Jian Song
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
Geometry asks what shapes are possible and which ones are, in a precise sense, the most natural or balanced. For a given space, is there a way of measuring distances and angles on it that is canonical, meaning it is distinguished by a curvature condition rather than chosen arbitrarily?
- Fields
Mathematics & Statistics
Physics
- Topics
complex geometry
Kahler metrics
constant scalar curvature
- Grants
- 2 known grants, through Jun 2028
What they work on
- 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.
- Relate Archimedean and non-Archimedean theory by matching slopes of energy functionals along geodesic rays in the space of Kahler metrics with invariants of algebraic degenerations.
- Extend the stability analysis to weighted constant scalar curvature Kahler metrics and probe their connections to Kahler-Ricci solitons, extremal Kahler metrics and conformally Einstein-Maxwell Kahler metrics.
- Construct optimal degenerations of unstable polarized manifolds that admit no constant scalar curvature Kahler metric, as substitutes for a canonical metric.
- Develop geometric analysis on singular complex spaces, studying Riemannian properties of singular Kahler metrics and the moduli problems for Einstein manifolds that accompany them.
- Advance the analytic minimal model program with Ricci flow, analyzing how singularities form and describing them as global and local metric uniformization by solitons, and find criteria for solving global Hessian-type equations.
More about the research
These questions sit at the meeting point of differential geometry, which uses calculus, and algebraic geometry, which describes shapes by polynomial equations. Research in this lab builds the analytic tools, mostly from nonlinear partial differential equations, needed to decide when such canonical measurements exist, what happens when they break down at singular points, and how the answers can be read off from algebraic data.
Why it matters
This is foundational mathematics: it sharpens our understanding of which geometric spaces admit balanced, canonical shapes and of how geometry degenerates at singular points. Results would supply new tools for nonlinear partial differential equations and new bridges between differential and algebraic geometry, and because Kahler-Einstein metrics and Ricci flow appear in classification problems and in physical theories of spacetime and string compactifications, progress could inform work in those areas.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Charles Cifarelli
Stony Brook University, NY·Mathematics & Statistics
Construct new examples of non-compact singularity model geometries, namely Kähler-Ricci solitons and complete non-compact Calabi-Yau metrics, which describe how complex manifolds degenerate.
Xuwen Zhu
Northeastern University, MA·Mathematics & Statistics
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.
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.
The school
Rutgers University-New Brunswick: 57 labs on PI Pages, $837M a year on research, 713 research doctorates a year.
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