William Minicozzi
Massachusetts Institute of Technology, MA·Mathematics & Statistics
How do shapes evolve when they are smoothed over time, and where do they break down by forming sharp points or pinches? Geometric analysis uses calculus and partial differential equations (PDEs, equations describing how quantities change across space and time) to answer questions like these about curved spaces and surfaces.
- Fields
Mathematics & Statistics
- Topics
geometric analysis
partial differential equations
elliptic equations
- Grants
- 1 known grant, through Aug 2028
What they work on
- Establish gradient estimates and monotonicity formulas for harmonic functions and heat equation solutions, clarifying how they depend on dimension and how elliptic and parabolic behavior relate.
- Study singularity formation in mean curvature flow and Ricci flow, including optimal regularity, uniqueness of blowups (zoomed-in limits at singularities), rigidity and quasi-local behavior.
- Prove strong half-space theorems for stable minimal hypersurfaces, which are higher-dimensional analogs of soap films. Also prove estimates for the distance between minimal hypersurfaces in high dimensions.
- Derive volume growth bounds and stable Bernstein theorems for minimal hypersurfaces confined to a slab in high dimensions.
More about the research
Such questions sit at the foundation of modern geometry. They also connect to models in physics, materials science and other fields where shape, motion and optimization matter.
Why it matters
This is pure mathematics. It aims to settle basic open problems about how geometric objects and their evolutions are structured, including how singularities form. The resulting methods strengthen the mathematical foundations behind models in physics and materials science. The planned real analysis textbook links classical analysis to optimization and machine learning.
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.
Christos Mantoulidis
Rice University, TX·Mathematics & Statistics
Investigate the structure of singularities, points where curvature blows up, in area-minimizing minimal surfaces and in mean curvature flow, where a surface evolves to reduce its area.
The school
Massachusetts Institute of Technology: 25 labs on PI Pages, $1.1B a year on research, 692 research doctorates a year.
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