Charles Cifarelli
Stony Brook University, NY·Mathematics & Statistics
Complex manifolds are higher-dimensional geometric spaces built using complex numbers, and two special kinds, Calabi-Yau manifolds and Kähler-Ricci solitons, appear throughout mathematics and physics. A central question is how such spaces can break down, developing singularities (points where the geometry degenerates).
- Fields
Mathematics & Statistics
Physics
- Topics
differential geometry
complex geometry
Kähler geometry
- Grants
- 1 known grant, through May 2028
What they work on
- 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.
- Describe the moduli of these model spaces, that is, classify the full family of possible Kähler-Ricci solitons and non-compact Calabi-Yau metrics.
- Analyze how singularities develop along the Kähler-Ricci flow, an equation that evolves the geometry of a complex manifold over time, using solitons as local models.
- Analyze degenerations of collapsing families of Calabi-Yau manifolds, using complete non-compact Calabi-Yau metrics as the models for the collapsing regions.
More about the research
This project studies how to build and classify the model shapes that describe these breakdowns, linking geometric analysis with algebraic geometry, topology and physics.
Why it matters
This is basic research in pure mathematics. Understanding how singularities form in Kähler geometry adds to knowledge connecting geometric analysis, algebraic geometry and topology, and the collapsing Calabi-Yau picture could eventually inform theoretical physics, where Calabi-Yau spaces have many applications.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
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.
Felix Janda
University of Illinois Urbana-Champaign, IL·Mathematics & Statistics·Early-career
Prove the localization formula for the logarithmic gauged linear sigma model (log GLSM), a curve-counting technique, and study the effective invariants that feed into it.
The school
Stony Brook University: 32 labs on PI Pages, $319M a year on research, 289 research doctorates a year.
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