Xuwen Zhu
Northeastern University, MA·Mathematics & Statistics
Many geometric spaces that come up in mathematics and physics are not perfectly smooth. They contain singularities, points where curvature blows up or the usual rules of calculus break down.
- Fields
Mathematics & Statistics
Physics
- Topics
differential geometry
geometric analysis
microlocal analysis
- Grants
- 2 known grants, through Aug 2028
What they work on
- Study how gravitational instantons, four-dimensional spaces that solve Einstein's equations, degenerate and how their moduli spaces are structured, using geometric microlocal analysis and PDE techniques.
- Build compactifications of the moduli space of constant-curvature metrics with conical singularities, and work out how these spaces are stratified and where they become singular.
- Analyze hyperbolic metrics with cusps and their spectral gaps, which measure how the vibration frequencies of the surface are spaced, using geometric resolutions of the singularities.
- Develop perturbation theory and singular uniformization results for singular metrics, guided by links between algebraic geometry, differential geometry and PDE analysis.
More about the research
Understanding these singular spaces is a central problem in differential geometry, the study of curved shapes using calculus. It connects to algebraic geometry, number theory, representation theory, and physics models such as gauge theory, string theory and electromagnetic systems. This research develops analytic tools to study singular geometric shapes. It focuses on spaces that form when smooth objects degenerate and on spaces used as building blocks to glue larger ones together.
Why it matters
This is basic mathematics. It adds new analytic methods for understanding how geometric spaces behave at and near singularities, and how families of such spaces fit together in moduli spaces. These results could eventually inform the mathematical side of theoretical physics, including gauge theory, string theory and models of electromagnetic systems, as well as related questions in algebraic geometry and number theory.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Laura Fredrickson
University of Oregon, OR·Mathematics & Statistics
Verify the Gaiotto–Moore–Neitzke conjectural description of the hyperkähler metric on Hitchin moduli spaces, especially near singular fibers, using constructive analytic techniques and geometric microlocal analysis.
Jian Song
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
Study the Yau-Tian-Donaldson conjecture for constant scalar curvature Kahler metrics on projective manifolds, testing uniform K-stability for models as a sufficient condition for existence.
Peter Hintz
Pennsylvania State University, PA·Mathematics & Statistics
Prove nonlinear stability of Kerr black holes, the standard model of a rotating black hole, for all spins below the maximum, showing that small disturbances decay over time, using geometric and microlocal analysis of wave equations.
The school
Northeastern University: 27 labs on PI Pages, $347M a year on research, 251 research doctorates a year.
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