Stanford University, CA·Mathematics & Statistics
Study questions connected to stable minimal hypersurfaces, building on the four-dimensional proof that such surfaces are flat, with the goal of resolving the stable Bernstein problem in higher dimensions.
Jonathan Zhu
University of Washington, WA·Mathematics & Statistics·Early-career
Analyze singularities of mean curvature flow near cylindrical and conical models, testing uniqueness of blow-up limits when curvature is present in the interior or at the boundary.
Xin Zhou
Cornell University, NY·Mathematics & Statistics
Prove the existence of closed minimal surfaces with controlled genus, or number of holes, in three-manifolds, with the goal of resolving Yau's Four Minimal Spheres Conjecture.
The school
Cornell University: 37 labs on PI Pages, $1.6B a year on research, 537 research doctorates a year.
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